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Series: Analysis Seminar

In any standard course of Analytical Mechanics students are indoctrinated that a Lagrangian have a profound physical meaning (kinetic energy minus potential energy) and that Lagrangians do not exist in the case of nonconservative system. We present an old and regretfully forgotten method by Jacobi which allows to find many nonphysical Lagrangians of simple physical models (e.g., the harmonic oscillator) and also of nonconservative systems (e.g., the damped oscillator). The same method can be applied to any equation of second-order, and extended to fourth-order equations as well as systems of second and first order. Examples from Physics, Number Theory and Biology will be provided.

Series: PDE Seminar

In this talk we will discuss some recent existence and regularity results for equilibrium configurations of epitaxially strained crystalline films.

Series: Other Talks

2:00 pm Tuesday, September 7, 2004

Applied & Computational Math Seminar: The fundamental science in nanotechnology

by Zhong Lin (ZL) Wang (Center for Nanoscience and Nanotechnology, School of Materials Science and Engineering, Georgia Tech) in Skiles 269

Global nanotechnology initiative is inspiring a lot of fundamental research. Nanotechnology is one of the two major strategic research and education directions at Georgia Tech. We have over 100 faculty who are actively engaging in various research in nanotechnology, including but not limited to, synthesis and properties of nanomaterials, novel techniques for nano-scale characterization, multi-length-scale modeling and simulation, nanoelectronics, catalysis, nanomedicine, nanodevices, and nanosystem integration and packaging. It is generally believed that nanotechnology will touch every corner of our life, from health to energy and to environment. This presentation will focus on what is nanotechnology, what is the fundamental science in nanotechnology, what are the challenges in developing nanotechnology, and what are the potential and impacts of nanotechnology. About the speaker: Zhong Lin Wang received his Ph.D. in Physics from Arizona State University in 1987. He is currently a Regents' Professor, and the Director of the Center for Nanoscience and Nanotechnology in the Georgia Institute of Technology. He is in the world's top 25 most cited authors in nanotechnology for the last decade (ISI). His publications have been cited over 6000 times. He has received the 2001 S.T. Li prize for Outstanding Contribution in Nanoscience and Nanotechnology, the 2000 Georgia Tech Faculty Research Award, and the 1999 Burton Medal from Microscopy Society of America. His most recent research focuses on oxide nanobelts and nanowires, in-situ techniques for nano-scale measurements, self-assembly nanostructures, fabrication of nano devices, and properties of magnetic nanostructures.

12:00 pm Wednesday, September 8, 2004

Research Horizons Seminar: Recent Results on Interval Graphs

by Tom Trotter (School of Mathematics, Georgia Tech) in Skiles 255

Interval graphs arise naturally as models for computer memory storage problems and in other settings where resources are allocated from a linear array. Not surprisingly, interval graphs have been widely studied in graph theory and combinatorics. Here we discuss continuing progress on a problem which is now 30 years old: How well does the First Fit Algorithm (Greedy) do in coloring interval graphs. The presentation will include a discussion of why the problem is interesting, what has been done to date, what techniques have been developed and applied, and what is left to do.

3:00 pm Wednesday, September 8, 2004

Analysis Seminar: Construction of Compactly Supported Pre-Wavelets in Sobolev Spaces

by Ming-Jun Lai (Department of Mathematics, University of Georgia) in Skiles 269

We show how to construct compactly supported prewavelets in Sobolev spaces. The method work in the multivariate setting. In particular, compactly supported prewavelets in L_2 can be constructed using the method. However, it yields a pre-Riesz basis for Sobolev space in the multivariate setting.

3:00 pm Thursday, September 9, 2004

Stochastics Seminar: Functional samples and bootstrap for predicting sulfur dioxide levels

by Serge Guillas (School of Mathematics, Georgia Tech) in Skiles 269

In this study, enhancements of several functional techniques are given in order to forecast sulfur dioxide levels near a power plant. The data are considered as a time series of curves. Assuming a lag one dependence, the predictions are computed using the functional kernel (with local bandwidth) and the linear autoregressive Hilbertian model. We carry out the estimation with a so-called historical matrix, which is a subsample that emphasizes uncommon shapes. A bootstrap method is introduced to evaluate the range of the forecasts, which uses Fraiman and Muniz's order for functional data. Finally, we compare our functional techniques with neural networks and semi-parametric methods, and find that the former models are often more effective. (Joint work with B. Fernandez de Castro and W. Gonzalez Manteiga)

1:00 pm Friday, September 10, 2004

Graph Theory Seminar: Curvature of infinite graphs and Higuchi's conjecture

by Csaba Biro (School of Mathematics, Georgia Tech) in Skiles 255

The speaker will present a paper by DeVos and Mohar that proves a conjecture of Higuchi, stating that every locally finite 3-connected planar graph with positive curvature at every vertex is finite.

3:00 pm Friday, September 10, 2004

Geometry and Topology Seminar: Approximations of short maps by PL-isometries

by Svetlana Krat (School of Mathematics, Georgia Tech) in Skiles 269

A PL-map between two polyhedral spaces is a continuous map which, for certain triangulation of the spaces maps every simplex from the first space to a simplex in the target space in a linear fashion. It has been known due to V. Zalgaller that any 2 or 3-dimensional polyhedral space admits a PL-isometry into \Bbb R^2 (\Bbb R^3 respectively). Our work in this direction began with a generalization of Zalgaller's proof for any dimension n. A short map is a map that does not increase distances between points. It is obvious that a map that can be C^0 approximated by path-wise isometries (and, in particular, PL-isometries) is a short map. It was established then that any short map from a 2-dimensional polyhedral space to \Bbb R^2 can be approximated by PL isometries. An application of this result is a solution of one of V. Arnold's problems (known as a "Can one make a dollar bigger?" problem): is it possible to "fold a rectangle and increase its perimeters"? (We formalize this by asking if there exists a PL-isometry from the rectangle to a plane such that the perimeter of the image is greater than that of the rectangle.) These results were obtained in collaboration with D. Burago and A. Petrunin.

4:00 pm Friday, September 10, 2004

Combinatorics Seminar: New Cheeger bounds on eigenvalues of Markov chains

by Ravi Montenegro (School of Mathematics, Georgia Tech) in Skiles 269

Variations on Cheeger's inequality are used in many areas of Mathematics to bound the smallest non-trivial eigenvalue of the Laplacian for a variety of structures (graphs, reversible Markov chains, manifolds, among others). In the Markov chain setting this involves finding the worst cutsets of a weighted graph. We show a generalization in which cutsets for a range of set sizes can be used, not just the worst case. Our methods are sharp on several Markov chains, lead to bounds on the largest eigenvalue as well, and also bound rates of convergence for non-reversible Markov chains.

4:30 pm Monday, September 13, 2004

CDSNS Collqouium: Nonlinear stability for multidimensional fourth order shock fronts

by Peter Howard (Mathematics, Texas A & M University) in Skiles 269

We consider the question of stability for planar wave solutions that arise in multidimensional conservation laws with fourth order regularization only. Such equations arise, for example, in the study of thin films, for which planar waves correspond with fluid coating a pre-wetted surface. An interesting feature of these equations is that both compressive and under compressive planar waves arise as solutions (compressive or under compressive with respect to asymptotic behavior relative to the un-regularized hyperbolic system), and numerical investigation by Bertozzi, Munch, and Shearer indicates that under compressive waves can be nonlinearly stable. Proceeding with pointwise estimates on the Green's function for the linear fourth order convection-regularization equation that arises upon linearization of the conservation law about the planar wave solution, we establish that under general spectral conditions consistent with our motivating thin films equations compressive waves are stable for all dimensions d > 1 and under compressive waves are stable for dimensions d > 3. We also consider a weaker spectral criterion for which we can establish nonlinear stability for compressive waves in dimension d = 1 and dimensions d > 3 and under compressive waves in dimensions d > 5. The case of stability for under compressive waves in the thin films equations for the critical dimensions d = 1 and d = 2 remains an interesting open problem.

2:00 pm Tuesday, September 14, 2004

Applied & Computational Math Seminar: Theoretical Approaches to Plankton Community Ecology

by Christopher Klausmeier (School of Biology, Georgia Tech) in Skiles 269

Phytoplankton form the base of most aquatic food webs, play major roles in global biogeochemical cycles, and are responsible for around half the Earth's primary production. Because they are small, fast-growing, and relatively simple, they also make an ideal system for understanding the organization of ecological communities. Today I'll talk about mathematical approaches to understanding three ways phytoplankton communities are organized: in space, in time, and in chemical composition. I'll end with an advertisement for participation in an undergraduate training grant at the interface of mathematics and biology.

3:00 pm Tuesday, September 14, 2004

Algebra and Topology Seminar: Topological complexity of Pfaffian sets

by Thierry Zell (School of Mathematics, Georgia Tech) in Skiles 269

In this talk, I will introduce the notion of Pfaffian function due to Khovanskii, and describe the notion of relative closure of a semi-Pfaffian couple, a construction proposed by Gabrielov to describe the o-minimal structure generated by Pfaffian functions. I will then discuss effective upper bounds for the Betti numbers of such sets in terms of the complexity of their description.

4:00 pm Tuesday, September 14, 2004

PDE Seminar: Approaching non quasiconvex problems of the calculus of variations with affine boundary data

by Giovanni Pisante (School of Mathematics, Georgia Tech) in Skiles 255

We study the existence of minimizers for problem of the type inf { int_\Omega f(Du(x))dx: u = \phi on \partial\Omega } where \phi is an affine function. We will approach this problem using the theory of differential inclusions, and we will see that this method allows us to get a general theorem which contains as subcases many well known examples that in literature were solved with ad-hoc methods.

12:00 pm Wednesday, September 15, 2004

Research Horizons Seminar: Quaternions and Octonions

by John Pearson (School of Mathematics, Georgia Tech) in Skiles 255

The speaker will introduce the quaternions and the octonions. These are interesting extensions of the complex numbers since the quaternions are not commutative and the octonions are not even associative. Specifically, I will study the problem of multiplication of vectors in vector spaces over the real numbers. This talk should be easily accessible to anyone with a knowledge of basic linear algebra.

3:00 pm Wednesday, September 15, 2004

Analysis Seminar: Hankel Operators and product BMO

by Michael Lacey (School of Mathematics, Georgia Tech) in Skiles 269

The Nehari theorem characterizes the boundedness of Hankel operators of the disc in the complex plane. We extend this theorem to the setting of a product of discs. The classical methods of proof do not work in this setting. Our proof uses the product BMO theory of Chang and Fefferman.

3:00 pm Thursday, September 16, 2004

Stochastics Seminar: Inequality-based perturbation theory for Markov chains: recent advances

by Alexander Yu Mitrophanov (School of Biology, Georgia Tech)

Perturbation bounds are useful when it is necessary to bound the uncertainty in the output of a mathematical model given the uncertainty in the parameter values. Such situations arise when numerical values for the parameters are taken from experimental data. In this talk I overview my results concerning perturbation bounds for Markov chains, focusing on the connection between the sensitivity to perturbations and the rate of convergence to stationarity. I have developed a new convergence-based approach which produces much tighter uniform sensitivity bounds than the techniques previously known. Originally, my results in this direction were obtained for finite, homogeneous, continuous-time Markov chains having a unique stationary distribution, but they can be generalized to nonhomogeneous chains, as well as discrete-time chains with general state space. I demonstrate the important role of the spectral gap of the generator in the sensitivity analysis for continuous-time Markov chains. The spectral gap is the primary factor which governs the sensitivity of the chain Rs distribution in the stationary state; for reversible chains, this is also true for the transient state. I also show that in some cases (e.g., for chains with a strongly accessible state) one can obtain sensitivity bounds in terms of the entries of the generator.

4:00 pm Thursday, September 16, 2004

Departmental Tea Social: Open discussions

by SoM Associates (School of Mathematics, Georgia Tech) in Skiles 236

An informal gathering for faculty, students and staff. Tea, coffee and cookies will be served.

1:00 pm Friday, September 17, 2004

Graph Theory Seminar: Gentle introduction to matroid theory

by Paul Wollan (School of Mathematics, Georgia Tech) in Skiles 255

This talk is intended as a "precolloquium" for the upcoming colloquium on September 23 and will be accessible to a general mathematics audience.

4:00 pm Friday, September 17, 2004

Combinatorics Seminar: The Theory of Excluded Matrices

by Adam Marcus (School of Mathematics, Georgia Tech)

This talk will discuss the theory of excluded matrices (equivalently, for you hard core graph-theory people, Turan-type problems on ordered bipartite graphs). Not many results exist in this area of research, but I will discuss what little is known. In particular, I will present a result by Gabor Tardos and myself on the special case when the excluded matrix is a permutation matrix and show how it relates to pattern avoidance in strings and permutation enumerations. Time permitting, I will mention generalizations and the myriad of open problems that exist.

4:30 pm Monday, November 8, 2004

CDSNS Colloquium: Bifurcations of travelling wave solutions in the discrete NLS equations

by Dmitry Pelinovsky (Mathematics, McMaster University) in Skiles 269

We study discrete NLS equations, which include the cubic NLS lattice with on-site interactions and the integrable Ablowitz--Ladik lattice. Standing wave solutions are known to exist in the discrete NLS equations outside of the finite spectral band. We study travelling wave solutions which have nonlinear resonances with unbounded linear spectrum. By using center manifold and normal form reductions, we show that a continuous NLS equation with the third-order derivative term is a canonical normal form for the discrete NLS equation near the zero-dispersion limit. Bifurcations of travelling wave solutions near the zero-dispersion limit are analyzed in the framework of the third-order derivative NLS equation. We show that there exists a continuous two-parameter family of single-humped travelling wave solutions in the third-order derivative NLS equation, when it is derived from the integrable Ablowitz-Ladik lattice. On the contrary, there are no single-humped solutions in the third-order derivative NLS equation, when it is derived from the cubic NLS equation with on-site interactions. Nevertheless, we show that there exists an infinite discrete set of one-parameter families of double-humped travelling wave solutions in the latter case. Our results are valid in the neighborhood of the zero-dispersion point on the two-parameter plane of travelling wave solutions.

10:00 am Tuesday, November 9, 2004

Special CDSNS Seminar: Rigorous Numerics for Partial Differential Equations using Conley Index Theory

by Ulrich Miller (TAMS, Cornell University) in Skiles 255

Computing explicit solutions for partial differential equations is usually a difficult task. Therefore one often uses the computer to calculate approximations, but in many cases it is not clear whether this numerical data really approximates a solution of the partial differential equation. We present a method, based on Conley index theory, that insures the existence of an equilibrium of a time dependent equation in a computed neighborhood. This technique was introduced by K. Mischaikow and P. Zglicszynski for steady states of the one dimensional Kuramoto-Shivashinsky equation for fixed parameter values. We improved this method in order to compute paths of equilibria for a two dimensional problem, namely the Cahn-Hilliard equation on the unit square. The Cahn-Hilliard equation, a parabolic equation of fourth order, was introduced as a model for the process of phase separation of a binary alloy at a fixed temperature. We summarize some analytical results on the set of equilibria and then present the results we obtained using the rigorous method, including secondary bifurcations from the known main branches of equilibria.

2:00 pm Tuesday, November 9, 2004

Applied & Computational Math Seminar: Small-scale biological-physical-chemical signals in the sea

by Jeanette Yen (School of Biology, Georgia Tech) in Skiles 269

Small-scale biological-physical-chemical signals in the sea Abstract: Plankton operate at low Reynolds numbers, generating watery signals that can be attenuated by viscosity and confused with small-scale turbulence. Yet messages are created, transmitted, recognized and perceived. These messages guide essential survival tasks of aquatic micro crustaceans. Cues created include those of escaping prey, lunging predators, attractive mates, and appropriate hosts. In this presentation, I describe some unusual and some typical examples of small-scale biological-physical- chemical signals in the sea that help to maintain the integrity of our aquatic ecosystems.

4:00 pm Tuesday, November 9, 2004

PDE Seminar: Singularities for minimal graphs on convex inner boundaries

by John McCuan (School of Mathematics, Georgia Tech) in Skiles 255

It is a well known fact that while Laplace's equation readily admits solutions with an isolated singularity, certain geometric curvature equations of mean curvature type do not. This situation persists to a certain degree at the boundary, where uniform L-infinity estimates can be obtained for solutions of the geometric equations, but linear equations still admit unbounded solutions (even with zero boundary conditions except at a single point). Minimal graphs with jump discontinuities have been constructed satisfying various boundary conditions, and recent techniques have allowed the construction of such graphs satisfying a contact angle condition on the sides of a wedge domain. Using the same techniques, we were recently able to construct minimal graphs with a jump discontinuity on the inner boundary curve of an annulus bounded by two convex curves.

12:00 pm Wednesday, November 10, 2004

Research Horizons Seminar: Unbalanced Ternary Expansion and Self-Affine Tiles

by Yang Wang (School of Mathematics, Georgia Tech) in Skiles 255

It is well known that every real number can be represented in base 3 using the digits {-1, 0, 1}. This is the balanced ternary expansion. Is it possible to use digits other than {-1, 0, 1}? For example, can we use {-1, 0, 531439}? We discuss this question. As it turns out, the question is linked to the study of self-affine tiles, which we overview in this talk.

4:00 pm Wednesday, November 10, 2004

Analysis Seminar: The median in the Gamma distribution

by Christian Berg (Institute for Mathematical Sciences, University of Copenhagen) in Skiles 269

The gamma distribution with parameter x > 0 has the density with respect to Lebesgue measure on (0,\infty) given by e^{-t}t^{x-1}/\Gamma(x). We consider the median m(x) of this distribution defined implicitly as \int_0^{m(x)}e^{-t}t^{x-1}/\Gamma(x)dt=\frac{1}{2}. J. Chen and H. Rubin conjectured in 1986 that the sequence m(n)-n, n\geq 1 decreases. In 1994 K.P. Choi found the asymptotic formula m(n) = n + \frac{2}{3} + \frac{8}{405n} - \frac{64}{5103n^2} + \frac{2944}{492075n^3} + \ldots and related the expansion to a problem of Ramanujan. Using this expansion the conjecture of Chen and Rubin has been established recently by H. Alzer. In joint work with Henrik L. Pedersen we have proved the stronger result that 0 < m'(x) < 1 for all x > 0. In addition we have found some uniform estimates of m(x) and the first 10 terms of the asymptotic series for m(x). The proof uses complex analysis.

4:15 pm Thursday, November 11, 2004

Departmental Tea Social: Open Discussions

by SoM Associates (School of Mathematics, Georgia Tech) in Skiles 236

An informal gathering for faculty, students, and staff. Tea, coffee and cookies will be served.

1:00 pm Friday, November 12, 2004

Graph Theory Seminar: Entropy and sorting: Part II

by Teena Carroll (School of Mathematics, Georgia Tech) in Skiles 255

I will talk on Kahn and Kim's "Entropy and Sorting" paper. This paper is one of the first where the entropy method is used. We can think of a partially ordered set as an interrupted sorting process- it is a natural question to ask how much of the work has already been done. This paper presents a relationship between number of linear extensions of a poset and its entropy, as well as a nice characterization of graph entropy in the context of a poset. My goal is to make the talk accessible, and I will give you the background on entropy that you will need. (Second talk on this subject).

3:00 pm Friday, November 12, 2004

Geometry and Topology Seminar: Homology of real and complex toric varieties

by Clint McCrory (Department of Mathematics, University of Georgia) in Skiles 269

If a complex variety is defined by real polynomial equations, then the set X(R) of real points is the fixed point locus of the action of complex conjugation on the set X(C) of complex points. It follows that the sum of the Betti numbers of X(R) with Z/2 coefficients is less than or equal to the corresponding sum for X(C) (Smith's inequality). If these numbers are equal one says X is maximal. Compact nonsingular toric varieties are maximal. I will discuss the problem of whether all compact toric varieties -- possibly singular -- are maximal. This involves understanding the interplay between the combinatorial geometry and the topology of toric varieties.

4:00 pm Friday, November 12, 2004

Combinatorics Seminar: On Sensitivity and Chaos

by Elchanan Mossel (Statistics, University of California, Berkeley) in Skiles 269

I will discuss some (very) recent results showing how techniques from the theory of Gaussian Hilbert spaces can be used in order to solve a number of open problems in discrete Fourier analysis. Joint work Ryan O'Donnell and Krzysztof Oleszkiewicz.

4:00 pm Friday, November 12, 2004

CDSNS Dynamics Seminar: Efficient and stable numerical methods for the generalized and vector Zakharov system

by Weizhu Bao (Department of Computational Science, National University of Singapore) in Skiles 255

In this talk, we present efficient and stable numerical methods for the generalized Zakharov system (GZS) describing the propagation of Langmuir waves in plasma. The key point in designing the methods is based on a time-splitting discretization of a Schroedinger-type equation in GZS, and to discretize a nonlinear wave-type equation by pseudospectral method for spatial derivatives, and then solving the ordinary differential equations in phase space analytically under appropriate chosen transmission conditions between different time intervals or applying Crank-Nicolson/leap-frog for linear/nonlinear terms for time derivatives. The methods are explicit, unconditionally stable, of spectral-order accuracy in space and second-order accuracy in time. Moreover, they are time reversible and time transverse invariant if GZS is, conserve the wave energy as that in GZS, give exact results for the plane-wave solution and possesses 'optimal' meshing strategy in 'subsonic limit' regime. Extensive numerical tests are presented for plane waves, solitary-wave collisions in 1D of GZS and 3D dynamics of GZS to demonstrate efficiency and high resolution of the numerical methods. Finally the methods are extended to vector Zakharov system for multi-component plasma and Maxwell-Dirac system (MD) for time-evolution of fast (relativistic) electrons and positrons within self-consistent generated electromagnetic fields.

4:30 pm Monday, November 15, 2004

CDSNS Colloquium: The study of Euler-Poisson equations

by Ling Hsiao (Chinese Academy of Sciences) in Skiles 269

The mathematical modelling and the corresponding mathematical analysis in semiconductor science have attracted a lot of attention in both applied mathematics and semiconductor physics. There are different kinds of models for which the common used PDE models can be divided in two classes, roughly speaking. Namely, kinetic models and fluid dynamic models. For each of these classes there are (semi)classical models and quantum models. This lecture is focused on the Hydrodynamic models which takes the form of Euler-Poisson equations.

3:30 pm Tuesday, November 16, 2004

Algebra and Topology Seminar: Immersed Lagrangian submanifolds with prescribed boundary data, and isolated Lagrangian singularities

by Tommaso Pacini (School of Mathematics, Georgia Tech) in Skiles 269

Many examples are known, in C^n, of (n-dimensional) Lagrangian submanifolds with isolated singularities: eg, cones. The goal of the seminar is to explain why one might want to desingularize them, and to present work in progress concerning how to build Lagrangian desingularizations via h-principle techniques.

4:00 pm Tuesday, November 16, 2004

PDE Seminar: The problems related to Euler-Poisson equations

by Ling Hsiao (Chinese Academy of Sciences) in Skiles 255

It is important to understand the relations between different mathematical models in semiconductor science. The relations are formally well understood. For instance,the passage from the Hydrodynamic models to Energy transport models or Drift-diffusion models is so-called relaxation limit,the way from the drift-diffusion model to the diffusion equation is given by a quasineutral limit,classical models are obtained from quantum models in so-called classical limit, etc. However, it is really a challenging problem to establish these limits rigorously. This lecture is focused on the relaxation limit problems related to Euler-Poisson equations and Quantum Euler-Poisson equations.

11:00 am Wednesday, November 17, 2004

Quantitative and Computational Finance Seminar: Energy Research: Opportunities and Challenges

by Edward V. Byrns Jr. (Citadel Investment Group) in ISyE Main Building 228

The purpose of this talk will be to present a broad overview of the Energy Quantitative Research effort at Citadel Investment Group, and highlight some challenges facing the practical application of finance theory to energy investing. Energy markets present a unique opportunity for financial analysis, since the underlying commodity economics cause many traditional finance assumptions to be violated. Instead, practical research solutions must address the fundamental economics and account for the shortcomings of established equity and fixed income theory. During this talk, we will attempt to highlight some of these issues and provide a discussion framework for application of textbook theory to the trading desk reality. Further, this talk will present a brief description of the overall hedge fund environment in which a research professional must operate. Founded in 1990, Citadel Investment Group is a world leader in alternative investments, with a team of over 800 people in five offices worldwide. Our research philosophy is to apply a systematic process driven approach to investing in order to advance the reliability and repeatability of high risk adjusted returns. Speaker Bio: Dr. Byrns is the Director of Energy Research at Citadel Investment Group in Chicago, IL. He received his PhD in Engineering, (GA Tech 1991), as well as a MS in Economics (GA Tech 1991), a MSE in Aerospace Engineering (GA Tech 1988) and a BSE in Mechanical and Aerospace Engineering (Princeton 1985). He has been actively involved in commodity research for almost 8 years, holding various staff and management positions at Williams Energy, Merchant Energy Group of the Americas, and Koch Industries. Prior to entering the commodity field, he worked as a consultant in Washington D.C.

12:00 pm Wednesday, November 17, 2004

Research Horizons Seminar: The Four-Color Theorem and Beyond

by Robin Thomas (School of Mathematics, Georgia Tech) in Skiles 255

The four-color theorem is simple to state, and yet very hard to prove. Some people say that it is an isolated result in mathematics, but I will argue that over the years it inspired many deep theorems and useful concepts, that it has interesting connections, and that its generalizations and their relatives will continue to shape up the future of graph theory.

4:00 pm Wednesday, November 17, 2004

Analysis Seminar: Nonlinear Hyperbolic Equations in Waveguides

by Jason Metcalfe (School of Mathematics, Georgia Tech) in Skiles 269

In this talk, we will look at existence results for quasilinear wave and Klein-Gordon equations with quadratic nonlinearities in infinite homogeneous waveguides. This is a joint work with C. D. Sogge and A. Stewart. We can handle both the case of Dirichlet boundary conditions and the case of Neumann boundary conditions. In the case of Neumann boundary conditions, we need to assume a natural nonlinear Neumann condition on the quasilinear terms. The results that we obtain are sharp in terms of the assumptions on the dimensions for the global existence results and in terms of the lifespan for the almost global results. For nonlinear wave equations, in the case where the infinite part of the waveguide has spatial dimension three, the hypotheses in the theorem concern whether or not the Laplacian for the compact base of the waveguide has a zero mode or not.

4:30 pm Thursday, November 18, 2004

COLLOQUIUM: Dynamics at the boundary of moduli space

by Laura DeMarco (University of Chicago) in Skiles 269

The moduli space of rational maps, M(d), is the collection of all holomorphic self-maps of the Riemann sphere of degree d > 1, modulo the action by conjugation of the group of M�bius transformations. I will discuss the limiting dynamics of rational maps at the boundary of M(d), from algebraic, geometric, and ergodic theoretic points of view. The ideas were motivated by relations to Teichm�ller theory, the study of moduli spaces in algebraic geometry, and the theory of entropy of a dynamical system.

**Refreshments will be served in 236 at 4:00**

1:00 pm Friday, November 19, 2004

Graph Theory Seminar: Random Planar graphs

by Stefanie Gerke (ETH Zurich) in Skiles 255

We consider random planar graphs on n labelled nodes and show that if the graph is picked uniformly at random then the expected number of edges is at least \frac{13}{7}n+o(n). We then concentrate on random labeled graphs R_{n,qn} on n nodes and \lfloor qn \rfloor edges where 1\leq q \leq 3. We show for example that asymptotically almost surely (that is with probability tending to one as n tends to infinity) R_{n,q} contains each given fixed connected planar subgraph and contains linearly many nodes of each given degree. We also show that the probability that R_{n,q} is connected is bounded away from one by a non-zero constant.

3:00 pm Friday, November 19, 2004

Geometry and Topology Seminar: Some Geometric Aspects of Quantum Topology

by Razvan Gelca (Texas Tech University) in Skiles 269

On the quantization of the moduli space of flat SU(2)-connections on the torus The talk is focussed on the joint result of the author and A. Uribe which shows that the Weyl quantization and the quantum group quantization of the moduli space of flat connections on the torus are unitarily equivalent. As an application we will show that the restriction of the Reshetikhin-Turaev modular functor to the torus can be recovered from the Weyl quantization.

4:00 pm Friday, November 19, 2004

Combinatorics Seminar: Spectra of random power law graphs

by Linyuan (Lincoln) Lu (University of South Carolina) in Skiles 269

In the study of the spectra of power law graphs, there are basically two competing approaches. One is to prove analogues of Wigner's semi-circle law while the other predicts that the eigenvalues follow a power law distribution. Although the semi-circle law and the power law have nothing in common, we will show that both approaches are essentially correct if one considers the appropriate matrices. We will prove that (under certain mild conditions) the eigenvalues of the (normalized) Laplacian of a random power law graph follow the semi-circle law while the spectrum of the adjacency matrix of a power law graph obeys the power law. Our results are based on the analysis of random graphs with given expected degrees and their relations to several key invariants. The spectrum distributions have direct implications to numerous graph algorithms such as randomized algorithms that involve rapidly mixing Markov chains, for example. This is joint work with Fan Chung Graham and Van Vu.

4:00 pm Friday, November 19, 2004

CDSNS Dynamics Seminar: Symmetric periodic, homoclinic and heteroclinic solutions

in reversible systems

by Daniel Wilczak (Jagiellonian University) in Skiles 255

We present a numerical method for proving the existence of infinitely many symmetric periodic, homoclinic and heteroclinic solutions in reversible systems. The method combines an abstract topological theorem with rigorous computations of the derivative of the Poincare map. The method has been successfully applied to the Michelson system and the Planar Restricted Circular Three Body Problem with the parameter values corresponding to the Oterma comet in Sun-Jupiter system.

4:30 pm Monday, November 22, 2004

CDSNS Colloquium: Emergent complexity and physics

by John Doyle (Electrical Engineering, Control & Dynamical Systems, and Bioengineering, Caltech) in Skiles 269

Part one (Monday) will focus on how the above views of "organized complexity" contrast sharply with the view of "emergent complexity" that is popular among physicists. While motivation will be drawn from biology and technology, greater emphasis will be on the model systems and phenomena, such as lattices, cellular automata, spin glasses, phase transitions, criticality, chaos, fractals, scale-free networks, self-organization, and so on, that have been the inspiration for the physics perspective. This has several potential benefits. One is that it seems to offer a novel way of teaching concepts and mathematics of organized complexity to a much broader audience while deferring the high level of domain detail currently necessary to understand the model systems from biological or technological networks. Another is that it provides apparently novel insights into RYF aspects of longstanding mysteries in physics, from coherent structures in shear flow turbulence and coupled oscillators, to the ubiquity of power laws, to the nature of quantum entanglement, to the origin of dissipation. Finally, the underlying mathematics may offer new tools to explore other problems in physics where RYF features may play a role, particularly involving multiple scales and organized structures and phenomena.

2:00 pm Tuesday, November 23, 2004

Applied & Computational Mathematics Seminar: Organized complexity and biology

by John Doyle (Electrical Engineering, Control & Dynamical Systems, and Bioengineering, Caltech) in Skiles 269

Part two (Tuesday) will describe qualitatively in as much detail as time allows these features of biological systems and their parallels in technology, and then discuss the mathematical challenges that this view of biological complexity implies. Much of this will be accessible to biologists, and will not depend critically on part one. A crucial insight is that both evolution and natural selection or engineering design must produce high robustness to uncertain environments and components in order for systems to persist. Yet this allows and even facilitates severe fragility to novel perturbations, particularly those that exploit the very mechanisms providing robustness, and this "robust yet fragile" (RYF) feature must be exploited explicitly in any theory that hopes to scale to large systems.

3:00 pm Tuesday, November 23, 2004

Department Colloquium: Subfactors and Planar Algebras

by Dietmar Bisch (Vanderbilt University) in Skiles 269

PLEASE NOTE UNUSUAL DAY AND TIME. Vaughan Jones introduced the theory of subfactors in the early 80's as a ``Galois theory'' for inclusions of certain algebras of operators on a Hilbert space. A subfactor can be viewed as a group-like object that encodes what one might call the generalized symmetries of the mathematical or physical situation from which it was constructed. To decode this information one has to compute a system of inclusions of finite dimensional algebras naturally associated to the subfactor. For instance, the Temperley-Lieb algebras arise as fundamental symmetries in this way. These algebras are examples of so-called planar algebras, a new algebraic-combinatorial structure that is intrinsic to the theory of subfactors. I will present some of the basic ideas in subfactor theory and describe what a planar algebra is. No prior knowledge of operator algebras is required for this talk. Refreshments will be served after colloquium in Skiles 236

4:00 pm Tuesday, November 23, 2004

PDE Seminar: C^1 Regularity for Infinity Harmonic Functions in Two Dimensions

by Ovidiu Savin (Department of Mathematics, UC Berkeley) in Skiles 255

A continuous function u:\Omega \to R, \Omega \subset R^n is said to be infinity harmonic if it satisfies \Delta_{\infty} u := \sum_{i,j=1}^n u_iu_ju_{ij} =0, in \Omega in the viscosity sense. This equation arises when considering optimal Lipschitz extensions from \partial \Omega to \Omega. An interesting question is to determine whether or not infinity harmonic functions are continuously differentiable. In this talk we show that in two dimensions infinity harmonic functions are actually C^1.

12:00 pm Wednesday, November 24, 2004

Research Horizons Seminar: Transport of nutrients in bones

by Guillermo Goldsztein (School of Mathematics, Georgia Tech) in Skiles 255

The goal of this talk is to describe an example of the use of mathematical modeling (and analysis) to understand problems in other sciences. We will discuss the transport of nutrients in bones.

4:30 pm Monday, November 29, 2004

CDSNS Colloquium: Systems of Integral Equations Related to the Weighted Hardy-Littlewood-Sobolev inequalities

by Congming Li (Applied Math, University of Colorado, Boulder) in Skiles 269

I will present the recent work (joint with W. Chen, C. Jin, J. Lim, and B. Ou) on some systems of integral equations related to the weighted Hardy-Littlewood-Sobolev (HLS) inequality. The focus is on the Euler-Lagrange equations of the HLS. We study the symmetry, monotonity, and regularity for the solutions. I will show you some key futures of the main technique--the method of moving planes for integral equations. I will also present a simple method for the study of regularity and obtain the optimal integrability interval for solutions of a class of systems equations as an application.

2:00 pm Tuesday, November 30, 2004

Applied & Computational Math Seminar: Total Variation Models for Wavelet Based Image Processing

by Hao Min Zhou (School of Mathematics, Georgia Tech) in Skiles 269

In this talk, I will present some total variation models for wavelet based image processing including compression and wavelet inpainting. For image compression, we use variational PDE techniques to modify the coefficients in the truncation process so that the oscillations are reduced in the reconstruction processes. In particular, we use minimization of total variation (TV), to select and modify the retained standard wavelet coefficients so that the reconstructed images have fewer oscillations near edges. The wavelet inpainting problem is closely related to the classical image inpainting, with the difference being that the inpainting regions are in the wavelet domain, that brings new challenges to the reconstructions, as there is no well defined inpainting region in the pixel domain, and the degradation is inhomogeneous. We propose new variational models, especially total variation minimization in conjunction with wavelets for the image inpainting problems in the wavelet domain. In both applications,The models lead to PDE's, which are Euler-Lagrange equations of the variational formulations, in the wavelet domain and can be solved numerically. The new models have direct by automatic control over the geometrical properties, such as edges, of the images (joint work with Tony Chan at UCLA and Jackie Shen at Minnesota).

3:30 pm Tuesday, November 30, 2004

Algebra and Topology Seminar: On thin position of knots in lens spaces.

by Kenneth Baker (Department of Mathematics, University of Georgia ) in Skiles 269

John Berge has a conjectured classification of knots in S^3 that admit Dehn surgeries yielding lens spaces. The corresponding knots in the lens spaces are 0- or 1-bridge braids with respect to the Heegaard torus of the lens space. Given a knot K' in S^3 of genus g with a Dehn surgery yielding a lens space Y of order r, if r \geq 4g-1 then we will sketch the key points of a proof that the corresponding knot K which is the core of the surgery solid torus) in Y has bridge number at most 3 with respect to the Heegard torus of Y. This is a work in progress.

4:00 pm Tuesday, November 30, 2004

PDE Seminar: Global Well-Posedness of the Viscous Boussinesq Equations

by Congming Li (University of Colorado) in Skiles 255

I will present the joint work with Tom Hou on the global well-posedness of the viscous incompressible Boussinesq equations in two spatial dimensions for general initial data in H^m with m\ge 3. It is known that when both the velocity and the density equations have finite viscosity, the Boussinesq system does not develop finite time singularities. We consider here the challenging case when viscosity enters only in the velocity equation, but there is no viscosity in the density equation. Using sharp and delicate energy estimates, we prove global existence and strong regularity of this viscous Boussinesq system for general initial data in H^m with m \ge 3.

12:00 pm Wednesday, December 1, 2004

Research Horizons: A Conley Index Based Multi-scale Data Structure

by Todd Moeller (Georgia Tech) in Skiles 255

I will discuss how topological theory can be used to create computational tools. A review of the Conley Index will be presented.

2:00 pm Wednesday, December 1, 2004

Graduate Committee Sponsored Seminar: Life after the Ph.D.: Now What? Career Choices, etc.

by Robin Thomas (School of Mathematics, Georgia Tech) in Skiles 255

3:00 pm Thursday, December 2, 2004

Stochastics Seminar: Central limit theorems in infinite dimensions - a new direction

by Joel Zinn (Department of Mathematics, Texas A & M University) in Skiles 269

After a review of the goals of the subject and many key results in the area, we'll indicate a new direction.

4:30 pm Thursday, December 2, 2004

Special Math Colloquium: Energy Aware Algorithm Design via Probabilistic Computing: From Algorithms and Models to Moore's Law and Novel (Semiconductor) Devices

by Krishna V. Palem (Electrical and Computer Engineering, Georgia Tech) in Skiles 269

The energy consumed by computations is becoming an increasing concern both within the context of high-performance systems well as embedded systems, on par with the past focus on raw speed or its derivative performance. In this talk, we will outline a novel framework for designing and analyzing algorithms wherein the figure of merit is the energy complexity - a measure of the (physical) energy consumed. Using the formulation of an energy-aware switch, and a network of such switches, fundamental limits will be established for the energy needed for switching deterministically, as well as energy savings derived from probabilistic switching, with a probability of being correct, p. For example, it is shown that using a switch modeled using an ideal (Boltzmann) gas, a single deterministic switching step for computing a BIT consumes at least (-(\kappa\cdot T \ln (2))) Joules of energy, whereas the energy consumed by a single probabilistic switching step to compute a PBIT can be as low as (-(\kappa\cdot T \ln (2p))) Joules. These results are developed within the context of an idealized switching device introduced here, including those constrained by the laws of classical (statistical) thermodynamics (of Maxwell, Boltzmann and Gibbs), as well as by the constraints of idealized semiconductor devices. Based on this notion of switching, models for algorithm analysis and design, as well as upper- and lower- bounds on energy complexity and hence, for the first time, asymptotic energy savings via the use of probabilistic computing will be established. Possible approaches to realizing these probabilistic switches using conventional CMOS technology, as well as their potential for accelerating the current semiconductor roadmap that is based on deterministic computing, including the projections implied by Moore's law, will be outlined. Finally, estimates of significant energy savings in the context of practical application workloads will be sketched, using the widely used and energy-friendly ARM processor's profile as a base-line for deterministic computing. This work draws upon basic concepts from computer science, microelectronics and classical statistical thermodynamics, and in the interest of being self-contained, the presentation will also include a brief survey of the relevant thermodynamics.

**Refreshments will be served in 236 at 4:00 pm**

3:00 pm Friday, December 3, 2004

Geometry/Topology: Total Positive Curvature and the Relative Isoperimetric Inequality

by Mohammad Ghomi (GA Tech) in Skiles 269

This is a report on recent joint work with Jaigyoung Choe and Manuel Ritore where we prove that if the boundary of a compact hypersurface in Euclidean n-space lies on the boundary of a convex body and meets that convex body orthogonally from the outside, then the total positive curvature of the hypersurface is bigger than or equal to half the area of the (n-1)-sphere. Also, as an application of this result, we prove that the area of a hypersurface which traps a given volume outside of a convex body in Euclidean n-space must be greater than or equal to the area of a hemisphere trapping the given volume on one side of a hyperplane.

4:00 pm Friday, December 3, 2004

Combinatorics Seminar: Concentration on the Discrete Torus

by Marcus Sammer (School of Math, Georgia Tech) in Skiles 269

The subgaussian constant of a cycle is useful in obtaining concentration inequalities on the n-dimensional torus, obtained as an n-fold product of the cycle. We find tight bounds on the subgaussian constant of a cycle, using a characterization (due to Bobkov-Gotze) of this constant in terms of a transportation inequality. The transportation viewpoint allows us to characterize the optimal Lipschitz functions on a cycle which attain the subgaussian constant. In particular we show that the subgaussian constant is equal to the related spread constant on cycles with an even number of vertices, while the two constants are different on odd cycles.

3:00 pm Monday, January 10, 2005

Algebra and Topology Seminar: Complexity and algorithms for semi-algebraic sets over quadratic maps

by Dmitrii V. Pasechnik (Tilburg University) in Skiles 269

A semialgebraic set S is said to be defined over a map if S is given by a formula of the form F(Q(X)), where Q : R^n -> R^k is a polynomial map and F(Y) is a quantifier-free Boolean formula with polynomial inequalities (with polynomials of degree at most d belonging to a subset of R[Y]of size s) as atoms. We concentrate on a nontrivial and important for applications case when Q is quadratic. It turns out that the behavour of S differs rather drastically from the behavour of a general n-variate semialgebraic set given by degree d polynomials. For instance, the sum of the Betti numbers of S is bounded by (sdn)^O(k), and a similar bound holds for the complexity of sampling in S (i.e. computing representatives of the connected components of S). References: Algorithms in real algebraic geometry, by S.Basu, R.Pollack and M.-F. Roy. Springer-Verlag 2003 Polynomial-time computing over quadratic maps I: sampling in real algebraic sets, by D. Grigoriev and D.V. Pasechnik. To appear in Computational Complexity, see also http://arxiv.org/abs/cs.SC/0403008 (Joint work with Dima Grigoriev)

4:30 pm Monday, January 10, 2005

CDSNS Colloquium: Incorporating aerosol-cloud interactions in global climate models: What is important and how can it be done?

by Athanasios Nenes (Earth & Atmospheric Sciences and Chemical & Biomolecular Engineering, Georgia Tech) in Skiles 255

The effects of aerosols on clouds (also known as the 'aerosol indirect effect') are recognized as one of the largest sources of uncertainty in assessments of anthropogenic climate change. This uncertainty arises from the complexity and range of scales involved in aerosol-cloud interactions. This talk will: i) address key sources of uncertainty that exist in current assessments of the aerosol indirect effect, and, ii) present the challenges and recent advancements in incorporating aerosol-cloud interactions within global climate models.

1:00 pm Wednesday, January 12, 2005

Noncommutative Geometry Seminar: Families index theorem in Haefliger cohomology

by Moulay Benameur, Professor (Univerity of Metz, France) in Skiles 269

The topological index of a leafwise Dirac type operator on a foliation can be defined in Haefliger cohomology as the Chern character of a Gysin map in K-theory. When the Novikov-Shubin invariants are larger than half the codimension, this topological index is related with the analytical index on the one hand and with the superconnexion index bundle on the other hand. (Joint work with James Heitsch)

4:00 pm Tuesday, January 18, 2005

PDE Seminar: Relativistic Euler Equations in (3+1)-Dimensional Spacetime

by Ronghua Pan [mail] (Georgia Tech) in Skiles 255

We study the local well-posedness and singularity formation of smooth solutions for the relativistic Euler equations in (3+1)-dimensional spacetime. The local well-posedness is established via a construction of a convex entropy if the initial data is in a sub-luminous region away from vacuum. However, the classical solutions are proved to blow up in finite time for any non-trivial finite initial energy or for infinite initial energy with large radial momentum. This is a joint work with Joel Smoller at University of Michigan.

12:00 pm Wednesday, January 19, 2005

Noncommutative Geometry and Mathematical Physics: A lower bound on the free energy of matter interacting with radiation

by Michael Loss [mail] (School of Math) in Skiles 269

4:00 pm Wednesday, January 19, 2005

Analysis Seminar: Lebesgue constants for Multiple Fourier Series

by Eli Liflyand (Bar Ilan University) in Skiles 269

4:30 pm Thursday, January 20, 2005

COLLOQUIUM: Singularities in complex dynamics

by Mattias Jonsson (University of Michigan) in Skiles 269

I will discuss how algebro-geometric methods can sometimes be used to study objects of nonalgebraic nature, e.g. certain dynamical systems. In dynamics one is often interested in asymptotic behavior as time evolves. For instance, given a polynomial map F:C2 --> C2 one may ask at what speed the orbit p, F(p), F(F(p)),..., Fn(p),... approaches infinity as n --> \infty if the original point p is chosen generically near infinity. This speed is governed by the behavior of \deg(Fn), the degree of the highest order term in Fn. For example, if F(X,Y) = (Y,XY), then \deg(Fn) gives the Fibonacci numbers, so in a suitable sense, the speed above equals the golden mean. A classical field of algebraic geometry is the study of singularities, such as the curve in C2 parameterized by t \mapsto (t2,t3), which has a cusp at the origin. It is known that singularities typically can be resolved, i.e. viewed as "shadows" of nonsingular objects; the cusp above is the shadow of the space curve t \mapsto (t,t2,t3). As I will explain, it turns out that a dynamic version of resolution of curve singularities can be used to understand the speed of convergence to infinity of polynomial maps of C2. As a consequence, the speed is always a quadratic integer.

**Refreshments will be served at 4PM in Room 236**

4:00 pm Friday, January 21, 2005

Combinatorics Seminar: A random tiling model for two dimensional electrostatics

by Mihai Ciucu (School of Mathematics, Georgia Tech) in Skiles 269

We consider triangular holes on the hexagonal lattice and we study their interaction when the rest of the lattice is covered by dimers. More precisely, we analyze the joint correlation of these triangular holes in a "sea" of dimers. We determine the asymptotics of the joint correlation (for large separations between the holes) in the case when one of the holes has side 1, all remaining holes have side 2, and the holes are distributed symmetrically with respect to a symmetry axis. Our result has a striking physical interpretation. If we regard the holes as electrical charges, with charge equal to the difference between the number of down-pointing and up-pointing unit triangles in a hole, the logarithm of the joint correlation behaves exactly like the electrostatic potential energy of this two-dimensional electrostatic system: it is obtained by a Superposition Principle from the interaction of all pairs, and the pair interactions are according to Coulomb's law. The proof involves combinatorics, hypergeometric functions, and Laplace's method for the asymptotics of integrals.

3:00 pm Monday, January 24, 2005

Algebra and Topology Seminar: Catalan's conjecture

by Matt Baker (School of Mathematics, Georgia Tech) in Skiles 269

Eugene Charles Catalan conjectured in 1844 that 8 and 9 are the only consecutive perfect powers of natural numbers. This was finally proved in 2002 by Preda Mihailescu using techniques from the theory of cyclotomic fields. I will give an overview of Mihailescu's proof without assuming any prior knowledge of algebraic number theory.

4:30 pm Monday, January 24, 2005

CDSNS Colloquium: Various Mathematical Aspects of Tiling Spaces

by Jean Bellissard (School of Mathematics, Georgia Tech) in Skiles 269

Tilings in a d-dimensional Euclidean space is the high dimensional version of coding. Various aspect of sets of tilings will be described. They are dynamical systems through the construction of the Hull and its transversal, they can be described through measure theory as well, they exhibit some Geometry, as foliated spaces or as inverse limit of Branched Oriented Flat Riemannian manifolds. Some open problems will be addressed concerning their metric and combinatorial properties, complexity, topological invariant and Noncommutative Geometry.

4:30 pm Tuesday, January 25, 2005

PDE Seminar: Bernoulli free boundary problems

by Prof. Diaraf SECK (Universit Cheikh Anta Diop (FASEG) , Dakar SENEGAL) in Skiles 255

We study the existence and uniqueness results of the Bernoulli free boundary problems (the exterior and interior cases ) for the p-Laplace operator. Using the shape optimization theory, the derivative with respect to the domain, we prove existence and uniqueness results and monotony results. And we show the existence of the free boundary problems. In the interior case, it is known that there is not always an existence result. We show an isoperimetric inequality. That is the optimal estimation for the upper bound of the Bernoulli constant.

12:05 pm Wednesday, January 26, 2005

Noncommutative Geometry (Working seminar): Algebraic Quantization of symplectic vector spaces

by Michael Burkhart (School of Math, Gatech) in Skiles 269

Using noncommutative geometry, I will present a quantization of the phase space of action-angle variables, in the case of the plane. The semiclassical limit will be well behaved under this picture.

3:00 pm Wednesday, January 26, 2005

Research Horizons: Semialgebraic sets and tame topology

by Thierry Zell [mail] (Georgia Tech) in Skiles 255

Semialgebraic sets are the subsets of real euclidean space defined using polynomial equalities and inequalities. They are the real counterpart to the complex constructible sets, and they come up naturally in a lot of engineering and computer science applications. The main part of the talk will be spent outlining a number of good properties that makes semialgebraic sets a good class of objects to study: sets defined from semialgebraic data are semialgebraic too, and their geometric properties are very simple. We will then discover that these properties follow from a more general approach to the notion of tame topology: the theory of o-minimal structures.

4:00 pm Wednesday, January 26, 2005

Analysis Seminar: A multivariable extension of the Fejer-Riesz theorem

by Jeff Geronimo (School of Mathematics, Georgia Tech) in Skiles 269

M Dritschel has recently shown that every strictly positive multivariable trigonometric polynomial can be factored into a sum of magnitudes square of polynomials. We will give a proof of this theorem.

4:15 pm Thursday, January 27, 2005

Math Department Tea:

in Math Department Lounge

This is the first math department tea, which is open to all students, faculty, and staff. Food and beverages will be served.

4:30 pm Thursday, January 27, 2005

COLLOQUIUM: A random tiling model for two dimensional electrostatics II: arbitrary charge distributions under periodic boundary conditions

by Mihai Ciucu (School of Mathematics, Georgia Tech) in 269

The correlation of holes on the triangular lattice can be defined, in analogy to considerations of Fisher and Stephenson on the square lattice, by including them in large hexagons that grow to infinity so that the holes remain near the center. In earlier work, we showed that if the holes are distributed symmetrically about a straight line, then for large distances between the holes the correlation behaves like the electrostatic energy of a two dimensional system of charges corresponding to the holes. However, since the dimer statistics is significantly distorted almost everywhere inside hexagonal regions, it arises as a desirable goal to define the correlation of holes in an alternate way, via regions that don't distort dimer statistics, and analyze its asymptotic behavior. In this talk we define such a correlation and prove that it also reduces to electrostatics in the scaling limit. Our proof applies to general, not necessarily symmetric distributions of the holes.

**Department tea will be begin at 4:15 in Room 236**

3:00 pm Friday, January 28, 2005

Geometry and Topology Seminar: Symplectic Calabi-Yau surfaces

by Tian-Jun Li [mail] (University of Minnesota) in Skiles 269

4:00 pm Friday, January 28, 2005

Combinatorics: Disjoint paths and cycles in group labeled graphs

by Paul Wollan (Georgia Tech) in Skiles 269

We consider a graph G where each edge is assigned a label from an abelian group A. The weight of a subgraph of G is simply the sum of the weights of it's edges. A recent result of Chudnovsky, Geelen, Gerards, Goddyn, Lohman, and Seymour shows that for any set of vertices X, either there exist k disjoint paths with non zero weight who intersect X exactly in their endpoints, or there exists a set of at most 2k-2 vertices whose removal eliminates all such paths of non-zero weight. We discuss recent work that utilizes this result to find paths with pre-specified ends whose lengths are pre-specified parities. We also present a new result showing that in highly connected graphs either there exist k disjoint cycles of non zero weight or there exists a set X of vertices whose size depends only on k and whose removal kills all non zero cycles in the graph. This is joint work with Kenichi Kawarabayashi.

3:00 pm Monday, January 31, 2005

Colloquium: Contact Geometry, Topology and Dynamics

by John Etnyre [mail] (University of Pennsylvania) in Skiles 269

Contact geometry is a venerable subject that arose out of the study of Geometric Optics in the 1800's. Through the years it has repeatedly cropped up in many areas of mathematics, but only in the past 30 years or so has it received serious attention. Recently there has been great progress in understanding contact structures. Depending on one's perspective, contact structures sometimes seem like topological objects, sometimes geometric objects and sometimes dynamical objects. In this talk I will begin by discussing how contact structures arise out of natural problems and how they have deep connections with topology and dynamics. Then after surveying a few topics about contact structures in low dimensions I will define contact homology in certain situations. Contact homology is a new invariant of contact structures (and/or certain submanifolds of them) that is similar, in spirit, to Gromov-Witten invariants of symplectic manifolds or Floer homology of Lagrangian submanifolds in symplectic manifolds. I then will proceed to discuss applications of contact homology, in particular, I will describe how it yields potentially new invariants of submanifolds of Euclidean space.

4:30 pm Monday, January 31, 2005

CDSNS Colloquium: Complexity functions and measures in dynamical systems

by Valentin Afraimovich (San Luis Potosi State University, Mexico) in Skiles 269

Complexity functions measure the amount of instability of orbits in dynamical systems. There are many notions of complexity, but it is supposed to be discussed in the talk mainly properties of the epsilon-complexity. Main results will be explained, measures of complexity will be introduced and some examples will be presented.

2:00 pm Tuesday, February 1, 2005

Applied & Computational Mathematics Seminar: The Black Scholes Barenblatt Equation for Options with Uncertain Volatility and Interest Rate - a PDE View

by Gunter Meyer [mail] (Georgia Tech) in Skiles 269

We employ the maximum principle to derive the nonlinear Black Scholes Barenblatt (BSB) Equation for the computation of attainable bounds on the price of financial options when volatility and interest rates are allowed to vary freely (but deterministically) within a prescribed range. We then illustrate how static hedging and the BSB equation can be used to narrow such bounds.

4:30 pm Tuesday, February 1, 2005

PDE Seminar: A Uniqueness Theorem for Nonlinear Reaction Diffusion Equations

by Prof. Xu-Yuan Chen, (Georgia Tech) in Skiles 255

It is well known that the Cauchy problem of the heat equation u_t=\Delta u has nontrivial classical solutions with zero initial data. The uniqueness for the heat equation only holds under some growth conditions on the solutions at space infinity. On the contrary, we will show that for a class of nonlinear reaction diffusion equations, the uniqueness of solutions to the Cauchy problem holds without any growth conditions. Our examples include u_t=\Delta u+u-u^3. The existence of solutions with singular initial data will also be discussed.

12:00 am Wednesday, February 2, 2005

Noncommutative Geometry And Mathematical Physics: No seminar this week

3:00 pm Wednesday, February 2, 2005

Research Horizons: Dynamics of Regulatory Networks: Nitrogen Catabolite Repression in Yeast

by Konstantin Mischaikow (School of Mathematics, Georgia Tech) in Skiles 255

Tremendous advances have been made in cataloguing the structures and motifs of genetic regulatory networks. However, our understanding of the implications of these structures on the dynamic response of the network is more limited. I will discuss our efforts to build a simple scalable model based on the Nitrogen Catabolite Repression (NCR) circuit in Saccharomyces cerevisiae and provide a mathematical analysis of its dynamics. In particular, time permitting I will focus on four topics: 1) Structure theorems, that is, mathematical results that allow one to make statements about the dynamics of a system without detailed knowledge of the nonlinear interactions of the system. 2) Some of the basic biology behind our model and how our model compares with experimental data. 3) Mathematical theorems about the asymptotic behavior of a sub-circuit of the NCR circuit. 4) Some tantalizing numerical results.

4:00 pm Wednesday, February 2, 2005

Analysis Seminar: The disproval of Fuglede's conjecture

by Mate Matolcsi (Renyi Institute, Budapest, Hungary) in Skiles 269

Fuglede's conjecture relates the notions of translational tiles and spectral sets in Euclidean spaces. Let T be a d-dimensional Lebesgue measureable set of finite non-zero measure. A discrete d-dimensional set L is said to be a spectrum of T if the characters corresponding to L (and restricted to T) form an orthogonal basis in the space of square-integrable functions on T. If such L exists, T is said to be spectral. The set T is said to be a (translational) tile if it is possible to tile the whole d-dimensional space with a family of translates t+T of T (ignoring overlaps and gaps of measure zero). Then, Fuglede formulated the following Conjecture: A set T of finite, non-zero Lebesgue measure is a tile if and only if it is spectral. Fuglede proved the conjecture in the special case when the spectrum L or the translation set T' is assumed to be a lattice. The general case of the conjecture was open for nearly 30 years, until last year Tao showed an example to disprove one direction of the conjecture in 5 and higher dimensions. Namely, he gave an example of a 5-dimensional spectral set which is not a tile. With a slight modification of Tao's arguments I reduced the dimension of the counterexample to 4, and subsequently (in a joint work with M. Kolountzakis) to 3. The 'tile --> spectral' direction, however, could not be tackled by Tao's arguments. A counterexample for this direction was found in dimension 5 during my visit to Prof. M. Kolountzakis at the University of Crete. The proof is mainly based on Fourier analysis and combinatorics. Recently, in a joint work Szilard Revesz and Balint Farkas reduced this dimension to 4, and subsequently Balint Farkas and I reduced the dimension further to 3. At present the Conjecture is still open in dimensions 1 and 2. In dimension 1, the 'tile --> spectral' part of the conjecture would follow from a particular number theoretic conjecture (which is far from being settled...)

4:30 pm Thursday, February 3, 2005

Colloquium: Job Candidate's talk: Wave equations with strong constraining potentials

by Prof. Chongchun Zeng [mail] (University of Virginia) in Skiles 269

In this talk, we consider a vector valued nonlinear wave equation of the unknown u(t, x) \in R^n. Suppose the energy density of the equation contains a nonlinear potential \frac 1{\epsilon^2} V(u) which achieves its minimal value 0 on a submanifold M in R^n. As \epsilon approaches 0, i.e. as this potential approaches infinity, we are interested in the convergence of finite energy solutions. Through a multi-scale formal asymptotic expansion involving rapid oscillations, J. Keller and K. Rubinstein (1991) found that the singular limits of those solutions satisfy a hyperboic PDE system. We rigorously justified this convergence procedure and the local well-posedness of this system. In particular, when the initial data is well prepared, the limit system reduces to the wave map equation (geometric wave equation) targeted on M. The comparison between the structures of the wave equation and the limit system and a more general picture of Hamiltonian PDEs (eg. Schrodinger maps) with strong potentials, will also be briefly discussed.

9:30 am Friday, February 4, 2005

Graph Theory: Forbidden Matroid Minors and Seymour's Conjecture

by Paul Wollan [mail] (Math, GT) in Skiles 255

Let C be a binary clutter and let M be the incident matrix with rows corresponding to the hyperedges of the clutter. Then C is ideal if the polyhedron Mx >= 1 has integral extremal points. The operations contraction and deletion exist for clutters, and the property of being ideal is then preserved under taking minors. Seymour conjectured that a clutter is not ideal if and only if it contains one of three forbidden minors. Cornuejols and Guenin proved that every non-ideal clutter contains one of a list of five minors (including the conjectured three). We present how the Cornuejols and Guenin proof first translates the problem into a question of forbidden matroid minors, and then finds the list of forbidden minors utilizing Seymours characterization of regular matroids. This talk will assume no previous familiarity with clutters.

2:00 pm Friday, February 4, 2005

Geometry and Topology: Quantization of Lie superalgebras

by Nathan Geer (Georgia Tech) in Skiles 269

For every semi-simple Lie algebra one can construct the Drinfeld-Jimbo algebra U. This algebra is a deformation Hopf algebra defined by generators and relations. To study the representation theory of U, Drinfeld used the KZ-equations to construct a quasi-Hopf algebra A. He proved that particular categories of modules over the algebras U and A are tensor equivalent. Analogous constructions of the algebras U and A exist for Lie superalgebra of type A-G. However, Drinfeld's proof of the above equivalence of categories does not generalize to Lie superalgebras. In this talk, we will discuss an alternate proof for classical Lie superalgebras. Our proof utilizes the Etingof-Kazhdan quantization of Lie (super)bialgebras. It should be mentioned that the above equivalence is very useful. For example, it has been used in knot theory to relate quantum group invariants and the Kontsevich integral.

4:00 pm Friday, February 4, 2005

Combinatorics: Reducibility for The Four-Color Theorem

by Serguei Norine (Georgia Tech) in Skiles 269

The Four-Color Theorem (4CT) states that every planar graph is 4-colorable. Known proofs of the 4CT consist of two steps - reducibility and discharging. While the discharging part of the proofs is essentially human-readable, the reducibility part is more problematic in the sense that it heavily depends on the use of computers. We explain the technique of reducibility as used in the proof of the 4CT by Robertson, Sanders, Seymour and Thomas and discuss recent improvements and new results. We will mention related reducibility notions for the cycle double cover conjecture and 5-flow conjecture. This is joint work with Robin Thomas.

1:00 pm Monday, February 7, 2005

Job Candidate Seminar: Model Selection via Information Criteria for Tree Models and Markov Random Fields

by Zsolt Talat (Hungarian Academy of Sciences) in Skiles 269

The concept of context tree, usually defined for finite memory processes, is extended to arbitrary stationary ergodic processes (with finite alphabet). The familiar BIC and MDL principles are shown to provide strongly consistent estimators of the context tree, via optimization of a criterion for hypothetical context trees of finite depth, allowed to grow with the sample size. Algorithms are provided to compute these estimators both off-line and on-line ways. For Markov random fields on the d-dimensional lattice with finite state space, I address the statistical estimation of the basic neighborhood, the smallest region that determines the conditional distribution at a site on the condition that the values at all other sites are given. A modification of the BIC is proved to provide strongly consistent estimation from observing a realization of the field on increasing finite regions, not assuming any prior bound on the size of the latter.

4:30 pm Monday, February 7, 2005

CDSNS Colloquium: Symbolic extensions and entropy structure

by M. Michael Boyle (Mathematics, University of Maryland) in Skiles 269

Let T be a selfhomeomorphism of a compact metric space X. The topological entropy of T is a crude measure of the complexity of T. The measure theoretic entropy, viewed as a function on the invariant Borel probabilities, indicates "where" in X this complexity occurs. Finally, the "entropy structure" theory of Downarowicz is a unified description of how this complexity appears as the system (X,T) is examined at finer and finer resolution, and provides a master functional-analytic invariant for general entropy theory. The entropy structure theory is based on the entropy theory of symbolic extensions of (X,T). I'll discuss all of this.

4:30 pm Tuesday, February 8, 2005

PDE Seminar: Some nonlinear problems in geometry and optics leading to Monge-Ampere equations

by Vladimir Oliker (Emory University) in Skiles 255

Many problems in geometry concerning existence of a closed hypersurface in Euclidean space with a prescribed curvature function require an investigation of a second order PDE of Monge-Ampere type. Similarly, the corresponding PDE's are of Monge-Ampere type in several classes of problems in optics which require determination of a convex hypersurface which for a given energy source will redirect and redistribute that energy in a prespecified manner. In my talk I intend to survey several such problems and describe geometric ideas (some of which go back to Minkowski) which allow to solve these equations (in weak sense) by purely geometric means. If time permits, I will also explain the connection of such problems to the Monge-Kantorovich theory.

12:00 pm Wednesday, February 9, 2005

Noncommutative Geometry and Mathematical Physics: The geometry and quantum topology of the WEAVE

by Stavros Garoufalidis [mail] (School of Mathematics, Georgia Tech) in Skiles 269

The volume conjecture connects quantum topology (ie the Jones polynomial of a knot) with Riemannian geometry (mostly hyperbolic, i.e. the geometry of Dehn fillings of the knot complement). Loosely speaking, it states that for every knot, and every fixed positive real number, the limit of the n-th colored Jones polynomial at exp(2 pi i a/n) exists and is given by the volume of the (1/a,0)-Dehn filling. In joint work with Thang Le, and using elementary calculus arguments, we will prove that (a) the limsup of the sequence above is finite, bounded above by a simple function on the number of crossings and (b) for every knot K, there is a positive number a(K) such that the volume conjecture holds for a in [0, a(K)). For a picture of the WEAVE.

3:00 pm Wednesday, February 9, 2005

Research Horizons: Extremal problems on hypergraphs

by Adam Marcus (Georgia Tech, School of Mathematics) in Skiles 255

Hypergraphs are a useful way of "visualizing" set systems: that is, a ground set and a collection of subsets. Many of the familiar problems from normal graph theory can be extended to this more general structure, and a particularly interesting extension is that of Turan-type problems. The general problem is easy to state: given n vertices and a collection P of "illegal" sub-hypergraphs, how many edges can we place on the n vertices while avoiding all of the structures in P? I will give a brief introduction to hypergraphs, and discuss one interesting problem in depth: when the structures in P are the set of "sunflowers." No prior knowledge will be expected.

4:30 pm Thursday, February 10, 2005

Note Time Change: Job Candidate Colloquium: : Resonance Phenomena in Nonlinear Dispersive Partial Differential Equations

by Eduard-Wilhelm Kirr (Department of Mathematics, University of Chicago) in Skiles 269

The resonance phenomena and stability of a periodically forced, linear oscillator is well understood. But the problem becomes quite difficult when the mechanical system has more than one degree of freedom and the forcing depends on the state of the system. Multiple scale analysis, Poincare continuation and KAM theory give only partial answers. My talk will focus on recent, rigorous results concerning systems with infinitely many degrees of freedom. I will briefly describe why such systems are ubiquitous in Quantum Mechanics, Statistical Physics and Optics where they are modeled by dispersive partial differential equations. A simplified mechanical example would be a mass-spring system attached to an infinitely long, tense string. The oscillations of the spring excite (resonantly) the string which carries the energy of the excitations to infinity. As a result one sees a decay of the amplitude with which the mass-spring system oscillates. I will present the mathematical techniques involved in proving that the same phenomenon occurs for the ground state of the cubic nonlinear Schroedinger equation subject to periodic in time perturbation, a result obtained in collaboration with S. Cuccagna and D. Pelinovsky. Then I will discuss the similarities and differences between this result and the ones for random and almost periodic perturbations of linear Hamiltonian partial differential equations obtained in collaboration with M. Weinstein. At the end I will mention some related open problems and argue that the above results and the mathematical techniques developed constitute a solid basis for attacking them.

9:30 am Friday, February 11, 2005

Graph theory: On multicommodity information flow in undirected graphs

by Kamal Jain [mail] (Microsoft) in Skiles 255

Li and Li conjectured that the maximum multicommodity information flow in an undirected graph is the same as the maximum multicommodity fluid flow (difference between information and fluid is that the different pieces of information can be mixed together and can be separated later whereas two different fluids can't be mixed together). I will show the truth of this conjecture on K_32 (complete bipartite graph on 3 vertices on one side and 2 on the other). An upper bound in K_32 can also be generalized to all directed bipartite graphs. This work is not technically complicated but involves ideas from very diverse research areas. It includes ideas from information theory, cryptography and combinatorics. My feeling is that the proof of the whole conjecture is not going to be technically complicated either but would require another idea, which we are missing now. This new idea could be from graph theory (since the underline object of the conjecture is a graph) or algebra or from both. The reason for this talk is to seek that missing idea. This is a joint work with Chou, Vazirani, Yeung and Yuval. The paper is available at http://www.cc.gatech.edu/~kjain/k32.pdf I will present all the basic ingredients to make sure that the talk is accessible to all. Still it might be helpful to glance through the paper beforehand.

3:00 pm Friday, February 11, 2005

Geometry and Topology: The rationality of three-dimensional Solv-manifolds

by Andy Putman [mail] (U. of Chicago and Georgia Tech) in Skiles 269

We study the growth series of the fundamental groups of three dimensional Solv manifolds. There is a natural metric on the elements in any finitely generated group, and hence one can consider the size c_k of the ball of radius k about the identity. The growth series is the power series with coefficients c_k. Our main theorem is that (after passing to a finite index subgroup) the growth series of a Solv manifolds is a rational function. Similar results are known for many other groups, but there is no known unifying "reason" for these results. We will state several open problems which attempt to shed some light on this odd state of affairs. Crucial to our approach is the theory of finite state automata. Since these are not in the standard tool box of most geometers, we will survey them in a non-technical way and attempt to show why they are natural objects to consider in this context.

4:00 pm Friday, February 11, 2005

Combinatorics: A Unification of Menger's Theorem, Edmonds' Theorem and Network Coding Theorem

by Kamal Jain (Microsoft) in Skiles 255

I will present a generalized theorem which has the classical Menger's theorem and Edmonds' theorem and modern Network Coding theorem as its special cases. I will try to include as much proofs as possible within the timeframe of the talk. Proofs are sophisticated but elementary enough to reach a wide audience. This is a joint work with Yunnan Wu and S.-Y. Kung from Princeton University.

11:15 am Monday, February 14, 2005

CNS Seminar: A Mathematical Framework for Deterministic Models of Chemical Reactions

by Mark Demers (School of Mathematics, Georgia Tech) in Howey W505

I will present a general mathematical framework for constructing deterministic models of chemical reactions. In such a model, an underlying dynamical system drives a process in which a particle undergoes a reaction (changes color) when it enters a certain subset (the catalytic site) of the phase space and (possibly) some other conditions are satisfied. This framework allows us to define the entropy of reaction precisely and does not rely on a stochastic mechanism to generate additional entropy. Rates of reaction are also handled in this framework, but are independent of the entropy of reaction. This is joint work with L. Bunimovich.

Refreshments will be provided

4:30 pm Monday, February 14, 2005

CDSNS Colloquium: A New KAM Theorem for Hamiltonian PDE in Higher Space Dimension

by Jiansheng Geng (Nanking University and Mathematics, Georgia Tech) in Skiles 269

The development of KAM theory has experienced half a century. From classical KAM theorem given by Kolmogorov in 1954, it was extended to finite dimensional case and later to Hamiltonian PDEs in one space dimension. It seems that there are unavoidable difficulties for KAM theory to apply to higher dimensional Hamiltonian PDEs. Recently, Bourgain took advantage of more powerful harmonic analysis techniques and results by Frohlich and Spencer to effectively control the inverse of Green function and obtain quasiperiodic solutions. But his methods can not give linear stability of these quasiperiodic solutions. In this lecture, we will discuss recent results obtained jointly with J. Yu regarding regularity and stability for Hamiltonian PDEs in higher space dimensions.

1:00 pm Tuesday, February 15, 2005

Job Candidate Seminar: Multivariate hypergeometric functions

by Laura Felicia Matusevich (University of Pennsylvania) in Skiles 255

In the late 1980s, Gelfand, Kapranov and Zelevinsky uncovered a connection between the classical hypergeometric functions and the theory of toric varieties. This surprising link to combinatorics and algebraic geometry can be exploited to obtain hypergeometric results. I will give a small survey of GKZ theory, ending in the recent solution (joint with Ezra Miller and Uli Walther) of a conjecture of Sturmfels concerning holonomic ranks of hypergeometric systems.

4:00 pm Tuesday, February 15, 2005

Job Candidate Seminar: Linearization coefficients, orthogonal polynomials, and free probability

by Michael Anshelevich (University of California, Riverside) in Skiles 269

The linearization coefficients for a polynomial basis {P_n} are the coefficients in the expansion P_n P_k = \sum c_{n,k,l} P_l. Surprisingly, these coefficients turn out to be positive integers for a number of classical families. For the Hermite polynomials, they can be found explicitly using probability theory. For other families, such as the Chebyshev polynomials of the 2nd kind, one can instead use a different "free" probability theory. In this talk, we will outline the proof of this result, introducing all the necessary ingredients along the way.

4:30 pm Tuesday, February 15, 2005

PDE Seminar: Low order regularizations for dynamic phase transitions

by Shaoqiang Tang (CalTech) in Skiles 255

Abstract: In the last a few decades, extensive explorations have been made on stationary phase transitions, e.g. theory of renormalization group. When dynamics is concerned, major difficulty comes from instabilities. Before the presence of a better approach from the perspective of physics, we aim at an attack on this challenging issue at phenomenological level. We shall investigate possible stabilizations, to substantiate our understanding of nonlinear interactions among instability and dissipative mechanisms. In particular, we shall propose a category of discrete BGK models for regularization. Suliciu model and Jin-Xin relaxation model are special cases. For Suliciu model, theoretical we obtain stability results under tri-linear structural relation. We further demonstrate numerically that low order dissipation mechanisms is capable to stabilize phase transition systems. Moreover, this approach applies to high space dimensions. With a relaxation model, numerical simulations produces interesting patterns. This may shed insight into further investigations on dynamic phase transitions, as well as related physical systems.

12:05 pm Wednesday, February 16, 2005

Noncommutative Geometry and Mathematical Physics: Helicity and linking invariants in hydrodynamics

by Rafal Komendarczyk [mail] (School of Math, Gatech) in Skiles 269

This is meant to be an introductory talk. I will define helicity H(X) of a volume preserving vector field X as a generalized Hopf invariant and describe its ergodic interpretation as an average linking number of trajectories in the flow. I will also show that H(X) bounds the kinetic energy of the fluid from below and consequently is an obstruction to energy relaxation. Finally, I will comment on "higher" order linking invariants. The long term goal is to investigate if these invariants are K- theoretic, and interpret them as cycles in cyclic cohomology of a suitable C^*- algebra. This would provide a possible ground for generalizations.

1:00 pm Wednesday, February 16, 2005

Job Candidate Seminar: Change-Point Problems in Sensor Networks

by Dr. Yajun Mei (Fred Hutchison Cancer Research Center) in Skiles 255

Change-Point problems have a variety of applications including industrial quality control, reliability, fault detection, surveillance, and security systems. By monitoring data streams which are generated from a process, we are interested in quickly detecting malfunctioning once the process goes out control, while keeping false alarms as infrequent as possible when the process is in control. The classical or centralized version of this problem, where all observations are available at a single, central location, is a well-developed area. In this talk, we investigate the decentralized version where the information available is distributed across a set of sensors. Each sensor receives a sequence of observations, and sends a sequence of sensor messages to a central processor, called the fusion center, which makes a final decision when observation are stopped. In order to reduce the communication costs, it is required that the sensor messages belong to a finite alphabet. In the decentralized change-point problem, the goal is to detect the change as soon as possible over all possible protocols for generating sensor messages and over all possible decision rules at the fusion center, under a restriction on the frequency of false alarms. We will present a general asymptotic theory, and provide procedures that are asymptotically optimal and easy to implement.

3:00 pm Wednesday, February 16, 2005

Research Horizons: The bispectral problem

by Plamen Iliev (Georgia Tech School of Math) in Skiles 255

The bispectral problem asks to classify operators possessing a certain symmetry. Different versions are closely related to PDEs and their symmetries, singular algebraic curves, classical orthogonal polynomials, representation theory, symmetric functions etc. I will describe the solution and open problems in a variety of situations.

4:00 pm Wednesday, February 16, 2005

Analysis Seminar: Which Weights on R admit Jackson Theorems?

by Doron Lubinsky (Georgia Institute of Technology, School of Mathematics) in Skiles 269

In about 1910, Bernstein posed the problem of approximating continuous functions defined on the whole real line by polynomials. To cope with the unboundedness of polynomials at infinity, he introduced a weight. He posed what became known as the Bernstein Approximation Problem. It was solved independently by Achieser, Mergelyan, and Pollard in the 1950's. Their solution showed for example, that the polynomials are dense for the weight exp(-|x|^a) iff a is greater or equal to 1. With that solved, the question of degree of approximation came to the fore, and has been explored actively since the 1970's. A key issue is the rate of weighted polynomial approximation, in terms of smoothness of the approximated function, the so-called Jackson Theorems. We discuss current progress in this topic, including the recent discovery that exp(-|x|) does not admit a Jackson-Favard theorem.

4:30 pm Thursday, February 17, 2005

COLLOQUIUM: Design and Optimization of a Solid State Qubit System

by Russel Caflisch (Mathematics Department, UCLA) in Skiles 269

This talk will describe the simulation, design and optimization of a qubit for use in quantum communication or quantum computation. The qubit is realized as the spin of a single trapped electron in a semi-conductor quantum dot. The quantum dot and a quantum wire are formed by the combination of quantum wells and gates. The design goal for this system is a "double pinchoff", in which there is a single trapped electron in the dot and a single (or small number of) conduction states in the wire. Because of considerable experimental uncertainty in the system parameters, the optimal design should be "robust", in the sense that it is far away from unsuccessful designs. We use a Poisson-Schrodinger model for the electrostatic potential and electron wave function and a semi-analytic solution of this model. Through a Monte Carlo search, aided by an analysis of singular points on the design boundary, we find successful designs and optimize them to achieve maximal robustness.

9:30 am Friday, February 18, 2005

Graph theory: Curves on Surfaces, String Graphs, and Crossing Numbers

by Daniel Stefankovic [mail] (University of Chicago) in Skiles 255

Computational topology is a classical algorithmic theory going back at least to Dehn (1912). While many of its central questions have been shown to be undecidable, we have found efficient algorithms for a number of classical questions involving curves on surfaces. These algorithms lead to the resolution of the complexity status of the "string graph problem," stated by Sinden in 1966. Classical algorithms for curves on surfaces work on explicit representations of curves. However applications often require that the curves be given by a compressed representation, such as, normal coordinates, or Dehn-Thurston parameterization. The compressed representation has size logarithmically smaller than the explicit representation. Which problems on simple curves can be solved in time polynomial in the size of the compressed representation? We will show that many problems, such as, computing the number of connected components, computing Dehn twists, and computing the geometric intersection number have polynomial-time algorithms. Our main tool are equations over monoids. A string graph is the intersection graph of a set of curves in the plane. Each curve is represented by a vertex, and an edge between two vertices means that the corresponding curves intersect. The string graph problem asks for an algorithm to recognize string graphs. The string graph problem was shown to be decidable by Pach and Toth and independently by Schaefer and the speaker in 2002; both papers put the problem in NEXP. Finally the problem was shown to be NP-complete by Schaefer, Sedgewick, and the speaker in 2003. Algorithms for simple curves on surfaces were an essential ingredient in the solution. The crossing number cr(G) of a graph G is the minimum number of crossings needed to draw the graph G. For the odd crossing number ocr(G) we minimize the number of pairs of edges that cross odd number of times. Is ocr(G)=cr(G) for every G? We will show how understanding curves on surfaces leads to progress on this question.

2:00 pm Friday, February 18, 2005

Frontiers of Physics: Branes and Gauge Theory Dynamics

by Amihay Hanany [mail] (MIT Center for Theoretical Physics) in Howey Lecture Hall L5

There are many types of extended objects in string theory which go under the name "branes". Thus, a pointlike object is a zero brane, a string is a one brane, a membrane is a two brane, etc. Branes have life of their own, and a quantum theory which describes their dynamics. Gauge fields live on them, similar to the gauge fields of the standard model of particle physics. These are fields like the electromagnetic field, the gluon field, and the W & Z gauge fields. There are also charged matter fields which have properties like the electrons, the quarks and the leptons. In a superstring theory the supersymmetry greatly simplifies the gauge dynamics, enabling us to understand better the dynamics of the brane gauge theories. The new insights offer a radically new perspective over the behavior of gravity in various dimensions: the world as seen from the eyes of a brane observer. In this talk we will get a glimpse to the wonderful world of branes. (Intended for a broad non-specialist audience.)

3:00 pm Friday, February 18, 2005

Job Candidate Seminar: How efficiently do 3-manifolds bound 4-manifolds?

by Dylan Thurston (Harvard University) in Skiles 269

It is known since 1954 that every 3-manifold bounds a 4-manifold. Thus, for instance, every 3-manifold has a surgery diagram. There are many proofs of this fact, including several constructive ones, but they do not bound the complexity of the 4-manifold. Given a 3-manifold M of complexity n, we show how to construct a 4-manifold bounded by M of complexity O(n2), for suitable notions of "complexity". It is an open question whether this quadratic bound can be replaced by a linear bound. The natural setting for this result is shadow surfaces, a representation of 3- and 4-manifolds that generalizes many other representations of these manifolds. One consequence of our results is some intriguing connections between the complexity of a shadow representation and the hyperbolic volume of a 3-manifold.

4:00 pm Friday, February 18, 2005

Combinatorics: Locally Testable Cyclic Codes

by Daniel Stefankovic [mail] (University of Chicago) in Skiles 255

Cyclic linear codes of block length n over a finite field F_q are the linear subspace of F_q^n that are invariant under a cyclic shift of their coordinates. A family of codes is good if all the codes in the family have constant rate and constant normalized distance (distance divided by block length). It is a long-standing open problem whether there exists a good family of cyclic linear codes (cf. [MS, p. 270]). A code C is r-testable if there exist a randomized algorithm which, given a word x in F_q^n, adaptively selects r positions, checks the entries of x in the selected positions, and makes a decision (accept or reject x) based on the positions selected and the numbers found, such that * if x then x is surely accepted; * if dist(x,C)>=eps.n then x is probably rejected. ("dist" refers to Hamming distance.) A family of codes is locally testable if all members of the family are r-testable for some some constant r. This concept arose from holographic proofs/PCPs. Goldreich and Sudan [GS] asked whether there exist good, locally testable families of codes. It is an open problem whether there exists a good locally testable. We will show: Theorem: There is no family of good locally testable cyclic codes.

12:00 pm Monday, February 21, 2005

Noncommutative Geometry and Mathematical Physics: Helicity and linking invariants in hydrodynamics Part II

by Rafal Komendarczyk [mail] (Gatech) in Skiles 269

This is the continuation of the talk given on Feb 16th. PLEASE NOTE THE UNUSUAL TIME !!

1:00 pm Monday, February 21, 2005

Job Candidate Seminar: Generating monotone sequences of die rolls

by Karen Ball (Department of Mathematics, Indiana University) in Skiles 255

Suppose you have two (possibly unfair) d-sided dice with faces labeled 1,...,d, and that these two dice have different shapes. We will investigate when and how we can jointly generate two sequences of numbers X = (...,X_{-1},X_0,X_1,...) and Y = (...,Y_{-1},Y_0,Y_1,...) so that X looks like a sequence of rolls of Die 1, Y looks like a sequence of rolls of Die 2, and X_i >= Y_i for all i with probability one. It is not difficult to characterize when such a joint process is possible and the natural construction is itself an independent and identically distributed process. In this work, we look for a stationary process (X,Y) as above, with the additional property that Y=f(X) is a deterministic function of X. The motivation for finding such an f comes from ergodic theory, where such functions are used to show when one dynamical system is a subsystem of another. We will also present a similar result for Poisson processes, which are a continuous-time analog of the die-rolling processes.

3:00 pm Monday, February 21, 2005

Algebra and Topology Seminar: Gromov-Witten invariants on Grassmannians

by Harry Tamvakis [mail] (Brandeis) in Skiles 269

The three point genus zero Gromov-Witten invariants of a Grassmannian X count the number of rational curves in X satisfying three natural incidence conditions. In joint work with Anders Buch and Andrew Kresch, we show that these numbers are equal (or closely related) to classical triple intersection numbers on a homogeneous space of the same Lie type as X. Our approach to this problem uses only basic geometry; my lecture will be accessible to graduate students.

4:30 pm Monday, February 21, 2005

CDSNS Colloquium: Critical Thresholds in Eulerian Dynamics

by Eitan Tadmor [mail] (Professor of Mathematics and Director for Scientific Computation and Mathematical Modeling (CSCAMM), Institute for Physical Science & Technology, University of Maryland at College Park) in Skiles 269

We study the questions of global regularity vs. finite time breakdown in Eulerian dynamics, u_t+u\cdot\nabla_x u=\nabla_x F, which shows up in different contexts dictated by different modeling of F's. To adders these questions, we propose the notion Critical Threshold (CT), where a conditional finite time breakdown depends on whether the initial configuration crosses an intrinsic, {\cal O}(1) critical threshold. Our approach is based on spectral dynamics, where the eigenvalues, \lambda:=\lambda(\nabla_x u) associated with the eigenpairs (\ell,r), are traced by the forced Riccati equation \lambda_t +u\cdot\nabla_x\lambda + \lambda^2 = \langle\ell, D^2F r\rangle. We shall outline three prototype cases. We begin with the n-dimensional Restricted Euler equations, obtaining [n/2]+1 global invariants which (i) precisely characterize the local topology of the 3D breakdown, and (ii) yield a surprising 4D global existence for sub-critical initial data. Next we introduce the corresponding n-dimensional Restricted Euler-Poisson (REP) system, identifying a set of [n/2] global invariants, which yield (i) sufficient conditions for finite time breakdown, and (ii) a remarkable characterization of two-dimensional initial REP configurations with global smooth solutions. And finally, we show that a CT phenomenon associated with rotation prevents finite-time breakdown. Our study reveals the dependence of the CT phenomenon on the initial spectral gap, \lambda_2(0)-\lambda_1(0).

2:00 pm Tuesday, February 22, 2005

Applied & Computational Math Seminar (Job Candidate): Geometric tools for high-dimensional data analysis

by Ann Lee (Yale) in Skiles 269

In many applied fields --- such as image analysis, information technology and biology --- one has to analyze noisy, but structured data, in very high dimensions (>1000 or even 10,000), often with a small number of samples. This "large p -- small N" regime presents challenges for data analysis and calls for efficient dimension reduction tools that take the inherent geometry of natural data into account. In the first part of my talk, I will describe a multi-scale orthogonal basis that can be used for feature extraction of smooth data (such as images and spectral measurements) as well as non-smooth data (such as DNA micro arrays and word-document arrays). I will then, in the second half of the talk, describe a general methodology for organizing high-dimensional data sets by embedding the data into Euclidean space via a non-linear diffusion map. Examples will be taken from image analysis, word-document clustering and spectroscopy.

4:30 pm Tuesday, February 22, 2005

PDE Seminar: Free Boundary Problems and Multidimensional Transonic Shocks

by Prof. Guiqiang Chen (Northwestern University) in Skiles 255

In this talk, we will first discuss some natural connections between multidimensional transonic shock waves and free boundary problems for the Euler equations for compressible fluid flow. Then we will present some new approaches developed recently for solving such free boundary problems through some concrete examples and address further applications in fluid dynamics. The examples and further applications especially include the existence and stability of multidimensional transonic shocks in steady compressible flow in the whole space R^n, n\ge 3, and past an infinite de Laval nozzle under the perturbation of the nozzle boundary. The nonlinear stability of multidimensional shocks in steady Euler flow past an infinite curved wedge or cone under the BV perturbation of the obstacle and the nonlinear stability of supersonic vortex sheets in steady Euler flow under the BV perturbation of the boundaries will also be addressed.

3:00 pm Wednesday, February 23, 2005

Research Horizons: Viscosity solutions of partial differential equations

by Andrzej Swiech (Georgia Tech School of Math) in Skiles 255

The notion of viscosity solution was introduced by M. G. Crandall and P. L. Lions on 1981 and over the years has become one of the fundamental tools of the modern theory of fully nonlinear first and second order partial differential equations. It has found applications in areas as diverse as deterministic and stochastic optimal control, image processing, moving fronts and phase transitions, statistical mechanics, economics, mathematical finance, and still continues its rapid development. This talk will be a brief introduction to viscosity solutions. I will define the notion of viscosity solution and I will talk about its history, basic results, and applications.

11:00 am Thursday, February 24, 2005

Nonlinear Science Seminar: POSTPONED: Details--TBA: Computer-Generated Animation of Fluids: An Applied Math Perspective

by Peter Mucha (School of Mathematics, Georgia Tech) in Physics N110

Computer graphics researchers and animators have embraced computational fluid dynamics (CFD) over the past decade. After summarizing a number of successfully implemented CFD techniques, we examine two recently addressed problems in detail: approximating melting by variable viscosities, and computing interactions between solid rigid bodies and fluids. In the latter, the Rigid Fluid method is used to efficiently animate the interplay between rigid bodies and viscous incompressible fluids with free surfaces. This technique uses distributed Lagrange multipliers to compute two-way solid-fluid couplings, alternating between treating the solids as if they were fluid and constraining the motions of those regions to obey rigid body motion. The method---straightforward to implement and incurring little computational overhead---generates realistic motion for both the solid objects and the fluid as they interact with one another. This talk will assume no detailed knowledge of fluid mechanics or CFD. Many movies will be shown. This talk represents work done in collaboration with Mark Carlson and Greg Turk

4:15 pm Thursday, February 24, 2005

Math Department Tea:

in Skiles 236

This is our math department tea, which is open to all students, faculty, and staff. Food and beverages will be served.

9:30 am Friday, February 25, 2005

Graph Theory: Forbidden Matroid Minors and Seymour's Conjecture (2)

by Paul Wollan [mail] (Math, GT) in Skiles 255

Let C be a binary clutter and let M be the incident matrix with rows corresponding to the hyperedges of the clutter. Then C is ideal if the polyhedron Mx >= 1 has integral extremal points. Seymour conjectured that a clutter is not ideal if and only if it contains one of three forbidden minors. We continue a discussion of the paper by Cornuejols and Guenin where they prove that every non-ideal clutter contains one of a list of five minors (including the conjectured three). We'll look at one aspect of the proof that could be improved to reduce the list. Previous exposure to clutters (including the first talk) will not be necessary.

3:00 pm Friday, February 25, 2005

Geometry and Topology: Knots of Constant Curvature

by Mohammad Ghomi [mail] (Georgia Tech) in Skiles 269

We will prove that there exists a knot of constant curvature in each isotopy class of closed embedded curves in Euclidean 3-space. Further we will show that two knots of constant curvature admit an isotopy through knots of constant curvature if and only if they are isotopic (through general knots) and have the same self-linking number. These results, which we prove using some convex integration techniques, generalize earlier work of Gluck and Pan on knots with non-vanishing curvature, and establish an h-principle in the sense of Gromov.

4:00 pm Friday, February 25, 2005

Combinatorics : On Long Arithmetic Progressions in Sets with Few Three-Term Arithmetic Progressions

by Ernie Croot (Georgia Tech) in Skiles 255

Let t be a density, which is a real number in (0,1]. Then, for all sufficiently large primes p the following holds: Let S be a subset of the integers modulo p having density at least t (that is, having at least t p elements) and having the least number of three-term arithmetic progressions modulo p among all other sets with at least tp elements. We will call such a set S a critical set for the density t. In this talk I will show that such a set S must contain an arithmetic progression of length at least (log p)^{1/4 + o(1)}, which is quite a bit longer than the best that is known for a generic set of density at least t (due to T. Gowers). The proof uses several combinatorial arguments, as well as harmonic analysis.

11:00 am Monday, February 28, 2005

Nonlinear Science: Singular surfaces: breakup, collapse & entrainment

by Wendy Zhang (Physics, J. Franck Inst., U. Chicago) in Howey N110

Surface tension can cause a deformed liquid drop to break into several droplets. A fluid interface subjected to external stresses spontaneously forms sharp points and thin, extended filaments and sheets. We examine two examples of singularity formation on the fluid interface: the breakup of a water drop in oil and the steady-state entrainment of water by oil. We show that the singularity-formation dynamics in these two systems are very different from previously studied situations, in which the dynamics near a surface singularity becomes scale-invariant and independent of large-scale conditions. In both examples here, conditions on the largest length- and time-scales have a significant effect on the dynamics near the singularity without destroying scaling behavior.

2:00 pm Monday, February 28, 2005

Job Candidate Seminar: Predictability of ensemble simulations

by Rafail Abramov (Courant Institute of Mathematical Sciences) in Skiles 255

Statistical ensemble simulations play an important role in predicting the behavior of chaotic / noise-driven processes in nature. It is therefore imperative to understand and quantify useful information in a forecast ensemble. Modern methods of estimating predictive skill in an ensemble are often based on its mean state and variance, thus not taking into account the shape of its distribution. Introduced is a novel information theory-based predictability approach via relative entropy which captures extra information in higher order statistical moments of a forecast ensemble.

4:30 pm Monday, February 28, 2005

CDSNS Colloquium: Synchrony in fiber laser arrays

by Slaven Peles (School of Physics, Georgia Tech) in Skiles 269

Mutual synchronization between coupled fiber lasers has been successfully demonstrated in a number of recent experiments. While these results may lead to a dramatic improvement in laser technology, the exact mechanism by which these lasers synchronize is not well understood. We use a recently proposed iterated map model for fiber laser arrays to help explain these phenomena. In particular, we look at synchronous solutions to the map when gain fields are constant. The stability of these solutions is analytically tractable for a number of different coupling schemes. We find that in most of the cases coherent states are either unstable or their stability is not robust enough for practical purposes. However, certain experimentally motivated coupling configurations lead to surprisingly robust stable coherent solutions. This coherence persists even beyond the pumping threshold at which the gain fields become time dependent.

11:00 am Tuesday, March 1, 2005

CNS Meeting: Effective computation of the dynamics around a two-dimensional torus of a Hamiltonian system

by Frederic Gabern (University of Barcelona) in Howey W505

3:00 pm Tuesday, March 1, 2005

Applied & Computational Math Seminar: A Mathematical Framework for Deterministic Models of Chemical Reactions

by Mark Demers (Mathematics, Georgia Tech) in Skiles 169

I will present a general mathematical framework for constructing deterministic models of chemical reactions. In such a model, an underlying dynamical system drives a process in which a particle undergoes a reaction (changes color) when it enters a certain subset (the catalytic site) of the phase space and (possibly) some other conditions are satisfied. This framework allows us to define the entropy of reaction precisely and does not rely on a stochastic mechanism to generate additional entropy. Rates of reaction are also handled in this framework, but are independent of the entropy of reaction. This is joint work with L. Bunimovich.

4:30 pm Tuesday, March 1, 2005

PDE Seminar: Wave breaking in a class of nonlocal dispersive wave equations

by Prof. Hailiang Liu (Iowa State University) in Skiles 255

The Korteweg de Vries (KdV) equation is well known as an approximation model for small amplitude and long waves in different physical contexts, but wave breaking phenomena related to short wavelengths are not captured in. We introduce a class of nonlocal dispersive wave equations which incorporate physics of short wavelength scales. The model is identified by the renormalization of an infinite dispersive differential operator and the number of associated conservation laws. Several well-known models are thus rediscovered. Wave breaking criteria are obtained for several typical models including the Burgers-Poisson system and the Camassa-Holm equation.

12:05 pm Wednesday, March 2, 2005

Noncommutative Geometry and Mathematical Physics seminar: Waves in random media: localization and a bound for the density of states inside a band

by Jeffrey Schenker (job candidate) [mail] (ETH, Zurich, Switzerland) in Skiles 269

I will discuss the propagation of waves in a weakly disordered environment. Currently there are very strong bounds describing spectral and dynamical localization at band edges, but the behavior in the center of a band, where diffusion is expected, remains poorly understood. I will first discuss the localization bounds, obtained in joint work with Aizenman, Friedrich, Hundertmark [Comm. Math. Phys. 244, p. 219] and Aizenman, Elgart, Naboko, Stolz [preprint]. Then I will outline the proof of a bound [to appear in Lett. Math. Phys.] on the difference between the densities of states with and without weak disorder. The proof makes use of a generalized Fourier transform, incorporating disorder translations, which may provide a convenient framework for more involved perturbative calculations inside a band.

2:00 pm Wednesday, March 2, 2005

Job Candidate Seminar: Stationary solutions for the Navier-Stokes system with random forcing in 2D and 3D

by Yury Bakhtin (Duke University) in Skiles 269

I will discuss randomly forced Navier-Stokes system in 2D and 3D. The problem of existence and uniqueness of statistically stationary solutions for randomly forced 2D Navier-Stokes system was intensively studied in the past decade. One of the successful approaches was the one due to E, Mattingly and Sinai. It involved a reduction to a finite-dimensional system with memory and analysis of this finite-dimensional non-Markov ("Gibbsian") system. I will give a new general result (joint with Jonathan Mattingly) for stochastic systems with memory and show that it is applicable to 2D Navier-Stokes. As for the 3D situation, not so much is known so far. I will give a new existence-uniqueness theorem for stationary solutions in 3D under some smallness conditions on the random forcing. The stationary solution will be constructed and studied with the help of a beautiful stochastic cascade construction due to Le Jan and Sznitman.

3:00 pm Wednesday, March 2, 2005

Research Horizons: Compactness: applications and examples

by Jean Bellissard (Georgia Tech School of Math) in Skiles 255

Compactness argument are used in many area of Mathematics including in Mathematical Physics, to get existence theorems that otherwise would be impossible to prove. We will investigate several examples and several theorems that are valid only on compact sets and used in various areas of mathematics.

4:00 pm Wednesday, March 2, 2005

Analysis Seminar: Refinable functions with non-integer dilations

by Yang Wang (School of Mathematics, Georgia Tech) in Skiles 269

Refinable functions with integer dilations have been studied extensively since the seminal work of I. Daubechies on compactly supported orthonormal wavelets. On the other hand, with non-integer dilations they have been studied since 1930's in connection with Bernoulli convolutions, by Erdos, Salem, Kahane, Solomyak and others. In this talk, I'll present some recent results, particularly the connection to the arithmetic properties of the dilations.

11:00 am Thursday, March 3, 2005

Nonlinear Science: Invariant curves of Hamiltonian-Hopf bifurcations of 4D symplectic maps

by Angel Jorba (Matem�tica Aplicada i An�lisi, U. Barcelona) in Howey N110

We give a numerical description of the neighbourhood of a fixed point of a symplectic map undergoing a transition from linear stability to complex instability,via the Hamiltonian-Hopf bifurcation. The numerical computation of the Lyapunov families of invariant curves near the fixed point shows how these families and their invariant manifoldsorganise the phase space around the bifurcation.

4:30 pm Thursday, March 3, 2005

Job Candidate Seminar: Invasion and persistence in flow-through systems

by Frithjof Lutscher (Center for Mathematical Biology, Dept. of Mathematical & Statistical Sciences, Univ. of Alberta) in Skiles 255

Individuals in rivers and streams are subject to unidirectional flow that potentially influences their movement. The question how populations persist in environments where individuals are washed out occurs in many situations (e.g. chemostat); in the context of stream ecology it has been called the "Drift Paradox". In this talk, we are particularly thinking of benthic invertebrates, phytoplankton, and periphyton. In addition to unidirectional flow, several processes such as groundwater exchange and runoff tend to create resource gradients along rivers and streams. We derive and analyze some biologically simple but mathematically tractable models for such systems. These models have the form of reaction-dispersal equations and reaction-diffusion systems. We show some results regarding traveling waves and spreading speeds, stability and critical domain size, as well as pattern formation through abiotic (flow rates) or biotic (competition, predation) factors. We interpret these results in terms of persistence and invasion criteria for rivers and streams and offer two different but mathematically equivalent explanations for the drift paradox. In some cases, mathematical modeling is complemented by research in small experimental streams.

4:00 pm Friday, March 4, 2005

CDSNS Seminar: On the fractalization of invariant curves in quasi-periodically forced 1-D systems

by Angel Jorba (University of Barcelona) in Skiles 255

We consider a discrete dynamical system such that one of the variables is a one-dimensional angle whose dynamics is a rigid irrational rotation. Our goal is to study the continuation w.r.t. parameters of the invariant curves and their bifurcations. In this continuation we keep fixed the rotation number of the curve to a Diophantine value (that is, in fact, the rotation angle of the angular variable of the system). It is well known that one of the obstructions for these continuation is a fractalization of the curve that can lead to its destruction. In the talk we will discuss some partial results of a longer project whose final goal is to explain the fractalization of invariant curves in the context of bifurcation theory. We will present both analytical and numerical results. This is joint work with J.C. Tatjer.

4:00 pm Friday, March 4, 2005

Combinatorics: Sharpening eigenvalue bounds on mixing by use of the spectral profile

by Ravi Montenegro (Georgia Tech) in Skiles 269

The second largest eigenvalue of a Markov chain transition matrix is widely used to study its rate of convergence to stationary. We present a self-contained proof of sharper mixing bounds by use of the Spectral profile, a generalization of eigenvalue bounds which considers set sizes as well. This leads to simple proofs of only a few lines each that the log-Sobolev constant and Nash Inequalities bound mixing time, avoiding the need for hypercontractivity methods which were required for past proofs. This is joint work with Goel and Tetali.

3:30 pm Monday, March 7, 2005

Job Candidate Seminar: Almost Periodicity and Weak Mixing in Topological Dynamics

by Alica Miller (University of California, Irvine) in Skiles 255

Almost Periodicity and Weak Mixing are among the most important properties of topological dynamical systems. Almost Periodicity was introduced by Birkhoff generalizing the work of Bohr on Almost Periodic Functions, while Weak Mixing was introduced by Furstenberg by adapting the analogous notion from Ergodic Theory. The two notions, in a way opposite to each other, are still intensively investigated. In this talk we discuss some of our results involving these notions and some relations to other areas. Motivated by an old, not yet completely resolved, question of Gottschalk, about the relation between Weak Mixing and Total Minimality of compact minimal flows, we establish, for a large class of acting groups, a precise position of Total Minimality with respect to both Weak Mixing and Almost Periodicity. We end with our refinement of a theorem by Bronstein, Holodenko and Egawa, which gives a surprising relation between Almost Periodicity and Weak Mixing.

4:30 pm Monday, March 7, 2005

CDSNS Colloquium: Global weak solutions to the Camassa-Holm equations

by Alberto Bressan (Pennsylvania State University) in Skiles 269

The Camassa-Holm equation can be written in the form of a scalar conservation law plus an integral source term that preserves the H^1 norm of the solution. Smooth initial data can lose regularity in finite time. However, solutions always remain Holder continuous. The talk will present two new methods for constructing a continuous semigroup of solutions to the Camassa-Holm equation. In one case, a suitable change of variables transforms the equations into a system which can then be solved by a contractive fixed point argument. A second approach starts with the analysis of a special class of solutions, called "multi-peakons". These depend on a finite number of parameters and can be constructed by solving a hamiltonian system of ODEs. One then introduces a suitable distance function, defined in terms of a problem of optimal transportation. Taking the closure of the dense set of multi-peakons solutions w.r.t. this metric, by continuous extension one obtains a semigroup of solution defined on the entire space H^1(R). (These works are in collaboration with A.Constantin and M.Fonte)

2:00 pm Tuesday, March 8, 2005

Job Candidate Seminar: Rigidity for substitution tiling spaces

by Charles Holton (Department of Mathematics, University of Texas) in Skiles 255

We consider tiling spaces generated by geometric substitutions. Examples to be described include the Penrose and pinwheel tilings, as well as the dual tilings constructed from symbolic substitutions of unimodular Pisot type. With Radin and Sadun, we proved an analog, valid for most substitution tiling spaces, of the Curtis-Lyndon-Hedlund theorem in symbolic dynamics. Under some additional conditions this implies a kind of rigidity for these spaces. We'll outline a proof of this and some related results.

9:00 am Wednesday, March 9, 2005

Geometry and Topology: Projectively invariant star products

by Dat Fox (Georgia Tech)

3:00 pm Wednesday, March 9, 2005

Research Horizons: Some problems in Numerical Dynamics

by Luca Dieci (Georgia Tech School of Math) in Skiles 255

The speaker will present some problems on which he has been working in the area of numerical dynamical systems. No background is assumed.

2:00 pm Thursday, March 10, 2005

Job Candidate Seminar: A Geometric View of the Long Time Dynamics of Superlinear

by Nils Ackermann (Universitaet Giessen, Germany) in Skiles 255

We are concerned with a class of semilinear parabolic equations on a bounded domain with (possibly indefinite) superlinear but subcritical nonlinearity. This type of equation admits blow-up in finite time for a large class of initial values. Assuming that 0 is an equilibrium we concentrate on the long time dynamics in the closure of the set of attraction of 0. Using the comparison principle and a detailed analysis of local and global (super-)stable manifolds at 0 we describe geometrical properties of the set of attraction. Combining this information with techniques originating in the calculus of variations we prove existence of equilibria and connecting orbits, and we also obtain information on nodal properties of equilibria.

3:00 pm Thursday, March 10, 2005

Stochastic Seminar: On the rate of convergence in some limit theorems for the spectra of random matrices

by A. Tikhomirov (St. Petersburgh and University of Minnesota) in Skiles 269

We discuss the rate of convergence of the expected and empirical spectral distribution function of random matrices with growing dimension in different probabilistic aspects. We consider the optimal bounds for the Kolmogorov distance between the expected distribution function of random matrices from the Gaussian Unitary Ensemble (Laguerre Unitary Ensemble) and semi-circle law distribution function (Marchenko-Pastur distribution function). We discuss the rate of convergence in probability and the rate of convergence almost surely of empirical spectral distribution functions for both Wigner matrices and sample covariance matrices as well.

3:00 pm Friday, March 11, 2005

Geometry and Topology: Symmetry of Canonical Heights of Periodic Points

by Tom Tucker (University of Rochester) in Skiles 269

Let f be a nonconstant rational map and let h_f be the canonical height associated to f. Let g be the map on the projective line that sends x to x^2; then the canonical height function h_g associated to g is the standard Weil height. We show that the following "symmetry relation" holds: the average of h_f on the periodic points of g is equal to the average of h_g on the periodic points of f. We also conjecture that the lim sup of |h_f(x) - h_g(x)| over all algebraic points x on the projective line is bounded by twice this average. The symmetry relation and the conjecture are both based on properties of the Arakelov-Zhang adelic pairing on canonically metrized line bundles associated to f and g. This represents joint work with Lucien Szpiro.

4:00 pm Friday, March 11, 2005

Combinatorics: On the Product of Two Sets in a Group

by Matt DeVos (Princeton) in Skiles 255

The Cauchy-Davenport theorem is the starting point for a large branch of combinatorial number theory. This theorem gives a natural lower bound on the size of the sumset A+B whenever A,B are subsets of the group of integers modulo a prime. Martin Kneser generalized this result to all abelian groups, and recently we have obtained a further generalization to arbitrary groups. The goal of this talk is to discuss connections between this problem and other branches of mathematics and to give a sketch of the main result.

1:00 pm Monday, March 14, 2005

Research Horizons: An elementary introduction to the Monge-Kantorovich mass transfer problem and its applications

by Giovanni Pisante (Georgia Tech School of Math) in Skiles 255

The Monge-Kantorovic mass transportation theory is an old subject born in 1781, in a geometrical framework, with a paper by G. Monge and then rediscovered by Kantorovich in the context of economics in the 50's. It concernes, roughly, how to transport a mass from one location to another, in such a way to keep the transportation cost to a minimum. After some years, in the late 80's, this basic mathematical problem and its connection with Monge-Ampere type equations started to be applied in various areas of sciences: in metereology, economics, optic. oceanography, PDEs, functional analysis, etc.. The aims of this talk are essentially two. The first one is to give a brief introduction on this topic, explaining what are the basic tools used in this theory, working on some simple examples in this theory. The second one is to give a flavour of its impact on the various areas of mathematics, looking in particular at some applications to PDEs.

3:00 pm Monday, March 14, 2005

Geometry/Topology Seminar: Symplectic Floer theories and the Jones polynomial

by Ciprian Manolescu [mail] (Princeton University/Clay) in Skiles 269

We will explain how the Jones polynomial of links appears in symplectic geometry. Seidel and Smith have used Floer homology for two Lagrangians in an affine variety Y to define an invariant of links. We describe a set of generators for the Seidel-Smith chain complex which can also be used to compute the Jones polynomial, or as a set of generators for the Heegaard Floer chain complex of the double branched cover.

4:30 pm Monday, March 14, 2005

CDSNS Colloquium: On mathematical theory of ratio-dependent ecological models

by Sze-Bi Hsu (Texas A & M University) in Skiles 269

Ratio-dependent ecological models can explain many phenomena that the classical prey-dependent models can't. For instance, the simultaneous extinction of both predators and prey, the outcomes dependence on initial populations. In this talk we shall discuss the mathematical analysis of three ratio-dependence ecological models: predator-prey models, one prey-two predators competition models and food chain models. We shall compare the mathematical results with those of classical prey-dependent models.

2:00 pm Tuesday, March 15, 2005

Applied & Computational Math Seminar: What can we learn from the topology of inaccurate biological networks?

by Debra S. Goldberg (Harvard Medical School) in Skiles 269

Many of the high-throughput experimental methods used to determine genome-wide data on genes or gene products (proteins) are both expensive and error-prone. Much of this data is naturally suited to description by a graph (called a network in many disciplines), with nodes representing genes or proteins. Despite noisy data, biologically relevant patterns can be discerned in the global network topology. In the first part of my talk, I will present methods that exploit the network topology to improve confidence assessment of individual protein interactions and predict unobserved interactions, and ultimately, to predict protein function. Understanding how proteins function is critical for finding new drug targets, and is essential if we hope to gain a systems-level understanding of any organism. In the second part of my talk, I will show that many networks in diverse fields have a more complex structure than previously recognized. This is joint work with Fritz Roth.

4:30 pm Tuesday, March 15, 2005

PDE Seminar: Nonlinear Dynamics of Traffic Jams

by Prof. Tong LI (Iowa University) in Skiles 255

A class of traffic flow models that capture the nonlinear dynamics of traffic jams are proposed. The self-organized oscillatory behavior and chaotic transitions in traffic systems are identified and formulated. The results can explain the appearance of a phantom traffic jams observed in real traffic flow. There is a qualitative agreement when the analytical results are compared with the empirical findings for freeway traffic and with previous numerical simulations.

12:05 pm Wednesday, March 16, 2005

Mathematical Physics: Quantum scattering on graphs and a solution to the Traveling Salesman

by Prof. Robert Schrader, [mail] (Department of Physics, Free University Berlin, Germany) in Skiles 269

On any metric graph all Laplace operators may be described in terms of local boundary conditions at the vertices. Viewed as a Schroedinger operator each such Laplace operator leads to a scattering theory in case the graph has infinite ends. The resulting on-shell S-matrix is always unitary and has both a matrix solution and a representation as a path space sum with contributions involving the total length of the path and the S-matrix associated to the individual vertices traversed. These representations allow for a new way of solving the Traveling Salesman Problem. In addition, for generic lengths and generic boundary conditions the inverse scattering problem may be solved uniquely up to gauge transformations in the interior. These results were obtained in collaboration with V. Kostrykin (Aachen).

4:00 pm Wednesday, March 16, 2005

Analysis Seminar: Redundancy and Localized Frames

by Chris Heil (School of Mathematics, Georgia Tech) in Skiles 269

Frames are particular types of sequences in a Hilbert space that have useful basis-like properties even though they may be redundant, or overcomplete. Such redundant systems offer advantages in applications, such as extra design flexibility or increased stability against noise or data loss. However, while redundancy has a clear qualitative meaning (there are "extra elements"), quantifying redundancy is a difficult problem. Seemingly trivial questions are completely open problems. For example, a slight generalization of the question "when is a frame a union of N orthonormal bases" is equivalent to the still unsolved 1959 Kadison--Singer conjecture. We will present this and other conjectures, and discuss recent progress in developing a quantified description of redundancy for the special class of "localized frames." As a special case, localized frames include Gabor frames, whose elements consist of time-frequency shifts of a single function, analogous to the notes of a musical score.

11:00 am Thursday, March 17, 2005

Nonlinear Science: CANCELLED

by Henry Greenside (Physics, Duke) in Howey N110

3:00 pm Thursday, March 17, 2005

Stochastic Seminar: Approximation to the Mean Curve in the LCS Problem

by Raphael Hauser (Oxford University, Computing Laboratory) in Skiles 269

The longest common subsquence (LCS) problem concerns a question from discrete probability that appears naturally in bioinformatics, speech recognition and other areas where hidden Markov models play a role: given two random sequences of length n, let L_n be the expected length of the longest subsequence that appears in both sequences. When n tends to infinity, (L_n)/n converges to the so-called Chvatal-Sankoff constant. This setup also has generalisations in which the sequences are not of equal length. The Chvatal-Sankoff constant affects certain phase-transition phenomena that depend on parameter choices in sequence alignment algorithms. The precise value of the Chvatal-Sankoff constant is not known, but there exist methods to generate upper and lower bounds. So far, these bounds yielded less than one correct digit of this constant even in the most simple case of random binary sequences obtained by flipping a perfect coin. In this work we analyse and apply a large deviation and Montecarlo simulation based method for the computation of improved upper bounds on the Chvatal-Sankoff constant for i.i.d. random sequences over a finite alphabet. Our theoretical results show that this method converges to the exact value when a control parameter converges to infinity. We also give upper bounds on the complexity for numerically computing the Chvatal-Sankoff constant to any given precision via the new method. Our numerical experiments confirm the theory and allow us to give new upper bounds that are correct to two digits. Joint work with Clement Durringer (Toulouse), Servet Martinez (Santiago de Chile) and Heinrich Matzinger (Georgia Tech)

9:30 am Friday, March 18, 2005

Graph Theory: Pentagon Coloring

by Matt DeVos in Skiles 255

Graph homomorphisms provide a natural and interesting generalization of graph coloring. In this talk, we will discuss some progress on two open problems concerning homomorphisms to a pentagon. First is a conjecture of Nesetril that every cubic graph of sufficiently high girth maps to a pentagon. For this problem we will present a computer based proof (joint with Robert Samal) that every cubic graph of girth 17 has a type of weak pentagon coloring. The second problem is a conjecture that every planar graph with no odd cycles of length <8 maps to a pentagon. Here we will sketch some recent work with Adam Deckelbaum which verifies the conjecture for graphs with no odd cycle of length <10.

3:00 pm Friday, March 18, 2005

Geometry and Topology: Projectively invariant star products

by Dan Fox [mail] (Georgia Tech) in Skiles 269

I will show that a (curved) projective structure on a smooth manifold determines on the Poisson algebra of smooth functions on the cotangent bundle, polynomial in the fibers, a natural star product which specializes in the projectively flat case to the one constructed previously by P. Lecomte and V. Ovsienko. A basic ingredient of the proof is the construction of projectively invariant multilinear differential operators on bundles of weighted symmetric $k$-vectors. The construction works except for a discrete set of excluded weights and generalizes the Rankin-Cohen brackets of modular forms.

4:00 pm Friday, March 18, 2005

Combinatorics: Sum-free Sets: Combinatorics, Number Theory, Fourier Analysis

by Seva Lev (University of Haifa, Israel) in Skiles 255

The set A of elements of an abelian group is called sum-free if it is "utmostly open under addition": that is, if the sum of two elements of A never belongs to A. A well-known phenomenon is that dense sum-free sets posses a rigid structure, usually being contained in a non-zero coset of a subgroup of small index or in a union of such cosets. We will review the present state of the art, discuss some new results, and outline the proof of our recent theorem establishing the structure of large sum-free subsets of the cyclic groups of prime order.

4:30 pm Monday, March 28, 2005

CDSNS Colloquium: Hyperbolic limit of relaxation models

by Stefano Bianchini (SISSA, Italy) in Skiles 269

We assume that the initial data (u_0,\epsilon u_{0,t}) are sufficiently smooth and close to (\bar u,0) in L^\infty, and have small total variation. Then we prove that there exists a solution (u^\epsilon(t),\epsilon u^\epsilon_t(t)) with uniformly small total variation for all t \geq 0, and this solution depends Lipschitz continuously in the L^1 norm w.r.t. time and the initial data. Letting \epsilon \to 0, the solution u^\epsilon converges to a unique limit, providing a relaxation limit solution to the quasilinear non conservative system u_t + A(u) u_x = 0. These limit solutions generate a Lipschitz semigroup \mathcal{S} on a domain \mathcal{D} containing the functions with small total variation and close to \bar u. This is precisely the Riemann semigroup determined by the unique Riemann solver.

2:00pm Tuesday, March 29, 2005

PDE Seminar: Contact Discontinuity for Gas Motions

by Prof. Feimin Huang (Chinese Academy of Sciences and Courant Institute) in Skiles 108B

The contact discontinuity is one of the basic wave patterns in gas motions. The stability of contact discontinuities with general perturbations is a long standing open problem. One of the reasons is that contact discontinuities are linearly degenerate waves in the nonlinear settings, like the Navier-Stokes equations and the Boltzmann equation. The nonlinear diffusion waves generated by the perturbations in sound-wave families couple and interact with the contact discontinuity and then cause analytic difficulties. Another reason is that in contrast to the basic nonlinear waves, shock waves and rarefaction waves, for which the corresponding characteristic speeds are strictly monotone, the characteristic speed is constant across a contact discontinuity, and the derivative of contact wave decays slower than the one for rarefaction wave. Here, we succeed in obtaining the time asymptotic stability of a damped contact wave pattern with an convergence rate for the Navier-Stokes equations and the Boltzmann equation in a uniform way. One of the key observations is that even though the energy estimate involving the lower order may grow at the rate (1+t)^{\frac 12}, it can be compensated by the decay in the energy estimate for derivatives which is of the order of (1+t)^{-\frac 12}. Thus, these reciprocal order of decay rates for the time evolution of the perturbation are essential to close the priori estimate containing the uniform bounds of the L^\infty norm on the lower order estimate and then it gives the decay of the solution to the contact wave pattern.

4:30 pm Tuesday, March 29, 2005

Job Candidate Seminar: Transport equations and oscillations for systems of conservation laws

by Athanasios Tzavaras (Department of Mathematics, University of Wisconsin-Madison) in Skiles 255

This talk will review how the transport (or semi-transport) equations obtained via the kinetic formulation for systems of two conservation laws can be used to provide information on cancellations and coupling of oscillations. For the equations on one-dimensional elastodynamics singular entropies and precise estimations on the Riemann function can be used to analyze compactness in the energy-norm setting.

3:00 pm Wednesday, March 30, 2005

Research Horizons: Realizable Dynamics in some Infinite Dimensional Systems

by Jack Hale (Georgia Tech School of Math) in Skiles 255

Consider an infinite dimensional dynamical systems generated by an evolutionary equation; for example, a quasilinear parabolic PDE of a delay differential equation. Also, suppose that there is a finite dimensional invariant manifold for the system. We discuss the realizable dynamics on this manifold as a function of the vector field in the evolutionary equation with special consideration being given to the number of equations in the delay differential case and the spatial dependence in the PDE case.

3:15 pm Wednesday, March 30, 2005

ACO Colloquium: An Approximate Koenig's Theorem for Edge-Coloring Weighted Bipartite Graphs

by Michel Goemans (MIT) in ISyE 228

Clos networks, introduced by Charles Clos in 1953, were the first non-blocking interconnection networks with a subquadratic number of links. In the last five decades, non-blocking interconnection networks have been much studied, and the setting in which different connections cannot share links is reasonably well understood. The situation in the multi-rate case in which different connections may use different amounts of bandwidth is quite different. The rearrangeability question --- whether having enough capacity at the inputs and outputs is enough to route every set of demands --- is not yet well understood, even for the simple 3-stage Clos networks. In graph theoretic terms, this question can be reformulated as a generalization of edge coloring bipartite graphs in which every edge has a weight in [0,1] and the coloring of the edges must satisfy that the sum of the weights of the edges of any color incident to a vertex v must be at most 1. In this talk, after introducing interconnection networks and their non-blocking behaviors, I will focus on this weighted edge-coloring problem for bipartite graphs. I will derive an approximate K\"onig's theorem and this will provide an upper bound on the number of middle switches needed in rearrangeable 3-stage Clos networks, improving previous bounds of Du et al. Our analysis is interesting in its own and involves a novel decomposition result for bipartite graphs and the introduction of an associated continuous one-dimensional bin packing instance which we can prove allows perfect packing.

4:30 pm Wednesday, March 30, 2005

Stelson Lecture I: The "Second Law" of Probability: Entropy Growth in the Central Limit Theorem

by Keith Ball (University College London) in Skiles 269

The famous second law of thermodynamics states that the entropy of a closed physical system increases with time. The convergence of simple thermodynamic systems toward equilibrium parallels the convergence of sums of independent random variables to the normal distribution: the convergence in the central limit theorem. It has long been believed that there should be an analogue of the second law for the central limit process. The problem was recently solved using a variational principle inspired by high-dimensional geometry. I will begin by recalling the second law and the central limit theorem and then provide a brief introduction to information theory and the Brunn-Minkowski inequality. Finally I will outline how they come together to yield a variational characterization of entropy and how this can be used to establish the second law for the central limit process.

4:30 pm Thursday, March 31, 2005

Stelson Lecture II: There are infinitely many irrational values of zeta at the odd integers

by Keith Ball (University College London) in Skiles 269

The values of Riemann's zeta function, zeta, at the (positive) even integers are rational multiples of integral powers of pi, so that their transcendence was established at the end of the 19th century with the transcendence of pi. The values at odd integers are harder to understand. 25 years ago Apéry proved that zeta(3) is irrational. This talk outlines the proof, found 5 years ago, that infinitely many of the odd-number values of zeta are irrational.

8:30 am Friday, April 1, 2005

QCFDay: Experience Quantitative Finance

in MARC Building

A day of connections & education about the Quantitative and Computational Finance community

http://www.qcf.gatech.edu/academic/qcfday05.html

4:00 pm Friday, April 1, 2005

2005 Evans/Hall Lecture at Emory U.: Angels, Devils and Demons

by Bela Bollobas (Cambridge University and The University of Memphis) in W201, Math & Science Ctr, Emory University

Many important and difficult questions concerning cellular automata, percolation and games can be formulated in rather colourful language: in this talk we shall present several attractive problems of this kind. Although these questions are easy to formulate and sound playful, many have only partial solutions even after years of concerted efforts by a host of mathematicians. This talk will be accessible to a very wide mathematical audience.

Reception: Follows the lecture at 5:00 in the Faculty/Staff Commons, E200, Mathematics & Science Center

Visitor Parking: Available in the B. Jones and Peavine lots

2:00 pm Monday, April 4, 2005

Research Horizons: Flat Surfaces

by Margaret Symington (Georgia Tech) in Skiles 255

The Gauss-Bonnet theorem is one those theorems that anyone on the street should know and love. OK, make that every mathematician. This beautiful theorem gives a relation between the curvature of a surface and its topology. I'll discuss the proof of the Gauss-Bonnet theorem and then give a simple construction showing that any surface with boundary admits a metric whose curvature is everywhere zero. (I don't know to whom the construction is due. Thurston has described it, but the construction was probably known well before him.)

3:00 pm Monday, April 4, 2005

Special Seminar: Excess Risk Bounds in Learning Theory

by Vladimir Koltchinskii (Department of Mathematics and Statistics, The University of New Mexico) in Skiles 255

Recently, many efficient algorithms of empirical risk minimization (ERM) over very large classes of decision functions, such as boosting (minimization over convex hulls) and kernel machines (minimization in reproducing kernel Hilbert spaces (RKHS)), have been suggested. A challenging problem is to develop sharp probabilistic bounds on the difference between the risk of a solution of ERM and the minimal risk, expressed in terms of relevant complexities of classes of decision functions. We will discuss a very general approach to this problem and applications of this approach, in particular, to learning problems in RKHS studied by a number of authors (Cucker and Smale; Blanchard, Bousquet and Massart; Mendelson among others). The results have a number of applications in Statistics and in Learning Theory, especially, to model selection problems for regression and classification.

4:30 pm Monday, April 4, 2005

CDSNS Colloquium (joint with Biology): DNA Minicircles: A Multi-Scale Hamiltonian Bifurcation Problem with Symmetry Breaking

by John Maddocks [mail] (Mathematics, Ecole Polytechnique Federale de Lausanne, Switzerland) in Skiles 269

DNA minicircles are a commonly adopted motif for the experimental examination of the sequence-dependent mechanical properties of DNA. I will describe how various mathematical techniques such as homogenization and Melnikov analysis arise in the computation of minicircle shapes.

2:00 pm Tuesday, April 5, 2005

Applied & Computational Math Seminar: Computing manifolds in dynamical systems

by Michael E. Henderson [mail] (IBM Research) in Skiles 269

I will describe approach to computing manifolds that are found in dynamical systems computations, and present several applications of the algorithm. One dimensional manifolds can be easily represented as points along a curve, with new points added at one of the two endpoints. For surfaces and higher dimensional manifolds this becomes an advancing front problem; a problem which seems to have a simple solution, which is notoriously difficult in the details. The approach I use is to represent the manifold as a set of points, together with balls in the tangent space of the manifold at each point. The projection of these balls onto the manifold gives a set of overlapping neighborhoods which covers part of the manifold. A continuation method for computation can be devised by finding a point near the boundary of this union (actually a point in the tangent space), projecting the point onto the manifold, and adding the point and ball to the union. This produces a well distributed set of points on the manifold, with higher density where the curvature is larger. It reduces to pseudo-arclength continuation for a curve, and so is in some sense the natural extension of that method to higher dimensions.

4:30 pm Tuesday, April 5, 2005

PDE Seminar: On a Multidimensional Model for the Dynamic Combustion of Compresssible Reacting gases

by Konstantina Trivisa (University of Maryland ) in Skiles 255

In this talk a multidimensional model will be introduced for the dynamic combustion of compressible reacting gases formulated by the Navier Stokes equations in Euler coordinates. For the chemical model we consider a one way irreversible chemical reaction governed by the Arrhenius kinetics. The existence of globally defined weak solutions of the Navier-Stokes equations for compressible reacting fluids is established by using weak convergence methods, compactness and interpolation arguments in the spirit of Feireisl and P.L. Lions. In addition, conditions on the initial data will be introduced yielding blow up of smooth solutions to the Navier-Stokes and Euler equations for compressible reacting gases.

12:00 am Wednesday, April 6, 2005

Mathematical Physics Seminar: Leaky graph resonances

by Pavel Exner >[mail] (University of Prague, Czech Republic) in Skiles 269

We discuss a model of "leaky quantum graphs" described formally by the Hamiltonians $-\Delta- \alpha \delta(x -\Gamma)$ with a singular attractive interaction supported by a graph $\Gamma$. A particular attention is paid to resonance effects in such structures: we present several numerical results based on approximation by point interactions, and an exactly solvable case which can be viewed as a caricature description of a wire and a dot.

12:00 pm Wednesday, April 6, 2005

Bioinformatics and Computational Biology Seminar: A New Class of Models for Evolution and Transport in Networks

by Leonid Bunimovich (School of Mathematics, Georgia Tech) in Cherry Emerson 320

Biological systems, and especially, biological networks, demonstrate very complicated and often surprising behavior that sometimes can not be captured by traditional mathematical models. I'll discuss in this talk a new class of models of networks with a rich variety of different behaviors including some entirely new types. This class of models, called Deterministic Walks in Random Environments, was not introduced and investigated for the sake of mathematical exercises. In fact, to the contrary, these models were introduced and independently (numerically) studied in communication theory, chemical kinetics, statisical and solid state physics, computer science, etc, where the corresponding objects moving in networks were called signals, waves, particles, ants, impulses, etc. Basically all the researchers reported some surprising results. Now many of these surprises are resolved but fortunately some new ones appear. I'll describe several simple models and their (completely undestood) possible behaviors. The main goal of this talk is to provide biologists with collection of phenomena (behaviors) that they may expect to observe and with simple models that reproduce such phenomena. There will be no formal Math, just pictures of what is going on will be shown. Listeners are expected to know though what 'probability' means.

3:00 pm Wednesday, April 6, 2005

Special Biology/Math Colloquium: Extracting Parametrizations of Coarse-Grain Sequence-Dependent Models of DNA Mechanics from Molecular Dynamics Simulations

by John Maddocks [mail] (Ecole Polytechnique Federale de Lausanne) in ES&T L1175

I will describe ongoing efforts to effect the passage from all atom Molecular Dynamic (MD) simulations of DNA fragments, which are of necessity still short both in duration of simulation and length of oligomer, to coarser grain sequence-dependent models involving rigid base, rigid base-pair, and continuum descriptions. These coarser grain models in principle allow quantitative, sequence-dependent modelling of experiments on DNA involving several tens to a few hundreds of base pairs, e.g. cyclization rates of minicircles, but detailed parametrizations of the energies in these models must be passed up from a finer resolution description. I will include a discussion of the Ascona B-DNA Consortium or ABC collaboration which involves ten labs that have combined computational resources to assemble a consistent data base of MD simulations of a library of oligomers containing multiple instances of all 136 independent sequence tetramers.

4:00 pm Wednesday, April 6, 2005

Analysis Seminar: Nonsymmetric Jack polynomials and Calogero-Moser models

by Charles Dunkl (Mathematics Department, University of Virginia) in Skiles 269

This is an overview of the technique of differential-difference operators associated with finite reflection groups in the context of complete integrability of certain Hamiltonians. The symmetric and hyperoctahedral groups are the main symmetry groups to be discussed. The wave functions for some Calogero-Moser models on the circle and on the line can be expressed in terms of nonsymmetric Jack polynomials, a family of orthogonal polynomials of several variables.

11:00 am Thursday, April 7, 2005

Nonlinear Science: Quantization of irreversible chaotic maps

by Karol Zyczkowski [mail] (Institute of Physics, Jagiellonian University, Cracow, Poland ) in Howey N110

We analyze a quantum analogue of an irreversible generalization of the model of classical baker map on the torus, in which the images of two parts of the phase space do overlap. This transformation is irreversible and cannot be quantized by means of an unitary Floquet operator. We construct and investigate the corresponding quantum system as a completely positive map defined by a set of measurement (Kraus) operators and acting in the space of density matrices [1]. The quantum dynamics is non-unitary and an initially pure state suffers decoherence, which may be quantified by the von Neumann entropy of the state. We demonstrate that the initial rate of the von Neumann entropy growth depends on the KS-entropy of the classical system, provided the measurement operators have a well-defined classical limit adjusted to the classical dynamics [2]. [1] A. Lozinski, P. Pakonski and K. Zyczkowski, "Irreversible Quantum Baker Map", Phys. Rev. E 66, 065201(2002). [2] R. Alicki, A.Lozinski, P. Pakonski and K.Zyczkowski, "Quantum dynamical entropy and decoherence rate" J. Phys.A 37, 5157(2004).

3:00 pm Thursday, April 7, 2005

Stochastic Seminar: Tempering stable processes

by Jan Rosinski (University of Tennessee) in Skiles 269

Tempered stable processes were introduced in statistical physics to model turbulence. They were also introduced in mathematical finance to model stochastic volatility. A classical tempered stable Levy process combines both the alpha--stable and Gaussian trends. In a short time frame it is close to an alpha--stable process while in a long time frame it approximates a Brownian motion. We consider a more general and robust class of multivariate tempered stable processes and distributions that ranges from light to heavy tail models. Under this extension, a short time behavior of a tempered alpha--stable process is still alpha--stable but a long range one can be Gaussian or another stable. We provide probabilistic representations of tempered stable processes which explain how the tempering happens, giving an insight into the structure of such processes. Our representations can also be used for computer simulation.

4:00 pm Friday, April 8, 2005

Combinatorics: Approximating Infinite Graphs by Finite Graphs

by Russell Lyons (Indiana University at Bloomington) in Skiles 255

Suppose that G is an infinite connected vertex-transitive graph of finite degree, like the nearest neighbor graphs of Euclidean lattices or like a regular tree. Are there finite graphs H that look very much like G? Of course, this depends on what ``look like" means. A large ball in a Euclidean lattice does look a lot like the whole lattice from the viewpoint of most of its vertices, since most of its vertices are far from the boundary of the ball. This is not true in a 3-regular tree, however. On the other hand, 3-regular graphs of large girth do look like a tree. But there are also transitive graphs that don't look like any finite graph. We shall discuss recent work with David Aldous that characterizes which infinite graphs have good finite approximations. In particular, all Cayley graphs do. This fact resolves a question of Weiss and completes the resolutions of conjectures of Kaplansky, Gottschalk, and Connes (for discrete groups), as well as the Determinant Conjecture, which implies conjectures of Schick and Lueck.

11:00 am Monday, April 11, 2005

PhD Defense: Aspects of Mass Transportation in Discrete Transportation Inequalities

by Marcus Sammer (School of Mathematics, Georgia Tech) in Skiles 269

1:00 pm Monday, April 11, 2005

Job Candidate Seminar: Regularity of Classical Hyperbolic Gauge Field Theories

by Jacob Sterbenz (Princeton University) in Skiles 269

I will discuss some current and ongoing work on the local and global (in time) regularity properties of gauge field equations on Minkowski space. These include the Maxwell-Klein-Gordon and Yang-Mills equations, as well as "generic" model equations which display the type of quadratic interactions typical to these field theories. The techniques in this area range from exploiting the global conformal geometry of Minkowski space, to microlocal constructions based on decompositions much finer than the classical Littlewood-Paley theory, as well as the construction of global oscillatory integrals with phase functions taking values in compact Lie groups. This is an exciting are of research which is undergoing rapid development, and I will try to give an indication of some of the very difficult open questions which are still on the periphery of current methods.

3:00 pm Monday, April 11, 2005

Geometry/Topology Seminar: Torsion of the Khovanov homology

by Alexander Shumakovitch (Dartmouth College) in Skiles 269

Given a diagram $D$ of an oriented link $L$ in the 3-sphere, one can assign to it a family of Abelian groups $H_{i,j}(D) using a construction due to Mikhail Khovanov. These groups are defined as homology groups of an appropriate (graded) chain complex $C(D)$, and their isomorphism classes depend on the isotopy class of $L$ only. The graded Euler characteristic of $C(D)$ is a version of the Jones polynomial of$ L$. Although the ranks of the Khovanov homology groups have many remarkable properties, their torsion appear to be even more fascinating. In this talk, we prove several properties of this torsion, discuss methods of its calculation and finally formulate several conjectures about it.

4:30 pm Monday, April 11, 2005

CDSNS Colloquium: Finding Travelling Waves on Lattices: Advanced-Retarded Functional Differential Equations

by Tony Humphries (McGill University) in Skiles 269

Advanced-Retarded Functional Differential Equations arise in a surprisingly wide range of applications, most recently receiving significant attention because travelling wave solutions to lattice differential equations are defined by FDE boundary value problems on an unbounded domain. The presence of advances as well as delays makes the analysis of these problems tricky. Moreover we are interested in the phenomemon of propagation failure, where a standing rather than travelling wave is observed, represented by a singular limit in the FDE. Analytical studies would be greatly aided by the existence of good numerics. To solve numerically the problem must first be approximated on a bounded domain. We discuss this truncation problem, and the definition of suitable boundary functions in some detail. A collocation boundary value problem method is used to compute solutions. Travelling waves in a spatially discrete Nagumo equation illustrate the issues that arise.

2:00 pm Tuesday, April 12, 2005

Applied & Computational Math Seminar: Stress-driven grain boundary diffusion: modelling, analysis and numerical methods

by Jon Wilkening (Courant) in Skiles 269

Microchips often fail when the metallic interconnects between transistors and diodes on the chip degrade due to extremely high current densities. The physics of this process is quite interesting; it is a non-local moving interface problem involving elastic deformation and diffusion. Stress singularities can develop which make boundary conditions difficult to understand and numerical simulation difficult to implement reliably. After describing the model, I will outline our recent proof of well-posedness, which uses techniques from semigroup theory and requires an analysis of a type of Dirichlet to Neumann map involving the equations of elasticity. I will also briefly describe my recent work on computing stable asymptotics for singularities of Agmon-Douglis-Nirenberg elliptic systems near corners and interface junctions, and show how to adjoin these singular functions to the finite element basis to accurately and efficiently resolve stress singularities without mesh refinement.

4:30 pm Tuesday, April 12, 2005

PDE Seminar: Subelliptic convexity and fully nonlinear PDEs on the Heisenberg group

by Guozhen LU (Wayne State University) in Skiles 255

In this talk, we review some results in recent years on convexity in the subelliptic setting, and properties of convex functions, and fully nonlinear subelliptic equations on the Heisenberg group or more general settings.

4:30 pm Tuesday, April 12, 2005

Special CDSNS Colloquium: Generalizations of the Jewett-Krieger theorem

by Tomasz Downarowicz (Wroclaw University of Technology, Poland) in Skiles 269

By an "assignment" we mean a "function" defined on the extreme points of an abstract simplex, whose values are ergodic measure-preserving transformations modulo isomorphism. We elaborate about the following question: What assignments occur in topological (minimal) systems. We consider both the invertible and noninvertible case. To illustrate: does there exist a minimal system with exactly two ergodic measures: one isomorphic to the rotation by alpha, the other to the rotation by beta? Or, does there exist a minimal system with all ergodic measures being Bernoulli and arranged as an interval parametrized increasingly by the entropy?

3:00 pm Wednesday, April 13, 2005

Research Horizons: Can you compute?

by Stavros Garoufalidis (Georgia Tech) in Skiles 255

Knots, like prime numbers, are intuitive objects, and there are several (in fact too many) algebraic invariants to tell them apart. We will focus on 2 notoriously difficult invariants to compute: (a) a 2-variable polynomial of a knot (the A-polynomial) that comes straight from algebraic geometry of plane curves, currently unknown for most of the 9 crossing knots. (b) the recursion polynomial for the colored Jones function, that comes out of quantum topology, currently unknown for most of the 8 crossing knots. If you can compute either one of them, you'll understand quantum topology, hyperbolic geometry, alg. geometry (of plane curves), the missing Galois theory (that you always wanted to learn, but found no reason to know) and you'll also get a good thesis!

4:00 pm Wednesday, April 13, 2005

Analysis Seminar: Density of irregular wavelet systems

by Gitta Kutyniok (Justus-Liebig-Universit�t Giessen, Mathematisches Institut) in Skiles 269

Density conditions have recently turned out to be a useful and elegant tool for studying irregular wavelet systems. In this talk we will discuss necessary and sufficient density conditions on the set of parameters for an irregular wavelet system to form a frame. In particular, we will derive a necessary condition on the relationship between the affine density, the frame bounds, and the admissibility condition. Several implications of this relationship will be studied. Moreover, we will prove that density conditions can also be used to characterize existence of wavelet frames, thus serving in particular as sufficient conditions.

3:00 pm Thursday, April 14, 2005

Stochastic Seminar: Spline estimation of eigenfunctions of compact operators

by Jianhua Huang (Texas A&M University) in Skiles 269

We consider the estimation of eigenfunctions of a compact operator in a Hilbert space. Estimators are constructed using polynomial splines. The statistical properties of the spline estimators are studied. Our theoretical framework allows us to provide a unified treatment of various multivariate data analysis problems, including nonlinear correlation, additive principal component analysis, alternating conditional expectation, and sliced inverse regression. This is a joint work with Zhihua Qiao.

11:00 am Monday, April 18, 2005

Nonlinear Science: The hydrogen atom in crossed fields: Examining the classical phase space

by Stephan Gekle (University of Stuttgart) in Howey W505

We analyze the phase space structure of the crossed-fields hydrogen above the saddle point energy using periodic orbit search and identify approximate torus structures within the regular regions of phase space.

3:00 pm Monday, April 18, 2005

Geometry/Topology Seminar: Periodic points of p-adic rational maps

by Juan Rivera-Letelier (Catholic University of Northern Chile) in Skiles 269

4:30 pm Monday, April 18, 2005

CDSNS Colloquium: How Smooth is Your Wavelet: A Dynamical Systems Approach

by Howie Weiss (Pennsylvania State University) in Skiles 269

The study of wavelet regularity is currently a major topic of investigation having important applications. Except in trivial cases, there are no closed form expressions for compactly supported wavelets, and estimating the regularity is nontrivial. In this talk I will discuss our unified approach to wavelet regularity via thermodynamic formalism. Along the way I will present an algorithm for the wavelet regularity which we prove converges with super-exponential speed. The algorithm uses approximations to the Ruelle-Fredholm determinant of the transfer operator acting on the Bergman space of analytic functions. The first part of the talk will provide a quick introduction to wavelets.

11:00 am Tuesday, April 19, 2005

Graph theory: Menger's theorem for infinite graphs

by Eli Berger [mail] (Institute for advanced study) in Skiles 255

We prove an old conjecture of Erdos, saying that Menger's theorem is valid also for infinite graphs, in the following strong form: given sets A and B of vertices in a graph (possibly directed, possibly infinite), there exists a family P of disjoint A-B paths, and an A-B separating set S, such that S consists of a choice of precisely one vertex from each path in P. The talk will describe the history of the problem and the main ideas of our proof. This is joint work with Ron Aharoni.

4:30 pm Tuesday, April 19, 2005

PDE Seminar: Blow up of BV- norms for Non-smooth Measure preserving Transport

by Tao Luo (Georgetown University) in Skiles 255

In this talk, I will first review some results on the transport equations with non-smooth coefficients of Diperna-Lions, Colombini-Lerner, Ambriosio, Depaul and Columbini-Luo-Rauch. Then I will sketch a proof of the Blow up of BV-norms when the coefficients are not Lipshitzean. This is a joint work with F. Columbini and J. Rauch.

3:00 pm Wednesday, April 20, 2005

Research Horizons: Long-time existence of nonlinear wave equations via Keel-Smith-Sogge estimates.

by Jason Metcalfe (Georgia Tech) in Skiles 255

A key estimate in past studies of boundary value problems for nonlinear wave equations is a weighted mixed-norm estimate that is due to M. Keel, H. Smith, and C. Sogge. In this talk, we will outline the original proofs of the KSS estimate as well as give the proof of long-time existence for a model nonlinear wave equation. A new proof of the KSS estimate has recently (within the last 6 months) been devised that is suitable for perturbed wave equations and curved backgrounds. We will discuss this progress as well as a few of the many problems that this more flexible approach seems to make accessible.

4:00 pm Wednesday, April 20, 2005

Analysis Seminar: Extensions of Positive Definite Functions on Groups

by Mihaly Bakonyi (Department of Mathematics, Georgia State University) in Skiles 269

In the first part of the talk I will present an extension of Krein's theorem, stating that every positive definite operator-valued function on a symmetric interval of an ordered Abelian group can be extended to a positive definite function on the entire group. In the second part, I will show that every positive definite operator-valued function on words of length <= m of the free group with n generators can be extended to a positive definite function on the whole group. Several related results will also be presented, including factorization of positive polynomials in noncommutative variables.

11:00 am Thursday, April 21, 2005

CNS Seminar: Computer-Generated Animation of Fluids: An Applied Math Perspective

by Peter Mucha (Mathematics) in Howey N110

Computer graphics researchers and animators have embraced computational fluid dynamics (CFD) over the past decade. After summarizing a number of successfully implemented CFD techniques, we examine two recently addressed problems in detail: approximating melting by variable viscosities, and computing interactions between solid rigid bodies and fluids. In the latter, the Rigid Fluid method is used to efficiently animate the interplay between rigid bodies and viscous incompressible fluids with free surfaces. This technique uses distributed Lagrange multipliers to compute two-way solid-fluid couplings, alternating between treating the solids as if they were fluid and constraining the motions of those regions to obey rigid body motion. The method---straightforward to implement and incurring little computational overhead---generates realistic motion for both the solid objects and the fluid as they interact with one another. This talk will assume no detailed knowledge of fluid mechanics or CFD. Many movies will be shown. This talk represents work done in collaboration with Mark Carlson and Greg Turk.

3:00 pm Thursday, April 21, 2005

Job Candidate Seminar: Statistical Problems in Diffusion Tensor Imaging

by Vladimir Koltchinskii (Department of Mathematics and Statistics, The University of New Mexico) in Skiles 269

We consider the problem of estimation of integral curves of a vector field observed at discrete locations in a bounded region in {\Reals}^d with random noise. We suggest a nonparametric approach to this problem, show the asymptotic normality of the estimated integral curves and develop statistical tests for hypotheses that the true integral curve reaches specified areas in the space. The problems of this nature are of interest in diffusion tensor imaging, a brain imaging technique based on measuring the diffusion tensor at discrete locations in the cerebral white matter, where the diffusion of water molecules is typically anisotropic. The diffusion tensor data is used to track white matter fibers from the initial location following the dominant orientations of the diffusion. Our approach brings more rigorous statistical tools in the analysis of this problem.

9:30 am Friday, April 22, 2005

Graph theory: The Birkhoff-Lewis equations

by Serguei Norine [mail] (Math, GT) in Skiles 255

The Birkhoff-Lewis equations were introduced in 1946 in an attempt to solve the Four Color Conjecture by applying methods of real and complex analysis to chromatic polynomials. The Four Color Conjecture has since become a theorem. However, the methods used in the proof seem not to apply to its strengthenings, which are formulated in terms of chromatic polynomials. This is one of the motivations for our interest in the approach of Birkhoff and Lewis. We will give an introduction to the subject following papers of Birkhoff and Lewis, and Tutte.

4:00 pm Friday, April 22, 2005

Combinatorics: Sequences with an Additive Property

by Kevin O'Bryant (U. C. San Diego) in Skiles 255

B = {0,1,2,3,5,...} has the property that each integer n > 0 has an even number of representations of the form n = s^2 + b, s non-negative and b in B. Moreover, B is the unique subset of the naturals (including 0) with this property. I will prove some properties of B, and replace {s^2 : s at least 0} with other polynomial sequences and finite sequences, and will explain the origin of the problem in partition theory.

11:00 am Monday, April 25, 2005

Nonlinear Science: Double ionization in strong laser fields: The classical origin of correlated electrons

by Bruno Eckardt (Marburg and U Maryland) in Howey W505

Experiments on double ionization of Argon and Neon in strong laser fields show that the electrons emerge with momentum components parallel to the field axis that are very similar and with perpendicular momenta that are opposite in sign but also of similar magnitude. A classical analysis of this process suggests that the origin of this correlation can be found in an extension of Wanniers 1953 analysis for double ionization to allow for the presence of an electric field.

1:00 pm Monday, April 25, 2005

Nonlinear Science: Energy stability and finite amplitude thresholds

by Shreyas Mandre (University of British Columbia) in Howey N110

Some fluid mechanical systems exhibit transition to non-trivial flows even when the base state is linearly stable. In such cases, it is believed that although all infinitesimal perturbations about the base state decay, perturbations of finite size may grow. At the same time, there are energy methods which prove monotonic decay of every perturbation in some parts of the parameter space. I will present an extension of these energy stability methods to determine thresholds on the size of perturbations that may grow. This extension helps us to systematically study the way in which nonlinearities may play a role. I will demonstrate this method using some toy models.

4:30 pm Monday, April 25, 2005

CDSNS Colloquium: Rigorous Numerics using Conley Index Theory

by Stanislaus Maier-Paape (Institute for Mathematics [Aachen, Germany]) in Skiles 269

In this talk it will be shown how the Conley index based method of Mischaikow and Zgliczy\'nski may be used to obtain computer aided existence proofs for equilibria of the Cahn-Hilliard equation on the square. Furthermore, we improve this method to a tool that may proof branches of equilibria. Even uniqueness of equilibria is proveable under more restrictive assumptions. For several branches concrete calculations were carried out and the pattern of the derived equilibria will be presented.

11:05 am Tuesday, April 26, 2005

Nonlinear Science: Dynamical-systems models of coherent structures in wall-bounded turbulence

by John F. Gibson (Worcester Polytechnic Institute) in Howey N110

In near-wall regions, turbulent flow is dominated by a few energetic structures. Aubry, Holmes, Lumley, and Stone used proper orthogonal decomposition and Galerkin projection to develop low-dimensional dynamical systems models of coherent structures in the turbulent boundary layer. We examine Aubry et al.'s derivation in order to understand the models' lack of predictive accuracy (specifically, the treatment of boundary conditions, mean flow, and unresolved modes) and reformulate the problem for the more tractable case of plane Couette flow. Here we find that model accuracy is limited by the slow convergence of Galerkin projection and eddy-viscosity modeling. Yet numerical simulations show that the flow's evolution is largely determined by moderately low-dimensional states, suggesting that accurate dynamical systems models are possible, in principle.

2:00 pm Tuesday, April 26, 2005

Applied & Computational Math Seminar: A network analysis of committees in the United States House of Representatives

by Mason Porter (Mathematics) in Skiles 269

Network theory provides a powerful tool for the representation and analysis of complex systems of interacting agents. Here we investigate the United States House of Representatives network of committees and subcommittees, with committees connected according to ``interlocks'' or common membership. Analysis of this network reveals clearly the strong links between different committees, as well as the intrinsic hierarchical structure within the House as a whole. We show that network theory, combined with the analysis of roll call votes using singular value decomposition, successfully uncovers political and organizational correlations between committees in the House without the need to incorporate other political information.

4:30 pm Tuesday, April 26, 2005

PDE Seminar: On the Alexandrov type inequalities for reflector problem

by Qingbo Huang (Wright State University) in Skiles 255

The Alexandrov inequality and the interior gradient estimate are important in the study of the Monge-Ampere equation. However, it turns out that establishing the inequalities of these types in the setting of the reflector problem is much more difficult than that for the Monge-Ampere equation. In this talk, we will discuss some recent joint work with Caffarelli and Gutierrez on this problem.

12:00 pm Wednesday, April 27, 2005

PhD Defense: Bifurcations, Normal Forms and their Applications

by Jian Chen (School of Mathematics, Georgia Tech) in Skiles 255

The thesis consists of two parts, which are only loosely connected. The first part is a study of an ecological model with one herbivore and N plants. The system has a new type of functional response due to the speculation that the plants compete each other and have different levels of toxin which inhibit the herbivore to eat up to a certain amount. We first derive the model mathematically and then investigate, both analytically and numerically, the possible dynamics from this model, including the bifurcations and chaos. We also discuss the conditions on which all the species can coexist. The second part is a study on the normal form theory. In particular, we study the relations between the normal forms and the first integrals in analytic vector fields. We are able to generalize one of the Poincare?s classical results on the nonexistence of first integrals in an autonomous system. To be precise, we find a formula which can determine the maximum number of the first intergrals in an analytic quasi-periodic vector field. Then in the space of analytic autonomous systems in C^&ob;2n&cb; with exactly n resonances and n functionally independent first integrals, we obtain some results related to the convergence and generic divergence of the normalizations. Lastly we have a new proof for the necessary and sufficient conditions of a planar Hamiltonian system having an isochronous center.

3:00 pm Wednesday, April 27, 2005

J Ford Commemorative Lecture: How does flow in a pipe become turbulent?

by Bruno Eckhardt (Philipps-U. Marburg) in Howey Lecture Hall 1

The routes to turbulence in many flows comprise series of transitions that introduce more and more temporal and spatial variations until the spatially and temporally disordered state we call turbulent is obtained. Flow in a pipe does not fit into this scheme, as theory and experiment do not show any sharp transitions. Building on experience in dynamical systems we have proposed a scenario that involves certain types of waves and a strange saddle. This model is in good agreement with experiments and numerical simulations.

3:00 pm Wednesday, April 27, 2005

Research Horizons: Numerical Simulation of Stochastic Differential Equations

by Christel Hohenegger (Georgia Tech) in Skiles 255

Since the work of Einstein and Langevin, Brownian motion has been used in a variety of physical and biological systems to model diffusion or molecular fluctuations. This has led to the study of stochastic differential equations. In this talk we will give a brief introduction to Ito calculus, the analytical tool necessary to develop numerical schemes for stochastic equations. Emphasis will be put on the intuition rather than on the very technical proofs. Through examples of various stochastic differential equations, we will show why most of the simplest numerical schemes for classical (i.e deterministic) differential equations are not good enough for stochastic systems. We will then present the ideas on how to develop more accurate and convergent schemes for stochastic differential equations. No knowledge of stochastic calculus will be assumed.

4:00 pm Wednesday, April 27, 2005

Analysis Seminar: Some applications of Green's functions to number theory

by Matt Baker (School of Mathematics, Georgia Tech) in Skiles 269

We will begin by giving a simple example to illustrate Arakelov's profound insight that Green's functions on a Riemann surface are complex analytic analogues of intersection numbers on an arithmetic surface. Then we'll discuss some analytic methods for bounding Green's functions from below "on average", together with some applications to the dynamics of rational functions and to the arithmetic of torsion points on elliptic curves.

4:30 pm Thursday, April 28, 2005

Colloquium: Rotating Fluids with Self Gravitation in Bounded Domains

by Joel Smoller [mail] (University of Michigan) in Skiles 269

We study the steady solutions of Euler-Poisson equations in bounded domains with prescribed angular velocity. This models a rotating Newtonian star consisting of a compressible perfect fluid with given equation of state $P=e^S\rho^{\gamma}$. When the domain is a ball and the angular velocity is constant, we obtain both existence and non-existence theorems, depending on the adiabatic gas constant $\gamma$. In addition we obtain some interesting properties of the solutions; e.g., monotonicity of the radius of the star with both angular velocity and central density. We also prove that the radius of a rotating spherically symmetric star, with given constant angular velocity and constant entropy, is uniformly bounded independent of the central density . This is physically striking and in sharp contrast to the case of the nonrotating star. For general domains and variable angular velocities, both an existence result for the isentropic equations of state and non-existence result for the non-isentropic equation of state are also obtained.

4:00 pm Friday, April 29, 2005

ACO Colloquium: Cake-cutting using a Polytopal Sperner Lemma

by Francis Su (Harvey Mudd College) in Skiles 255

Sperner's lemma is a combinatorial statement about labeled triangulations of simplices which was used in the 1960's and 1970's as the basis of constructive fixed point algorithms. In this talk we will present a generalization of Sperner's lemma to polytopes. We also give two proofs of this theorem, one non-constructive and one constructive. We then discuss several new applications of Sperner's lemma and its relatives to the problem of "fair division": how to divide an object fairly among several parties. Applications include envy-free cake-cutting, rent-division, and consensus-halving problems.

3:00 pm Monday, May 2, 2005

Special CDSNS Colloquium: Bifurcation of an Asset Pricing Model

by Duo Wang (Beijing University (visiting Mathematics, Northwestern University)) in Skiles 269

4:00 pm Monday, May 2, 2005

ACO Colloquium: Information Theory and Probability Estimation

by Alon Orlitsky (ECE and CSE, Georgia Tech; UCSD) in Skiles 255

Standard information-theoretic results show that data over small, typically binary, alphabets can be compressed to Shannon's entropy limit. Yet most practical sources, such as text, audio, or video, have essentially infinite support. Compressing such sources requires estimating probabilities of unlikely, even unseen, events, a problem considered by Laplace. Of existing estimators, an ingenious if cryptic one derived by Good and Turing while deciphering the Enigma code works best yet not optimally. Hardy and Ramanujan's celebrated results on the number of integer partitions yield an asymptotically optimal estimator that compresses arbitrary-alphabet data patterns to their entropy. The same approach generalizes Fisher's seminal work estimating the number of butterfly species and its extension authenticating a poem purportedly written by The Bard. The talk covers these topics and is self contained. Joint work with Prasad Santhanam, Krishna Viswanathan, and Junan Zhang.

4:30 pm Monday, May 2, 2005

CDSNS Colloquium: Quasiperiodic dynamics in Bose-Einstein condensates in periodic lattices and superlattices.

by Martijn van Noort (Mathematics, Imperial College) in Skiles 269

This talk is about quasiperiodic dynamics in a model for Bose-Einstein condensates in periodic lattices and superlattices, using KAM theory. We look for (standing) waves, which can be modelled by a parametrically forced Duffing equation that describes the spatial dynamics of the condensate. For shallow-well, intermediate-well, and deep-well potentials, we find invariant tori and Aubry-Mather sets, which shows that one obtains mostly quasiperiodic dynamics for condensate wave functions of sufficiently large amplitude, where the minimal amplitude depends on the experimentally adjustable BEC parameters. Our approach is applicable to periodic superlattices with an arbitrary number of rationally dependent wave numbers. This is joint work with Mason Porter, Yingfei Yi and Shui-Nee Chow.

2:00 pm Tuesday, May 3, 2005

Stochastic Seminar: Some Applications of Entropy Tensorisation

by Andreas Maurer (Munich, Germany) in Skiles 269

The talk is concerned with consequences of the subadditive property of entropy. As a first application the well known Hoeffding-Azuma-McDiarmid 'Bounded Difference Inequality' is derived. Another consequence of tensorisation is a general concentration theorem, which can be compared to applications of Talagrand's convex distance inequality. The theorem is used to give an improved upper tail bound for optimal travelling salesmen tours.

3:00 pm Wednesday, May 4, 2005

Research Horizons: An entropy dissipation-entropy estimate for a thin film type equation

by Suleyman Ulusoy (Georgia Tech) in Skiles 255

Thin film equation is a hot topic in which many people are working.There are many open problems and many resent results.I will first give some background on the problem and akso the good references then I will present our joint work with Prof.Carlen which will be published in the June issue of Communications in Pure and Applied Analysis. We prove a lower bound on the rate of relaxation to the equilibrium in the H^1 norm for a thin film equation. We find a two stage relaxation, with power law decay in an initial interval, followed by exponential decay, at an essentially optimal rate, for large times. The waiting time until the exponential decay sets in is explicitly estimated.

4:00 pm Wednesday, May 4, 2005

Analysis Seminar: A d-bar approach to orthogonal polynomials

by Peter Miller (Department of Mathematics, University of Michigan) in Skiles 269

The orthogonality conditions defining orthogonal polynomials on the unit circle can be translated into the conditions of a Riemann-Hilbert problem of finding an appropriate analytic factorization of a certain 2x2 matrix function on the unit circle in which the measure of orthogonality and the degree n of the polynomial in question appear explicitly. This formulation makes possible the asymptotic analysis of orthogonal polynomials in the limit of large degree, in a universal way that does not depend on details of the measure of orthogonality. In the simplest setting when the measure is absolutely continuous with analytic density, an easy analytic deformation of the Riemann-Hilbert problem reduces the calculation to summing a Neumann series for n large enough (more importantly, this gives an asymptotic series in powers of 1/n). For nonanalytic densities the technique must be modified. This talk will present joint work with Ken McLaughlin on a generalization of the asymptotic analysis to nonanalytic weights. Here the approach based on Riemann-Hilbert problems must be replaced by an approach based on d-bar problems for nonanalytic functions. In particular, we obtain new detailed asymptotic results for the orthogonal polynomial zeros trapped inside the unit circle when there are jump discontinuities in derivatives of the weight function.

3:00 pm Thursday, May 5, 2005

Stochastic Oral Comprehensive Exam: An Interesting Characterization of the n-Sphere via Random Hyperplanes: Some Results of Artstein, Friedland, and Milman

by Trevis Litherland (School of Mathematics, Georgia Tech) in Skiles 269

3:00 pm Wednesday, July 13, 2005

Analysis Seminar: From Fractals to Superfractals

by John Hutchinson (Mathematical Sciences Institute, Australian National University) in Skiles 269

Iterated Function Systems [IFSs] and random IFSs are used to generate deterministic and random fractals. Such IFS fractals have had many applications in engineering, economics and in the physical and biological sciences. In addition, IFSs provide a convenient mathematical tool for the study of fractals sets and measures. I will briefly review these ideas and some applications. In recent work with Michael Barnsley and Orjan Stenflo we have developed a more general notion of V-variable fractals, generated by a family of one or more IFSs, and intermediate between deterministic and standard random IFS fractals. The generation process is by means of a fast Monte Carlo Markov Chain [MCMC] algorithm. These new notions appear to open up many modelling and application possibilities. In applications where IFS fractals have already been used, the "controlled variability" of V-variable fractals offer further modelling advantages. I will briefly discuss applications to computer graphics and, on the speculative side, a fractal analogue of how genetic information and mutations can be passed from one "generation" of V-variable fractals to the next. The integer parameter V in "V-variable" determines the number of distinct "shapes" or "forms" (up to rescaling and possibly other transformations) at each level of magnification of the fractals generated. An interesting fact is that the infinite set of V-variable fractals generated by a family of IFSs is a deterministic IFS fractal in its own right, which we call a superfractal. "Points" on the superfractal are V-variable fractals. The MCMC algorithm used to generate V-variable fractals is the standard random iteration algorithm (or Chaos Game) for the corresponding superfractal. Large V allows one to rapidly compute approximations to standard IFS random fractals. On the mathematical side, I will discuss how dimensions can be computed using products of random matrices and ideas related to notions from statistical mechanics. I will discuss the key ideas in an informal geometric manner with many graphics and examples.

2:00 pm Friday, July 22, 2005

Special Analysis/Numerical Analysis Seminar: FEM-BEM coupling for electro-magnetic problems in R^3

by Ernest Stephan (Institute of Applied Mathematics, Hannover University, Germany) in Skiles 255

Electro-magnetic interface problems in 3D (eddy current, time harmonic scattering) are considered in a variational form where the exterior problem is substituted by boundary integral equations on the surface of the scatterer. This coupled formulation is discretized with finite elements (Nedelec edge elements) in the scatterer and with boundary elements (Raviart-Thomas elements) on its surface. Standard Galerkin and mixed formulations are discussed together with a posteriori error estimates and adaptive refinement algorithms. Both h- and p- versions are analyzed, where in the first case higher accuracy is achieved by reducing the mesh size h, and in the second case by enlarging the polynomial degree p. Numerical experiments support our theoretical results.

1:00 pm Thursday, August 4, 2005

REU MiniConference Presentation: Orthogonal and biorthogonal polynomials

by Ioana Soran (School of Mathematics, Georgia Tech) in Skiles 269

The talk will begin with a brief presentation of orthogonal polynomials and some of their properties, as well as some relevant examples. Then it will attempt to explain what biorthogonal polynomials are and present the differences between the two types. Finally, the results of the research project will be presented, accompanied by a hand-out that professor Lubinsky and I wrote.

1:30 pm Thursday, August 4, 2005

REU MiniConference Presentation: Carmichael Numbers in Abelian Extension Fields of Q

by Ander Steele (School of Mathematics) in Skiles 269

A Carmichael number is a composite number n such that a^n\equiv a mod n for all integers a. These numbers present a major problem for Fermat-like primality tests. In this paper, we look to Abelian extensions of Q to distinguish Carmichael numbers from primes. We generalize the concept of a Carmichael number to these extension fields and prove a natural generalization of Korselt's Criterion. We provide an algorithm for identifying composite numbers using a Fermat-like test in appropriate extension fields and show that, given certain conditions on n, that n cannot be Carmichael in all Abelian extensions of Q.

2:30 pm Thursday, August 4, 2005

REU MiniConference Presentation: On-line coloring of interval graphs

by Bill March (School of Mathematics, Georgia Tech) in Skiles 269

3:00 pm Thursday, August 4, 2005

REU MiniConference Presentation: An optimization problem for resistive networks

by Matthew Tanzy (School of Mathematics, Georgia Tech) in Skiles 269

4:00 pm Thursday, August 4, 2005

REU MiniConference Presentation: Basic Digit Sets: Investigations and Applications

by Charles Martin (School of Mathematics, Georgia Tech) in Skiles 269

Call a set of integers D basic for base beta if every integer n has a unique representation n=sum d_i beta^i where each d_i is in D. We proceed first by giving an algorithm to test basic digit sets. From there, we form conjectures of infinite families of basic digit sets, proving one of them. We then propose and analyze a cryptographic scheme based on these sets. Mentioned briefly will be the possibility of generating new basic digit sets from previously determined ones, via polynomials, as well as possible extensions of Benford's law. There are also a few open questions to be mentioned.

4:30 pm Thursday, August 4, 2005

REU MiniConference Presentation: Small roots of polynomials modulo squarefree numbers

by Brian Williams (School of Mathematics, Georgia Tech) in Skiles 269

We examine the distribution of the roots to polynomial congruences modulo squarefree N. We obtain lower bounds for the maximum number of roots contained in various small intervals of N through the use of polynomial constructions and upper bounds through the use of divisibility arguments. Finally we conjecture a bound on the number of roots in the interval N^t, t<1, for all polynomials mod squarefree N.

1:00 pm Friday, August 5, 2005

REU MiniConference Presentation: Determining lower bounds for differences of powers of 2 and 3 using Pade Approximations

by Brian Swanagan (School of Mathematics, Georgia Tech) in Skiles 269

Several methods will be explored for finding lower bounds for differences of powers of 2 and 3. They involve Pade approximations of logarithms, contradictions brought on by Roth's Theorem, alterations of Bennett's Theorem, and manipulation of formulas involving Pade approximations for (1-z)^k. Some of the above approaches, however, are incomplete or ineffective for which the reasons shall be explained in an attempt to spark possible intuitions for overcoming the difficulties involved.

1:30 pm Friday, August 5, 2005

REU MiniConference Presentation: A Local Community Detection Algorithm for Weighted Real-World Networks

by Arthur Friend (School of Mathematics, Georgia Tech) in Skiles 269

I will give a quick introduction to network theory and introduce the problem of community and hierarchy detection in real-world networks. Here, the use of real-world data requires a computational approach, rather than the analytical mathematics which can be used on idealized regular or random networks. I will describe the algorithm we developed, which measures the clustering of collections of nodes to detect communities on weighted networks. The method's output consists of a one-parameter family of "dendrograms", or trees, revealing the hierarchical community structure of a given network. We can then single out desirable dendrograms based on their "modularity", a particular measure of the quality of a partitioning of a network. As an example, I will apply this method to study the the community structure of committees and subcommittees in the U.S. House of Representatives. I will then compare these results to those obtained through other, well-established algorithms.

2:30 pm Friday, August 5, 2005

REU MiniConference Presentation: Automated reasoning assistant GOEDEL and Peano's arithmetic

by Claudia Huang (School of Mathematics, Georgia Tech) in Skiles 269

3:00 pm Friday, August 5, 2005

REU MiniConference Presentation: A geometric method for measuring the structure of optical fibers

by Alan Diaz (School of Mathematics, Georgia Tech) in Skiles 269

Optical fibers often consist of several different interior layers with different refractive indices. People who work with fiber optics would find it useful to know the exact dimensions of the various layers, which cannot be precisely controlled during the manufacturing process. It is not a simple matter to measure the the shape of the interior regions visually because of the complicated refraction patterns that occur. It is possible to probe the fiber's structure by shining a laser orthogonal to the fiber's axis of symmetry and measuring where it is emitted and at what angle. It is interesting to determine how much information can be deduced about the structure of the fiber via such laser probing. We present our results for some simple cases and also describe some more complicated cases which have so far proved too difficult to solve.

4:00 pm Friday, August 5, 2005

REU MiniConference Presentation: Simulation of plankton ecosystem dynamics in oceans

by Carina Saxton (School of Mathematics, Georgia Tech) in Skiles 269

4:30 pm Friday, August 5, 2005

REU MiniConference Presentation: Numerical Analysis of an Epsilon-Difference Equation

by Nick Cotton (School of Mathematics, Georgia Tech) in Skiles 269

A brief introduction to epsilon-difference equations, details on the numerical analysis, the problems that were run into and what was done to overcome the difficulties of this particular numerical problem. Also some of the potential repercussions of our findings.

3:30 pm Friday, August 19, 2005

Topology Seminar: Integral TQFT and Perturbative expansion

by Gregor Masbaum (Paris VII)

We show that the integral SO(3)-TQFT's studied in previous joint work with Pat Gilmer have a perturbative expansion as the order of the root of unity goes to infinity. We obtain a new 'universal' representation of the Torelli group which gives a TQFT interpretation of Ohtsuki's power series invariant of homology spheres. As a byproduct, we obtain a purely skein-theoretical construction of this invariant.

4:00 pm Thursday, August 25, 2005

Department Tea:

in Math Department Lounge

This is our first math department tea, and is open to all faculty, students, and staff. Coffee, tea, and pasteries will be served. It is a great chance for new faculty and students to meet other members of the department.

2:00 pm Friday, August 26, 2005

Nonlinear Science: Bose-Einstein Condensates in Optical Lattices and Superlattices

by Mason Porter [mail] (Caltech, Department of Physics and Center for the Physics of Information) in Howey N110

Over the past several years, the study of Bose-Einstein condensates (BECs) has become one of the most important areas of atomic and molecular physics. Their study has begun to yield an increased understanding of superfluidity and superconductivity, and their eventual engineering applications also hold great promise. In this talk, I will discuss my recent research on the macroscopic dynamics of coherent structures in BECs loaded into lattice and superlattice potentials, for which I employ methods from dynamical systems and perturbation theory.

4:00 pm Friday, August 26, 2005

Combinatorics: On a Combinatorial Method for Counting Smooth Numbers in Sets of Integers

by Ernie Croot (School of Mathematics, Georgia Tech) in Skiles 255

A number n is said to be y-smooth if it has no prime divisors exceeding y. There are many well-known, unsolved problems in number theory concerning how many y-smooths there are in certain sets of integers. For example, consider the set of all numbers in {1,2,...,N} of the form p-1, where p is prime. How many such numbes are N^(1/2) - smooth? In this talk I will describe a new way to count smooth numbers in such sets of integers by developing a structure called a "local global set."

4:30 pm Tuesday, August 30, 2005

PDE Seminar: Variational Principle for General Diffusion Problems

by Prof. Adrian Tudorascu (Georgia Tech.) in Skiles 255

We employ the Monge-Kantorovich mass transfer theory to study existence of solutions for a large class of parabolic partial differential equations. We deal with inhomogeneous diffusion problems (of Fokker-Planck type) with explicitly time-dependent nonlinearities. This work greatly extends the applicability of known techniques based on constructing weak solutions by approximation with time-interpolants of minimizers arising from Wasserstein-type implicit schemes. It also generalizes previous results by the authors, where proof of convergence in the case of a right hand side in the equation is given by these methods. To help us prove existence of weak solutions we establish an interesting and, to our knowledge, novel maximum principle. This involves comparison with the solution of a corresponding homogeneous, stationary problem.

4:00 pm Wednesday, August 31, 2005

Analysis Seminar: Bernstein Constants and Entire Functions of Exponential Type

by Doron Lubinsky (School of Mathematics, Georgia Tech) in Skiles 269

In papers published in 1913 and 1938, S. Bernstein showed that approximation of the function f(x)=|x|, x in [-1,1], by polynomials, leads to approximation of |x| on the whole real line by entire functions of exponential type. Ever since then, the values of the Bernstein constants, and the identity of the entire functions that appear, have eluded investigators. We discuss progress towards finding the Bernstein constants and the entire functions that appear in them.

12:00 pm Thursday, September 1, 2005

Applied & Computational Mathematics: Cocoon Bifurcations in 3-dimensional reversible vector fields

by H. Kokubu (Kyoto) in To Be Announced

3:00 pm Thursday, September 1, 2005

Stochastics Seminar: Some Generic Concentration Inequalities

by Christian Houdré (School of Mathematics, Georgia Tech) in Skiles 269

11:00 am Friday, September 2, 2005

A&CM Semiar: time reversal of waves and applications to imaging.

by HongKai Zhao (UC Irvine) in Skiles 269

In this talk I will discuss about a recent physical experiments for waves: time reverse a recorded wave signal and send it back. We will study the property of auto focusing on source or targets. In particular the super-resolution phenomenon in random medium will be analyzed. Then I will talk about some applications in imaging.

3:30 pm Friday, September 2, 2005

Algebra-Geometry-Topology Seminar: How to build aspherical manifolds.

by Igor Belegradek (GaTech) in Skiles 269

In this expository talk I shall discuss some methods of constructing aspherical manifolds, i.e. manifolds with contractible universal covers. The examples I shall describe are close cousins of hyperbolic and flat manifolds, yet they are build from cubes in a purely combinatorial fashion.

4:30 pm Tuesday, September 6, 2005

PDE Seminar: On Constrained Optimization in the Wasserstein Metric

by Adrian Tudorascu (Georgia Tech. ) in Skiles 255

We prove the monotonicity of the second-order moments of the discrete approximations to the heat equation arising from the Jordan-Kinderlehrer-Otto (JKO) variational scheme. This issue appears in the study of constrained optimization in the 2-Wasserstein metric performed by Carlen and Gangbo via a duality argument. A direct argument, via Lagrange multipliers, is provided here.

11:00 am Wednesday, September 7, 2005

ACO Colloquium: Algorithmic Self-Assembly: Models and Problems

by Ashish Goel (Stanford University) in Skiles 255

DNA Self-assembly has emerged as an important technique for molecular computation and nano-technology. At these scales, self- assembly is governed by simple (and local) probabilistic rules for growth, making it amenable to algorithmic techniques. We will discuss two important challenges in algorithmic self-assembly: robustness and efficiency. This talk will present recent results, and also attempt to provide a road-map of open problems.

4:30 pm Wednesday, September 7, 2005

Analysis Seminar: Convergence and divergence of multilinear averages in ergodic

by Ciprian Demeter (UCLA) in Skiles 269

We analyze general multilinear averages in measurable dynamical systems and prove their almost everywhere convergence when the coefficient matrix has special rank properties. The positive results are contrasted with some negative ones, when the input functions are in $L^p$ spaces with $p$ sufficiently close to 1. This is joint work with Terence Tao and Christoph Thiele.

3:00 pm Thursday, September 8, 2005

Stochastic Seminar: Mode Estimation for Functional Random Variable and its Application for Curves Classification

by S. Dabo-Niang (University of Lille, France) in Skiles 269

We investigate a non parametric estimate of the mode of a density function of a functional random variable, that is a variable taking values in some infinite dimensional space. The strong consistency of the estimate is shown and an almost sure rate of convergence is given. Special attention is paid to the links between the rates of convergence of our estimate and the probabilities of small balls in the infinite dimensional space. In particular, the key practical question of bandwidth choice is linked with this concentration notion. In addition to these theoretical results, we show how our methodology can be used for unsupervised curves classification through modal curves estimation. A real functional dataset, of chemiometric interest, is treated. As by-products we give some new results about the nonparametric estimation of some infinite dimensional density function.

3:00 pm Friday, September 9, 2005

Combinatorics: Enumeration of perfect matchings of a new family of graphs constructed from the hexagonal lattice graph

by Mihai Ciucu (School of Mathematics, Georgia Tech) in Skiles 255

We consider a family of cylindrical graphs constructed from the hexagonal lattice graph, and show how to count their perfect matchings using linear algebra.

3:30 pm Monday, September 12, 2005

Algebra-Geometry-Topology Seminar: Open books and contact structures

by John Etnyre (GaTech) in Skiles 269

3:30 pm Monday, September 12, 2005

Analysis Seminar: A New Proof of The Embedding Theorem

by Brett Wick (Vanderbilt) in Skiles 255 [Note, unsual time and room!]

In this talk, I will discuss some joint work with S. Petermichl and S. Treil concerning the Carleson Embedding Theorem. I will give a new proof of one part of the Carleson Embedding Theorem. Time permitting I will indicate how one can use this to say something about the Embedding Theorem on the unit ball in C^n.

4:30 pm Monday, September 12, 2005

CDSNS Colloquium: On the hyperbolicity of the real and complex Henon map

by Zin Arai [mail] (Kyoto University) in Skiles 255

In this talk, we propose a rigorous computational method to prove the uniform hyperbolicity of discrete dynamical systems. Applying the method to the real and complex H\'{e}non family, we prove the existence of many regions of hyperbolic parameters in the parameter space of the family.

4:30 pm Tuesday, September 13, 2005

PDE Seminar: Dynamic bifurcation and Stability for Rayleigh-Benard Convection

by Shouhong Wang (Indiana University) in Skiles 255

I shall present in this talk my recent work with Tian Ma on a new dynamic bifurcation and stability theory and its applications to Rayleigh-Benard convection problem. The bifurcation and stability theory is centered at a notion of bifurcation, called attractor bifurcation for nonlinear evolution equations. The main ingredients include the attractor bifurcation theory, together with new strategies for the center manifold reduction procedures. A recipe of the theory toward to applications is presented in this talk. For the application to the Rayleigh-Benard Convection, I shall focus on 1) the stability and bifurcation of solutions, and on 2) the structure of the bifurcated solutions in the physical space.

3:00 pm Thursday, September 15, 2005

Stochastics Seminar: Concentration property of spectral measure of large random matrices with stable entries

by Hua Xu (School of Mathematics, Georgia Tech) in Skiles 269

4:00 pm Thursday, September 15, 2005

Math Department Tea: Coffee and Tea

in Math Department Lounge

This is our second math department tea. Coffee, Tea, soft drinks, and food will be served. All math department staff, faculty and students are invited.

2:00 pm Friday, September 16, 2005

Stochastics Seminar : Estimation of densities of functions of several sample variables

by Evarist Giné (Department of Mathematics, University of Connecticut) in Skiles 269 (NOTE: Earlier time)

Whereas kernel estimators of densities of one random variable X cannot estimate the density of X at rates (in pr) larger than (nh_n)^{-1/2}, where h_n is the window size, usually a negative power of n, it turns out that natural kernel estimators of densities of functions of more than one sample variables, g(X_1, ... , X_m) can estimate the density (of g(X_1, ... ,X_m)), uniformly over R^d, and also in L_p( R^d,\lambda) for any p\ge 1, at the faster rate n^{-1/2}. Likewise for the a.s. rates. It is natural that densities of functions of several sample variables should be easier to estimate because of the regularity introduced by convolution, however, quantifying the effect of this regularity, which is what we do, requires some interesting methods such as moment estimates and exponential inequalities for U-statistics. Examples include the inter-point distance g(x-1,x_2) = \|X_2-X_1\| and convolutions g(x_1, ... ,x_m) = x_1+ ... +x_m. Frees (JASA 1994) was first to make the observation for the estimation of a density at a fixed point, and Schick and Wefelmayer (J. Non-parametric Statist 2004) considered the case of sums. This is joint work with David M. Mason.

3:00 pm Friday, September 16, 2005

Combinatorics: Chess Tableaux

by Timothy Chow (Department of Mathematics, MIT) in Skiles 255

A chess tableau is a standard Young tableau (SYT) in which orthogonally adjacent entries have opposite parity. Remarkably, the number of 3xn chess tableaux is the same as several other quantities: the number of 3x(n-1) nonconsecutive tableaux (SYT in which i and i+1 never appear in the same row), the Charney-Davis statistic of a 3xn shape, and the number of Baxter permutations of n. Yet there is no obvious bijection between any two of these. Our main result is a pleasant but mysterious bijection between chess tableaux and nonconsecutive tableaux with three rows. Bijections with the Charney-Davis statistic remain an open problem. In the last part of the talk we present and explain a recreational application of our results: the composition of two chess problems (one by Noam Elkies) dedicated to Richard Stanley on his 60th birthday. The definition of a Young tableau and (for the last part of the talk) knowledge of how chess pieces move are sufficient background for the talk; other terminology will be explained. This is joint work with Ken Fan and Henrik Eriksson.

2:00 pm Monday, September 19, 2005

Theory Group Seminar: On the Complexity of Numerical Analysis

by Eric Allender (Rutgers University) in College of Computing Bldg 102

In this talk, we consider two quite different approaches toward understanding the complexity of fundamental problems in numerical analysis: (a) The Blum-Shub-Smale model was defined in order to incorporate the real numbers into a computational model, and to provide a framework for classifying the complexity of continuous functions and sets of real numbers. This model gives us the complexity class "polynomial time over the reals", denoted P_R. An important subclass of P_R (denoted P^0_R) arises when machines in this model are provided only with algeraic real numbers.(b) We define a discrete computational problem entirely within the realm of discrete computation (the ``generic task of numerical analysis'') to focus attention on the computational complexity of the job confronting the designers of numerically stable algorithms. We show that both of these approaches hinge on the question of understanding the complexity of the following problem, which we call PosSLP: Given a division-free straight-line program producing an integer N, decide whether N>0. More precisely, P^PosSLP = the Boolean part of P^0_R, and PosSLP is poly-time equivalent to the "generic task of numerical analysis". We show that PosSLP lies in the counting hierarchy. Combining our results with work of Tiwari, we show that the Euclidean Traveling Salesman Problem lies in the counting hierarchy -- the previous best upper bound for this important problem (in terms of classical complexity classes) being PSPACE. This is joint work with Peter Buergisser, Johan Kjeldgaard-Pedersen, and Peter Bro Miltersen.

3:30 pm Monday, September 19, 2005

Topology seminar: Integral TQFT and Perturbative expansion

by Gregor Masbaum (Paris VII) in Skiles 269

We show that the integral SO(3)-TQFT's studied in previous joint work with Pat Gilmer have a perturbative expansion as the order of the root of unity goes to infinity. We obtain a new 'universal' representation of the Torelli group which gives a TQFT interpretation of Ohtsuki's power series invariant of homology spheres. As a byproduct, we obtain a purely skein-theoretical construction of this invariant.

4:30 pm Monday, September 19, 2005

CDSNS Colloquium: Asymptotic Speeds of Spread and Traveling Waves for Monotone Semiflows with Applications

by Xiaoqiang Zhao [mail] (Memorial University of Newfoundland) in Skiles 255

The theory of asymptotic speeds of spread and monotone traveling waves is established for a class of monotone discrete and continuous-time semiflows and is applied to a functional differential equation with diffusion, a time-delayed lattice population model and a reaction-diffusion equation in an infinite cylinder. This talk is mainly based on a recent joint work with Xing Liang.

4:30 pm Tuesday, September 20, 2005

PDE Seminar: Problems in the calculus of variations associated to droplet formation

by Maria da Conceicao Carvalho (Georgia Tech.) in Skiles 255

On a mesoscopic scale, the equilibrium configurations of statistical mechanical systems in contact with a heat bath are found by minimizing a free energy function that is nonconvex if the system has a phase transition. Determining the geometric nature of the minimizers is a problem of considerable mathematical and physical interest. Detailed solutions will be provided for certain well known models in this talk, which is based on joint work with Carlen, Esposito, Lebowitz and Marra.

2:00 pm Wednesday, September 21, 2005

Research Horizons: Orthogonalities, polynomial and other

by Doron Lubinsky (Georgia Tech) in Skiles 269

Orthogonal polynomials appear in many contexts, and their asymptotics are widely used in mathematical physics. The study of these asymptotics is a hot topic now, with some groups using Riemann-Hilbert techniques, others using operator theory, and older groups sticking to Bernstein-Szego methods. We discuss some recent developments. There are other types of "polynomials" where orthogonality plays a role: Muntz polynomials, biorthogonal polynomials, ... . We discuss some recent activity in these areas too.

4:00 pm Wednesday, September 21, 2005

Analysis Seminar: On Possible 2-Variable Reflection Coefficient

by Jeffrey Geronimo (School of Mathematics, Georgia Tech) in Skiles 269

4:00 pm Thursday, September 22, 2005

Math Department Tea:

in Math Department Lounge

Coffee, tea, soft drinks, and food will be served. All math department students, staff, and faculty are welcome to attend.

11:00 am Friday, September 23, 2005

A&CM : Error analysis on image inpainting, and new approaches on video dejittering

by Sung Ha Kang (University of Kentucky) in Skiles 269

In resent years, there have been many developments on computational approaches to image inpainting problems. In this talk, I will consider the error estimation for image inpainting problems by considering harmonic and TV inpainting methods. In addition, as a separate problem I will present couple of new approaches on video dejittering problems.

4:05 pm Friday, September 23, 2005

ACO Colloquium: Additive Approximation for Edge-Deletion Problems

by Benny Sudakov (Mathematics, Princeton University) in Skiles 255

A graph property is monotone if it is closed under removal of vertices and edges. In this talk we consider the following edge-deletion problem; given a monotone property P and a graph G, compute the smallest number of edge deletions that are needed in order to turn G into a graph satisfying P. We denote this quantity by E_P(G). Our first result states that for any monotone graph property P, any \epsilon > 0 and n-vertex input graph G one can approximate E_P(G) up to an additive error of \epsilon n^2. Given the above, a natural question is for which monotone properties can one obtain better additive approximations of E_P(G). Our second main result essentially resolves this problem by giving a precise characterization of the monotone graph properties for which such approximations exist. We will show that for any dense monotone property, that is, a property for which there are graphs on n vertices with \Omega (n^2) edges that satisfy it, it is NP-hard to approximate E_P(G) up to an additive error of n^{2-\delta}, for any fixed positive \delta. The proof requires several new ideas and involves tools from Extremal Graph Theory together with spectral techniques. Interestingly, prior to this work it was not even known that computing E_P(G) precisely for dense monotone properties is NP-hard. We thus answer (in a strong form) a question of Yannakakis raised in 1981. (Joint work with N. Alon and A. Shapira.)

4:30 pm Tuesday, September 27, 2005

PDE Seminar: Nonlinear Instability for the Navier Stokes Equations

by Susan Friedlander (U of Illinois-Chicago) in Skiles 255

It is proved that linear instability implies nonlinear instability for the Navier Stokes equations in L^p, p > 1. The result holds in all spatial dimensions and both finite domains and R^n. The method of proof uses a bootstrap argument.

2:00 pm Wednesday, September 28, 2005

Research Horizons: Two Dimensional Corners

by Bill McClain (Georgia Tech) in Skiles 269

We're finding a bound on the answer to the following question but stated in a finite field setting: How big must a subset of the square integer lattice {1,...,N} x {1,...,N} be in order to guarantee it has a corner? That is, a triple of points (x,y), (x+d,y) and (x,y+d) for some d > 0.

4:00 pm Wednesday, September 28, 2005

Analysis Seminar: Space filling curves and geodesic laminations

by Victor Sirvent (School of Mathematics, Georgia Tech) in Skiles 269

In this talk we shall associate space filling curves to connected fractals, obtained as the fixed point of an iterated function systems (IFS) satisfying certain conditions, mainly the common point property. These curves are Holder continuous and measure preserving. To these space filling curves we associate geodesic laminations satisfying among other properties that points joined by geodesics have the same image in the fractal under the space filling curve. The laminations help us to understand the geometry of the curves. We define an expanding dynamical system on the laminations.

3:00 pm Thursday, September 29, 2005

Stochastic Seminar: Model Selection via Information Criteria for Tree Models and Markov Random Fields

by Zsolt Talata (School of Mathematics, Georgia Tech) in Skiles 269

The concept of context tree, usually defined for finite memory processes, is extended to arbitrary stationary ergodic processes (with finite alphabet). The familiar BIC and MDL principles are shown to provide strongly consistent estimators of the context tree, via optimization of a criterion for hypothetical context trees of finite depth, allowed to grow with the sample size n as o(log n). Algorithms are provided to compute these estimators both off-line and on-line ways. For Markov random fields on Z^d with finite state space, we address the statistical estimation of the basic neighborhood, the smallest region that determines the conditional distribution at a site on the condition that the values at all other sites are given. A modification of the BIC is proved to provide strongly consistent estimation from observing a realization of the field on increasing finite regions, not assuming any prior bound on the size of the latter. Stationarity of the Markov field is not required, and phase transition does not affect the results.

4:00 pm Thursday, September 29, 2005

Math Department Tea:

in Math Dept. Lounge (Skiles 236)

Food and beverages will be served. All math department faculty, staff, and students are welcome to attend.

3:00 pm Friday, September 30, 2005

Center for Signal and Image Processing Seminar Series: Orthogonal polynomials in one and two variables with applications

by Jeff Geronimo (School of Mathematics, Georgia Tech) in GCATT Building, Room 325

Polynomials orthogonal on the unit circle have played a useful role in various problems in engineering such as AR models, stability and spectral factorization, and the development of fast algorithms for the inversion of positive Toeplitz matrices. I will review the one variable theory and then discuss progress in extending these results to two variables. I will also mention some open problems.

3:05 pm Friday, September 30, 2005

Algebra-Geometry-Topology Seminar: The uniform Schanuel conjecture over the reals by Kirby and Zilber.

by Thierry Zell (GaTech) in Skiles 269

Schanuel's conjecture (SC) is a central problem in transcendental number theory, with a very simple statement: it says that if (a_1, ..., a_n) are complex numbers which are linearly independent over Q, the transcendence degree of Q(a_1, ..., a_n, exp(a_1), ..., exp(a_n)) over Q is at least n. This conjecture has surprisingly rich consequences, both in and outside of number theory. My talk aims to present the following result due to Kirby and Zilber: when restricted to the reals, Schanuel's conjecture is equivalent to a uniform version of itself (USC). Their proof is a nice (and short: 2 pages) illustration of the powerful methods derived from the theory of o-minimal structures. For the record, Zilber first proved a similar result holds over the complexes, but the complex case required to assume some still open Diophantine conjectures (about intersections of complex varieties with tori). The proof over the reals does not require any additional assumption. The talk will be completely elementary.

4:05 pm Friday, September 30, 2005

Combinatorics Seminar: The Green-Tao Proof of Arithmetic Progressions in the Primes

by Ernie Croot (School of Mathematics, Georgia Tech) in Skiles 255

In this talk I will give a rough outline of the recent proof due to B. Green and T. Tao that the primes contain arbitrarily long arithmetic progressions. Depending on the amount of time I have, I may also sketch a proof due to B. Green (which was the one of the starting ideas for the Green-Tao proof) that any positive density subset of the primes contains a three-term progression.

4:30 pm Monday, October 3, 2005

CDSNS Colloquium: Hamiltonian-preserving schemes for the Liouville equation with discontinuous Hamiltonians

by Shi Jin [mail] (University of Winsconsin) in Skiles 255

When numerically solving the Liouville equation with a discontinuous potential, one faces the problem of severe time step restriction, and the inconsistency to the constant Hamiltonian which is related to the problem of how the weak solution should be defined for such linear hyperbolic equations with singular coefficients. In this talk, we present a class of Hamiltonian-preserving schemes that are able to overcome these numerical deficiencies. The key idea is to build into the numerical flux the behavior of a classical particle at a potential barrier. We establish the stability theory of these new schemes, and analyze their numerical accuracy. Numerical experiments are carried out to verify the theoretical results. This method can also be applied to the level set methods for the computations of multivalued physical observables in the semiclassical limit of the linear Schrodinger equation with a discontinuous potential. For wave equations with discontinuous local speeds, this leads to numerical schemes consistent with Snell's Law of Refraction.

11:00 am Wednesday, October 5, 2005

Research Horizons: Geometric Means of Positive Operators

by William Green (Georgia Tech) in Skiles 269

We give a quick description of spectal theory for bounded normal operators. We then use this theory to show how to define the geometric and harmonic means of positive operators. When properly interpreted, many results for numbers (e.g., the arithmetic-geometric-harmonic mean inequality) remain true for positive operators.

11:30 am Wednesday, October 5, 2005

OIT Brown Bag Series: Learn How to Maximize IT Services

by Lew Lefton, Justin Filoseta, Diego Remolina (School of Mathematics and IBB, Georgia Tech) in Savant 308

IT staff have a number of responsibilities to juggle, from IT security to basic help desk support, with only so many hours in the day to complete them. Two of these necessary tasks that can eat away at precious time are the setup of new desktops/laptops and the repair of spyware/virus-compromised systems. This workshop will be given by departmental IT support providers from The School of Mathematics and the Institute of Bioengineering & Biosciences. They will explain the system they developed to perform the automated and unattended installation of operating systems, applications, and security patches. In particular, they will provide a look ?under-the-hood? of their procedures and technologies which are used to install fully patched and configured Windows and Linux systems. You will leave this workshop with ideas about how to implement a similar system in your own department.

4:30 pm Wednesday, October 5, 2005

Physics Colloquium: Molecular Motors: Observation and Theory

by Michael E. Fisher (University of Maryland and Institute for Physical Sciences and Technology) in Physics Lecture Room 3

12:00 pm Thursday, October 6, 2005

Applied and Computational Mathematics Seminar: Molecular Seisemology: An Inverse Problem in Nano-Biology

by E. Boczko (Vanderbilt) in Skiles 269

3:00 pm Thursday, October 6, 2005

Algebra-Geometry-Topology Seminar: Defining relations of the quantum (super)algebras and $Z/3Z$-super quantum group

by Hiroyuki Yamane (Osaka University) in Skiles 255

In this talk, I begin with the definition of the Lie superalgebra (especially I explain how the Lie superalgebras osp(m|n)'s include the Lie algebra so(n) and sp (n)) I give the defining relations of the (affine) Lie superalgebras of type A-G. Next, I explain a Lusztig-type definition of the quantum superalgebra and give the defining relations of the (affine) quantum superalgebras. Finally, I introduce a $Z/3Z$-super quantum group and give its universal R-matrix.

3:00 pm Thursday, October 6, 2005

Stochastic Seminar: Optimal Aggregation in Sparse Classification Problems

by Vlad Koltchinskii (School of Mathematics, Georgia Tech) in Skiles 269

Binary classification (pattern recognition) problems can be often formulated as (penalized) risk minimization with respect to a convex loss function over a linear span of N given base functions. The risk is then replaced by its empirical version based on n i.i.d. training examples. We consider such problems in the cases when N is very large, but when the minimum of the true risk functional is attained at a sparse linear combination of the base functions. We prove inequalities showing that in this case with high probability the empirical risk also attains its minimum at an "approximately sparse" linear combination, which allows one to bound nicely the accuracy of the solution.

3:05 pm Thursday, October 6, 2005

Graph Theory: Graphs of Bounded Rank-width

by Sang-il Oum [mail] (Math, GT) in Skiles 170

In this talk, I would like to survey the current status of my research on clique-width and rank-width. Rank-width was defined by Seymour and myself to make it easy to investigate clique-width. We will briefly discuss why clique-width was defined and is useful. Then we will discuss the following three topics: (1) Poly-time algorithm to recognize graphs of rank-width at most k for fixed k. (2) Relations to binary matroids and "vertex-minors" of graphs. (3) Seese's conjecture on decidability of monadic second-order logic on graphs.

4:30 pm Thursday, October 6, 2005

Colloquium: Exploring Critical Singularities via Spherical Models: Beguiling but Wayward

by Michael Fisher [mail] (University of Maryland) in Skiles 269

3:30 pm Monday, October 10, 2005

Topology: Conformal Metrics and True "Gradient Flows" for Curves

by Anthony Yezzi (GT) in Skiles 269

The problem of finding shapes in images is a long standing and far-fetched one. This problem is related and fundamental to such issues as image segmentation, shape analysis, shape optimization, etc. Following the introduction of snakes by Kass, Witkin and Terzopoulos, a method known as active contours has played a prominent role. These active contours are closed planar curves that, driven by the minimization of suitable energies, move to achieve desireable segmentations of the image (foreground/background partitioning of the image domain). In recent years, the latest trend in active contour research seems to be that of incorporating global shape priors into the active contour paradigm. This has brought up non-trivial questions such as how to define an "average shape" or how to characterize "variations in shape". All of these questions ultimately lead to a more basic and fundamental question of how to define a Riemannian Geometry in the space of curves. Almost two decades of literature on variational approaches to active contours suggests a consistent metric on the space of curves and refers to a variety of evolution models as "gradient flows". We will show how to adjust this metric through a conformal factor to correct its pathological properties and thereby allow us to compute geodesics, etc. A particularly nice property of this new class of metrics is that, due to their conformal structure, all variational active contour models that have been called "gradient flows" in the past will constitute gradient flows with respect to these new metrics after appropriate reparameterization in time.

4:30 pm Monday, October 10, 2005

CDSNS Colloquium : Wasserstein kernels for the classical one-dimensional, one-phase Stefan problem

by Adrain Tudorascu (Georgia Institute of Technology) in Skiles 255

Arising from Optimal Mass Transportation methods applied to PDE's, the concept of Wasserstein kernel associated with one-dimensional diffusion problems of parabolic type with no-flux BC is introduced. As an interesting application, we provide an existence and regularity result for the one-phase Stefan problem. The long time behavior of solutions is addressed by related means.

2:00 pm Wednesday, October 12, 2005

Research Horizons: Fourier's Law: a challenge for theorists.

by Federico Bonetto (Georgia Tech) in Skiles 269

Since the creation of Statistical Mechanics more than a century ago, there have been many attempt to derive the law of heat conduction from the first principles of mechanics, without much success. We will try to review the model and methods involved in this research.

4:00 pm Wednesday, October 12, 2005

Analysis Seminar: Factorization of Hessenberg matrices, spectral transforms and orthogonal polynomials

by Francisco Marcellan. (Departamento de Matematicas, Universidad Carlos III de Madrid) in Skiles 269

In this talk a survey about recent results related to LU and QR factorization of Hessenberg matrices associated with the multiplication operator in terms of polynomial bases orthogonal with respect to different inner products will be presented. In particular, the case of inner products defined by measures supported on the real line [1] and the unit circle [3] will be analyzed.The connection with the perturbation theory of these measures will be stated.Finally,some perturbation problems related to the CMV representation [2] for the multiplication operator will be considered. Notice that this last representation is given in terms of an orthogonal basis of Laurent polynomials. The interest of such an approach is emphasized in the recent monograph by B. Simon [4]. Some references

[1] M. I. Bueno, F. Marcellan, Darboux transformations and perturbation of linear functionals, Linear Algebra and Applications 384 (2004), 215-242.

[2] M. J. Cantero, L. Moral, L. Velazquez, Five-diagonal matrices and zeros of orthogonal polynomials,Linear Algebra and Applications 362. (2003), 29-56.

[3] L. Daruis, J. Hern�ndez, F. Marcellan, Spectral transforms for Hermitian Toeplitz matrices. Submitted.

[4] B. Simon, Orthogonal Polynomials on the unit circle, Colloquium Publications, Vol 54.American Mathematical Society, Providence RI. 2005.

4:15 pm Wednesday, October 12, 2005

Combinatorics Seminar: Rank-width and Well-quasi-ordering of Skew-symmetric Matrice

by Sang-il Oum (School of Mathematics, Georgia Tech) in Skiles 255 (Note change in day and time)

Robertson and Seymour proved that a set of graphs of bounded tree-width is well-quasi-ordered by the graph minor relation. Extending their methods to matroids, Geelen, Gerards, and Whittle proved that a set of matroids representable over a fixed finite field are well-quasi-ordered if it has bounded branch-width. More recently, it was shown that a set of graphs of bounded rank-width (or clique-width) is well-quasi-ordered by the graph vertex-minor relation. We discuss a common generalization of the above three theorems in terms of skew-symmetric matrices over a fixed finite field under "pivot-minors". A pivot-minor of a skew-symmetric matrix M is a principal submatrix of a matrix obtained by principal pivoting transformation to M. As a useful tool, we define Lagrangian chain-groups, motivated by Tutte's chain-groups and Bouchet's isotropic systems. Roughtly speaking, Lagrangian chain-groups correspond to equivalence classes of skew-symmetric matrices under principal pivoting transformation.

3:00 pm Thursday, October 13, 2005

Stochastic Seminar: Probabilistic models for biological sequences: uniform accuracy of the maximum likelihood estimates

by Svetlana Ekisheva (School of Biology, Georgia Tech) in Skiles 269

Probabilistic models for biological sequences (DNA and proteins) are frequently used in bioinformatics. For three types of the models, a sequence of i.i.d. random variables, a stationary Markov chain, and an HMM (hidden Markov model), we derive the lower bound for the rate of convergence in probability of the maximum error in estimating over all parameters of the model. This is a joint work with Dr. M. Borodovsky.

3:05 pm Thursday, October 13, 2005

Graph Theory: Graphs of Bounded Rank-width II

by Sang-il Oum [mail] (Math, GT) in Skiles 255

In this talk, I would like to survey the current status of my research on clique-width and rank-width. Rank-width was defined by Seymour and myself to make it easy to investigate clique-width. We will briefly discuss why clique-width was defined and is useful. Then we will discuss the following three topics: (1) Poly-time algorithm to recognize graphs of rank-width at most k for fixed k. (2) Relations to binary matroids and "vertex-minors" of graphs. (3) Seese's conjecture on decidability of monadic second-order logic on graphs.

4:00 pm Thursday, October 13, 2005

Math Department Tea:

in Math Lounge (Skiles 236)

All math department staff, students, and faculty are welcome to attend. Food and beverages will be served.

3:30 pm Friday, October 14, 2005

Algebra-Geometry-Topology Seminar: Helicity of vector fields on S^3

by Jason Parsley (University of Georgia) in Skiles 269

The helicity of a vector field measures the extent to which its flowlines wrap and coil around one another. Helicity is analogous to the writhing number of a curve, and is closely related to the linking number of two curves. On the three-sphere, we define helicity using an integral formula and show this is in accordance with the definition in Euclidean space. For a vector field V defined on a subdomain of the three-sphere, upper bounds on the helicity of V are established. We detail applications of helicity to geometric knot theory, plasma physics, and energy minimization problems for vector fields.

2:00 pm Wednesday, October 19, 2005

Research Horizons: Trigonometric polynomials on the bi-circle

by Jeff Geronimo (Georgia Tech) in Skiles 269

The theory and applications of orthogonal polynomials in several variables is still quite undeveloped. I will describe a recent application related to factorization of positive trigonometric polynomials on the bi-circle.

12:00 pm Thursday, October 20, 2005

Applied and Computational Mathematics Seminar: New physical insights from experimentally realistic numerical simulations: Spatiotemporal Chaos and BioNEMS

by Mark Paul (Virginia Tech) in Skiles 269

Laboratory experiments often do not meet the idealizations required by available theory making it difficult to compare experimental results with theoretical predictions. However, in many situations of engineering and scientific interest, it is now possible with efficient parallel programs and/or clever physically motivated numerical algorithms, to perform numerical simulations for precise experimental conditions allowing the link between theory and experiment to be made. We can gain new physical insights by exploiting numerical advantages, such as, for example: the ability to modify the physics in order to disentangle competing subtle effects, the capacity to measure quantities inaccessible to experiment, and the ability to investigate regimes beyond current experimental capabilities. In this talk this approach is used to gain new physical insight into the two physically diverse examples of spatiotemporal chaos in Rayleigh-Benard Convection and the stochastic Brownian motion of nanoscale cantilevers immersed in a viscous fluid for use in biofunctionalized nanoelectromechanical systems (BioNEMS) as a single molecule detector.

3:00 pm Thursday, October 20, 2005

Stochastic Seminar: Sensitivity of Hidden Markov Models

by Alexander Yu. Mitrophanov (School of Biology, Georgia Tech) in Skiles 269

It has long been known to the modeling practitioners that the behavior of hidden Markov models is usually more sensitive to perturbations in the emission probabilities than to perturbations in the transition probabilities and the initial distribution of the underlying Markov chain. However, no rigorous results proving this tendency have been obtained. In the talk, I will describe my work aimed at gauging this phenomenon. Our major result is a tight perturbation bound for hidden Markov models. The nature of the result is quite general, and our approach can be used to investigate the sensitivity of other stochastic processes, such as mixture processes and semi-Markov processes. Joint work with Alexandre Lomsadze (School of Biology, Georgia Tech) and Mark Borodovsky (Schools of Biology and Biomedical Engineering, Georgia Tech).

5:00 pm Thursday, October 20, 2005

Karlovitz Lecture: Mathematical Modeling for Environment, Medicine and Sport

by Alfio Quarteroni (Chair of Modelling and Scientific Computing, EPFL (Switzerland)and Professor at the Polytechnic of Milan (Italy)) in Weber (SST) Room 1

For details.

11:00 am Friday, October 21, 2005

Colloquium: The Unreasonable Affinity of Knot Theory and the Algebraic Sciences

by Dror Bar-Natan (University of Toronto, Department of Mathematics) in Skiles 269

In an article bearing a similar title [1], Eugene Wigner once wondered about the numerous and unexpected ways in which mathematics comes into play in the natural sciences. While our topic and claims are much humbler, we find ourselves puzzled by the unreasonable affinity of knot theory and certain parts of algebra. It figures that knot theory is related to 3-dimensional topology. But why on earth should the mundane study of tangled shoelaces and unwieldy piles of seamen's rope be related to the elegance and sophistication of the likes of Lie algebras, 6j-symbols and homological algebra? I'll tell you about the Jones polynomial and its relationship with Lie algebras and about Khovanov homology [2,3] for knots and its relationship with yet-to-be-explored parts of algebra. Everything will be very basic; fancy words like homology will be defined and a picture of Khovanov will be shown. The experts will be disapointed and everybody else will have fun.

1. Eugene Wigner, The Unreasonable Effectiveness of Mathematics in the Natural Sciences, Communications in Pure and Applied Mathematics 13-I (1960).

2. Mikhail Khovanov, A categorification of the Jones polynomial, arXiv:math.QA/9908171.

3. Dror Bar-Natan, Khovanov's Homology for Tangles and Cobordisms, arXiv:math.GT/0410495.

3:00 pm Friday, October 21, 2005

Special Colloquium: Numerical Modeling of Blood Flow Problems

by Alfio Quarteroni (EPFL/Switzerland and MOX-Politecnico di Milano/Italy) in Skiles 269

4:00 pm Friday, October 21, 2005

Combinatorics Seminar: The Möbius function of the n-composition poset

by Sarah Mason (Department of Mathematics, University of Pennsylvania) in Skiles 255

The n-composition poset is the partially ordered set whose elements are compositions of n into n parts, ordered by dominance. This poset occurs naturally in the theory of nonsymmetric Schur functions. To calculate the Möbius function of this poset, we make use of a method developed by Sagan and Vatter involving discrete Morse theory. This result can be extended to the poset of compositions of n into infinitely many parts.

3:15 pm Monday, October 24, 2005

Algebra-Geometry-Topology Seminar: Ozsvath-Szabo invariants of fiber sums

by Tom Mark (Southeastern Louisiana University) in Skiles 269

The symplectic normal connected sum, or fiber sum, of symplectic manifolds is a primary tool in the construction of "exotic" symplectic 4-manifolds. In this talk, we will focus on the case of a fiber sum along surfaces with trivial normal bundle, and the effect of this operation on the Ozsvath-Szabo invariants. After a brief introduction to Ozsvath-Szabo thoery, we will discuss a product formula, obtained in joint work with S. Jabuka, that essentially determines the behavior of the invariants under this operation. Special cases of our formula have analogues in Seiberg-Witten theory due to Taubes, Morgan-Szabo-Taubes, and others. In particular we will describe the situation in case the fiber summands do not have "simple type"--which so far is not fully understood in Seiberg-Witten theory--and if time permits give some examples and applications.

4:30 pm Monday, October 24, 2005

CDSNS Colloquium: Image Noise Removal Based on the Variational Approach and Wavelets

by Charles Chui (Univ. Missouri-St Louis /Stanford) in Skile 255

The background of this work is the standard problem of minimization of some total energy functional, but with specific choices of the internal energy density functions g(x). Our interest in this study is motivated by the search of effective solutions to certain inverse problems, particularly for real-time image noise removal for digital cameras. In general, depending on the objectives of the inverse problems under investigation, such as curve fitting, image noise removal, and feature extraction, the internal energy in our study is governed by g(|Lu|); with (Lu)(x) = u''(x), (Lu)(x,y) = (Grad u)(x,y), and Lu being some wavelet transform of u in any dimension. For digital image noise removal, in particular, a suitable choice of g(x) leads to the anisotropic diffusion model, the discretization of which, in turn, is relevant to the design of certain content-dependent filters, notably the bilateral filters. A natural generalization of this approach also gives rise to the notions of diffusion maps and geometric harmonics that constitute the foundation for the recent research investigations in diffusion wavelets for analyzing complex data in high dimensions. This is a joint work with Jianzhong Wang.

4:30 pm Monday, October 24, 2005

Algebra-Geometry-Topology Seminar: Knot invariants from contact homology

by Lenny Ng (Stanford Univeristy) in Skiles 269

I will describe a technique of constructing invariants of knots and other submanifolds through contact topology and holomorphic curves. The knot invariant can be described purely topologically, with no reference to contact geometry, and is fairly strong. In particular, it encodes the Alexander polynomial and is closely related to the A-polynomial.

4:30 pm Tuesday, October 25, 2005

PDE Seminar: Macro-Micro Models in Viscoelastic Materials

by Chun Liu (Penn State University) in Skiles 255

A unified energetic variational framework will be presented for "elastic" complex fluids. It highlights the competition of the kinetic energy and the elastic energies, through the transport of the internal elastic variables. In particular, we will focus on the multiscale coupling effects in these materials.

12:00 pm Wednesday, October 26, 2005

Applied and Computational Mathematics Seminar: Homology Computations on Braid Classes: algorithms and implications

by Sarah Day (Cornell) in Skiles 269

Braid classes arise naturally as representatives of sets of solution curves for systems given by, for example, scalar parabolic differential equations and twist maps. These systems also induce dynamics on the space of braid classes. Ghrist, van den Berg, and Vandervorst defined a Morse-Conley index for such braid classes, which they used to obtain forcing theorems for the dynamics of the original systems. In this talk, we will focus on a computational approach to studying indices of braid classes. We will also discuss a few examples, including a set of computations for 8-dimensional braid classes where preliminary computations yield unexpected results.

2:00 pm Wednesday, October 26, 2005

Research Horizons: What is Contact Geometry?

by John Etnyre (Georgia Tech) in Skiles 269

Contact geometry was born more than two centuries ago in the work of Huygens, Hamilton, Jacobi as a geometric language for optics. It was soon realized that it has applications in many other areas, including non-holonomic mechanics and thermodynamics. One encounters contact geometry in everyday life when parking a car, skating, using a refrigerator, or watching the beautiful play of light in a glass of water. Lie, Cartan, Darboux and many other great mathematicians devoted a lot of time to this subject. However, until very recently most mathematicians know little about contact geometry. In the last couple of decades contact geometry has taken a central place in low-dimensional topology and geometry. In this talk I will discuss some of the origins of contact geometry and hint at a few amazing recent developments.

4:00 pm Wednesday, October 26, 2005

Analysis Seminar: Correlation inequalities of Brascamp-Lieb type

by Eric Carlen (School of Mathematics, Georgia Tech) in Skiles 269

12:00 pm Thursday, October 27, 2005

Applied and Computational Mathematics Seminar: Computing Invariant Manifolds by Integrating Fattened Trajectories

by Mike Henderson (IBM) in Skiles 269

Some of the most interesting structure in dynamical systems is associated with invariant manifolds. These are invariant in the sense that if a point lies on the manifold the trajectory which contains the point also lies on the manifold. An invariant manifold can therefore be defined as a set of trajectories passing through points on a smooth manifold that is transverse to the flow, extending backwards and forward in time. For an invariant surface, two adjacent trajectories, with a mesh between, might be used to determine a small strip of the invariant manifold, except for the exponential divergence that is common in "interesting" flows. I will show how to reparameterize the surface along a trajectory so as to factor out the stretching, giving a locally Euclidean quadratic approximation to the surface at each point along the trajectory. This takes the form of an initial value problem involving the point and the derivatives of the surface, which can be integrated with the usual tools. We can calculate an expansion for the unstable manifold near a fixed point, and use it for initial points for integrating these "fattened" trajectories. By selecting an initial point from the boundary of the part of the manifold covered by the fattened trajectory, we can cover the unstable manifold with long trajectories, while avoiding an accumulation of trajectories near the fixed point.

1:00 pm Thursday, October 27, 2005

ACO Pizza Lunch Lecture: Forward, March?

by Peter Winkler (Dartmouth) in MiRC 101A&B

We describe some circumstances in which you can be sure that, if you can attain your goal at all, you can do so without making backward steps. Joint work with Graham Brightwell, LSE.

3:00 pm Thursday, October 27, 2005

Stochastic Seminar: The statistical distribution of floating-point numbers in scientific computations

by Ted Hill (School of Mathematics, Georgia Tech) in Skiles 269

Analysis of the average behavior of floating-point arithmetic algorithms (and in particular their average running time) requires information on how often various cases arise. For more than half a century, computer scientists have observed empirically that the statistical distribution of the fraction part of floating point numbers in scientific calculations is generally not uniform, as might be expected, but tends to follow a logarithmic distribution called Benford's Law (BL). The purpose of this survey talk is to review a few of the implications of BL in numerical calculations (e.g., for roundoff and overflow/underflow errors), and to outline several very recent analytical BL results by Berger, for recurrence relations; by Kontorovich and Miller, for geometric Brownian motion (hence stock market models); and by Berger and H, for Newton's Method that help explain the ubiquity of BL in scientific calculations. Examples, graphical heuristics, and open problems will be included, and the talk will be aimed for the non-specialist.

4:30 pm Thursday, October 27, 2005

Joint ACO-School of Mathematics Colloquium: Optimization using matrix and tensor approximations

by Ravi Kannan [mail] (Yale University) in Skiles 269

Linear Algebra gives us algorithms to find low-rank approximations to matrices. We will first survey results which use these approximations as well as some more combinatorial ones to solve several graph optimization problems. Other problems like maximizing the number of satisfied clauses in a Boolean formula involve tensors for which there is no nice Linear Algebra-like theory. We will discuss new algorithms for approximating tensors which can be applied to a broad class of these optimization problems.

2:00 pm Friday, October 28, 2005

GT Homecoming Seminars: Grand Challenges in Mathematics

by William Trotter (School of Mathematics, Georgia Tech) in Global Learning and Conference Center

It may be surprising to learn there are a number of easily stated and readily understood problems in the mathematical sciences whose resolution has the potential for revolutionizing life as we know it. Join in an informal discussion of some of these "grand challenges."

Registration and a full seminar schedule are available at gtalumni.org/homecoming.

3:30 pm Friday, October 28, 2005

Algebra/Geometry/Topology Seminar: Quadratic twists of elliptic curves

by Sungkon Chang (Armstrong State) in Skiles 269

The theory of elliptic curves plays a central role in number theory; for example, elliptic curves are a key ingredient in Wiles' proof of Fermat's Last Theorem. In this talk, we shall review some basic aspects of the theory of elliptic curves, and introduce our result on quadratic twists of an elliptic curve. The talk will be geared toward a general mathematical audience, including graduate students.

11:00 am Monday, October 31, 2005

CNS Seminar: Wavelet methods to analyze and compute turbulent flows

by Marie Farge & Kai Schneider (Ecole Normale Superieure & Universite de Provence) in Physics Building Room N110

Turbulence is characterized by its nonlinear and multi-scale behavior, self-organization into coherent structures and a generic randomness. The number of active spatial and temporal scales involved increases with the Reynolds number, therefore it soon become prohibitive for direct numerical simulation. However, observations show that for a given flow realization these scales are not homogeneously distributed, neither in space nor in time, which corresponds to the flow intermittency. To be able to benefit from this property, a suitable representation of the flow should reflect the lacunarity of the fine scale activity, in both space and time. A prominent tool for multiscale decompositions is wavelets. A wavelet is a well-localized oscillating smooth function, i.e. a wave packet, which is dilated and translated. The thus obtained wavelet family allows to decompose a flow field into scale-space contributions from which it can be perfectly reconstructed. Note that for finer scales the physical support of the basis functions is decreasing. The flow intermittency is reflected in the scarcity of the wavelet representation, i.e. only few coefficients, the strongest ones, are necessary to represent the dynamically active part of the flow. The Coherent Vortex Simulation (CVS) approach we have proposed is based on the wavelet filtered Navier-Stokes equations. At each time step the turbulent flow is split into two orthogonal parts, one corresponding to coherent vortices, which are kept, and the other to an incoherent flow, which is discarded. In the talk we will present first applications of the CVS filter to data computed by Direct Numerical Simulation (DNS) at high resolution (up to 2048^3 grid points). We will show that the coherent flow can be represented by few wavelet modes only, which are sufficient to fully reproduce the vorticity probability density function (PDF) and the energy spectrum. The discarded incoherent background flow, which is homogeneous, gaussian and decorrelated, corresponds to the turbulent entropy but has a negligible contribution to the energy. Finally, we present simulations of a time-developing turbulent mixing layer where the CVS filter is applied at each time step. The results show that CVS preserves the nonlinear dynamics of the flow, and that discarding the incoherent modes is sufficient to model turbulent dissipation. Related publications can be downloaded from the following web page: wavelets.ens.fr

3:00 pm Monday, October 31, 2005

Applied and Computational Mathematics Seminar: Centrifugal BioReactors: Theory and Implementation

by Heath Herman (KBI BioPharma, Inc.) in Skiles 255

A novel method by means of which dense arrays of microscopic particles, including living cells, may be effectively immobilized in a force field has been developed. The particles are placed in a centrifugal field where anti-radially directed liquid flows are used to generate buoyant forces counteracting the effects of �centrifugal� forces. When the effect of gravity on the particle array immobility is minimized, dense arrays of living, respiring cells can be maintained indefinitely if the flowing liquid is nutrient. While the theoretical basis for the immobilization process is understood, the relationships between microscopic flow velocity, particle array density, and confinement chamber shape as functions of rotational radius are not well understood. A more rigorous understanding of particle motion in such arrays might greatly extend the utility of the method.

3:30 pm Monday, October 31, 2005

Geometry-Topology Seminar: A dilogartithmic formula for the Cheeger-Chern-Simons class

by Christian Zickert (Columbia University) in Skiles 269

We introduce a new and efficient formula for computing the Cheeger-Chern-Simons class of a hyperbolic 3-manifold. A formula by Dupont computes the universal class using Rogers' dilogarithm, but his formula is only correct modulo Q\Z. In a recent paper Neumann extends the work by Dupont and obtains a formula without this indeterminacy. However, Neumann's formula is defined using very complicated combinatorial topology. We construct a similar formula which can be applied directly on a homology class in the bar complex. Here all geometry is replaced by homological algebra which vastly simplifies the proofs.

4:30 pm Monday, October 31, 2005

CDSNS Colloquium: Closed Characteristics on Non-Compact Hypersurfaces

by Robert Vandervorst [mail] (Vrije Universiteit Amsterdam) in Skiles 255

The Weinstein Conjecture for compact hypersurfaces of contact type states that such surfaces always contain a closed characteristic. Viterbo proved this conjecture right for hypersurfaces in $\re^{2n}$ (with the standard symplectic structure). For non-compact hypersurfaces in $\re^{2n}$ matters become more delicate. Without additional geometric requirements such surfaces need contain any closed characteristics. We try shed more light on the non-compact case by studying the special case of energy surfaces of mechanical Lagrangian systems. Due to the non-compactness an additional geometric condition will be introduced: coercivity. Under the assumption of coercivity a result on the existence of closed characteristics on non-compact energy surfaces will be obtained.

4:00 pm Tuesday, November 1, 2005

Emory University Colloquium: Running through the primes

by Jean-Pierre Serre (College de France) in Emory University, Auditorium, MSC Rm W201

This will be a general-audience talk about number theory, delivered by one of its main architects from the second half of the 20th century. Serre has won both of the top prizes in mathematics: the Fields Medal (1954, youngest recipient ever) and the Abel Prize (2003, the first time the prize was awarded). He has also received the Steele Prize for mathematical writing (1995) and the Balzan Prize (1985).

Refreshments at 3:30 p.m.

Coffee Room, MSC Rm W427

4:30 pm Tuesday, November 1, 2005

PDE Seminar: Uniqueness of Weak Solutions of the Navier-Stokes Equations of Multidimensional, Compressible Flow

by Prof. David Hoff (Indiana University) in Skiles 255

I'll describe a result on the uniqueness and continuous dependence on initial data of weak solutions of the Navier-Stokes equations of compressible flow in two and three space dimensions. The solutions considered include those displaying the generic singularities of the system but typically have small energy. I'll then discuss a program for selecting physically correct solutions from the set of large-energy weak solutions and will give a rigorous result implementing this program in one dimension. I hope to make the talk accessible to a general audience, including grad students in pde's and applied analysis.

2:00 pm Wednesday, November 2, 2005

Research Horizons: Introduction to Treewidth

by Torsten Inkmann (Georgia Tech) in Skiles 269

The treewidth of a graph is a parameter that is important in Graph Theory and Theoretical Computer Science, but also has applications in other areas like Logic or Numerical Linear Algebra. The concepts related to treewidth are a key ingredient to the proof of one of the deepest results in Graph Theory, they are linked to the most important open problem in Theoretical Computer Science, and more recently, their usability in practice has been investigated. I will try to survey some of the main ideas related to treewidth and their impact in both theory and practice. The talk will be elementary; in particular all necessary concepts from Graph Theory and Computer Science will be defined.

3:00 pm Wednesday, November 2, 2005

Analysis Seminar: Postponed

See instead the Emory University Colloquium by Jean-Pierre Serre on November 1, 2005 at 4:00.

11:00 am Thursday, November 3, 2005

ISyE Seminar: Inference for Quantile Regression Models

by Xuming He (University of Illinois at Urbana-Champaign) in Executive Classroom, ISyE Building

Quantile regression models are increasingly popular in a wide range of applications. It is easy to argue that the usual regression models that focus on conditional means are often inadequate to reflect inhomogeneity or to capture some interesting part of the population. As the quantile regression approach gains popularity in the econometrics, statistics and biostatistics literature, it is important that we have reliable inference tools. In this talk, I will review a number of existing methods for estimating standard errors and for constructing confidence intervals, and explain why it has been difficult for software developers to choose a default method. I will then introduce the Markov chain marginal bootstrap (MCMB) algorithm, and assess its performance in terms of accuracy, speed, and reliability. The MCMB algorithm is not about Bayesian computation, but it is especially appealing for handling high dimensional problems. The current version of the MCMB algorithm for quantile regression is available as an R package or a SAS procedure.

12:00 pm Thursday, November 3, 2005

Applied and Computational Mathematics Seminar: The Evolution of Topologically Complex Interfacial Morphologies during Coarsening

by P. Vorhees (Northwestern) in Skiles 269

Nature frequently produces two-phase mixtures with great topological complexity. Examples include the interfacial morphologies found following spinodal decomposition and order-disorder transformations in metals and polymers. These structures are highly interconnected and have both positive and negative mean curvatures. Through large-scale computer simulations we have examined the evolution of the three-dimensional interfacial morphology of systems undergoing spinodal decomposition and order-disorder transformations using the Cahn-Hilliard and Allen-Cahn equations. We characterize the morphology of the resulting interfaces using the probability of finding a patch of interface with a certain pair of principle curvatures, the interfacial shape distribution, and the genus. We find that the interfacial shape distributions become time independent under the appropriate scaling for systems evolving by both Allen-Cahn and Cahn-Hilliard dynamics. The interfacial shape distributions are, however, different. We will discuss the manner in which the interfacial shape distributions and genii evolve during coarsening.

3:05 pm Thursday, November 3, 2005

Graph Theory: On a graph packing conjecture by Bollobas, Eldridge and Catlin

by Gexin Yu [mail] (University of Illinois at Urbana-Champaign) in Skiles 255

Two graphs G and H pack if G and H can be embedded into the same vertex set such that the image of edge sets do not intersect. The concept of graph packing generalizes various extremal graph problems, including problems on fixed subgraphs, forbidden subgraphs, and equitable coloring. Graph packing results have also been widely applied to the study of computational complexity of graph properties. Bollobas and Eldridge, and independently Catlin, conjectured that if n-vertex graphs G and H, with maximum degree D(G) and D(H) respectively, satisfy (D(G)+1)(D(H)+1)<=n+1, then G and H pack. If true, this conjecture would be sharp, and would be a considerable extension of the Hajnal-Szemeredi theorem on equitable colorings. The conjecture has only been proved when one of the graphs is highly degenerate, or D(G)<=2, or D(G)=3 and n is huge. We prove that for n-vertex graphs G and H, if D(G), D(H)>=400, and (D(G)+1)(D(H)+1)<= 0.6n+1, then G and H pack. This is joint work with H. Kaul and A. Kostochka.

4:30 pm Thursday, November 3, 2005

Colloquium: Dynamics and Asymptotics of Delay-Differential Equations

by John Mallet-Paret [mail] (Brown University) in Skiles 269

Delay-differential equations describe dynamical processes in which the evolution of the system depends on its past history as well as on its present state. Such equations arise in a variety of scientific models, most notably in control theory and biology. A rich mathematical theory of delay-differential equations has been, and is being, developed. In this lecture we discuss several delay-differential equations from the theoretical viewpoint of dynamical systems. In particular, we study questions related to global dynamics, attractors, stability, and asymptotics of solutions. Of particular recent interest are systems in which the time-delay parameter is itself a variable, the so-called state-dependent delays. A variety of mathematical techniques, including bifurcation theory, degree theory, monotonicity methods, and geometric singular perturbation theory, are used in the analysis of these systems.

3:05 pm Friday, November 4, 2005

Combinatorics Seminar: Coloring powers of chordal graphs

by Daniel Kral (School of Mathematics, Georgia Tech and Charles University, Prague) in Skiles 255

The k-th power of a graph G is the graph with the same vertex set in which two vertices are adjacent if their distance in G is at most k. We prove that the k-th power of a chordal graph G with maximum degree D is O(k^0.5 D^{(k+1)/2})-degenerate for even values of k and O(D^{(k+1)/2})-degenerate for odd values of k. In particular, this bounds the chromatic number of the k-th power of G. The bound proven for odd values of k is the best possible (it exactly matches the lower bound) and the bound is assymptotically tight for even values of k. We will also discuss consequences of our results on the distance constraint labeling of graphs.

3:30 pm Monday, November 7, 2005

Algebra-Geometry-Topology Seminar: Distances of Heegaard Splittings

by Aaron Abrams (Emory University) in Skiles 269

I will discuss the complex of curves and the theorem of Masur-Minsky that it is Gromov-hyperbolic. Then I will give an application of this theorem to Hempel's theory of distances of Heegaard splittings.

4:30 pm Monday, November 7, 2005

CDSNS Colloquium: Correspondence maximization: a problem, an algorithm and some analysis

by Tomas Gedeon [mail] (Montana State University) in Skiles 255

We study the combinatorial problem of correspondence maximization which arises in applications like computer vision, motion plannning and automatic speech recognition. Given an input pattern and a set of reference patterns, the goal is to find a composition of transformations which gives the best fit between the transformed input patterns and one of the reference patterns. This problem is a discrete optimization problem, which we embed into continuous setting by paramterizing the set of transformations and then analyze the resulting constrained optimization problem. Eventhough the task of finding the global solution is NP-complete, we present several algorithms and analyze their performance.

1:00 pm Tuesday, November 8, 2005

GT/UGA/FRG Joint Analysis Semina: Local Tb Theorems and applications

by Steven Hofmann (University of Missouri, Columbia) in Skiles 269

The Tb Theorem, and its predecessor, the T1 Theorem, were introduced in large part to better understand the Cauchy integral operator on a Lipschitz curve. The rough idea of these theorems is that the L^2 boundedness of a singular integral T can be deduced from sufficiently good behavior of the operator on some suitable test function, namely, the constant function f(x) := 1, in the case of the T1 theorem, or a sufficiently non-degenerate function b, in the case of the Tb theorem. However, in some PDE applications, it may be easier to test the operator T locally (say on any given cube Q), on a test function b_Q that depends upon Q, rather than on a single, globally defined b. Or to be more precise, in the applications, it may be easier to find a family of local b_Q's on which T is well behaved, than it is to find a single global b for which Tb is nice. In this talk, we'll discuss some versions of local Tb theorems, as well as some applications to PDE.

4:30 pm Tuesday, November 8, 2005

PDE Seminar: A modified Perron's method for viscosity solutions and its applications

by Shigeaki Koike (Saitama University, Japan) in Skiles 255

Since H. Ishii established Perron's method for viscosity solutions in 1987, it has been a very convenient tool for existence results. In this talk, I will give a modification of Perron's method that applies for fully nonlinear PDEs with measurable ingredients. When the PDEs are uniformly elliptic, it is possible to show that such a solution constructed by our Perron's method is Holder continuous. I will mention some applications and explain a bit more about the advantage of our solutions.

2:00 pm Wednesday, November 9, 2005

Research Horizons: Deterministic Walks in Random Environments

by Leonid Bunimovich (Georgia Tech) in Skiles 269

Deterministic walks in random environments (DWRE) occupy an intermediate position between purely random (generated by random trials) and purely deterministic (generated by deterministic dynamical systems) models. DWRE were (independently) introduced as phenomenological models in statistical physics, material science, communication theory, theory of artificial life, computer science, etc. where the corresponding moving objects were called particles, waves, signals, ants, read/write heads of Turing machine, etc. DWRE demonstrate many unusual (surprising) types of behavior from the point of a general intuition based on complete understanding of (some) purely probabilistic and purly deterministic systems. Recently some models of DWRE were explicitely solved which allowed to build some basic intuition on their behavior. The interest to DWRE (even more recently) increased when it was shown that some problems of the random matrix theory and of the quantum field theory can be reduced to DWRE. Analysis of DWRE requires a combination of the methods from Probability, Dynamical systems, Combinatorics and Topology. However, to attend (and even understand) this talk requires just to attend.

4:00 pm Wednesday, November 9, 2005

Analysis Seminar: Distributional estimates for multillinear operators

by Dmitriy Bilyk (School of Mathematics, Georgia Tech) in Skiles 269

We provide a method allowing to deduce distributional inequalities from certain boundedness properties of a multilinear operator and its adjoints. In particular, we show that if an m-linear operator and all its adjoints have restricted weak type (1, ... ,1, 1/m) (which is the case for multilinear Calderon-Zygmund operators), then the distribution function of the operator applied to characteristic functions has exponential decay at infinity. We use these methods to obtain similar inequalities for the bilinear Hilbert transform.

12:00 pm Thursday, November 10, 2005

Applied and Computational Mathematics Seminar: Laboratory experiences for math majors

by John Pelesko (U Delaware) in Skiles 269

In this talk, we'll discuss the activities of the MEC Lab at the University of Delaware. In particular, an overview of our senior undergraduate capstone course will be presented. We'll discuss the pro's and con's of having math majors in a laboratory.

3:00 pm Thursday, November 10, 2005

Stochastic Seminar: Maximal inequalities and invariance principle for martingale-like sequences

by Magda Peligrad (Department of Mathematical Sciences, University of Cincinnati) in Skiles 269

For processes with short memory the theory of weak invariance principle is very well fine tuned under various mixing conditions. Unfortunately, there are several important examples, including some versatile time series, for which the mixing conditions are rather restrictive, they might not be satisfied or the mixing coefficients are not tractable. This is the reason why one of the new directions in modeling the dependence is to introduce, analyze, and obtain sharp results for new dependent structures, defined by either substantially reducing the classes of functions used in the definition of mixing coefficients or, by using innovative martingale-like conditions. In this talk, I shall survey some recent, "fine tuned" results on the central limit theorem and its invariance principle under martingale-like conditions. When applied to mixing sequences these results extend the sharpest known results for strongly mixing sequences and rho-mixing sequences, and at the same time have a broader class of applications, including linear processes and a large class of Bernoulli shifts. Since the key technique for obtaining invariance principles is the use of maximal inequalities for partial sums, we shall also survey several recent advances on this subject.

3:05 pm Thursday, November 10, 2005

Graph Theory: On the Reconstruction of Planar Graphs

by Mark Bilinski [mail] (Math, GT) in Skiles 255

We show that the planarity of a graph can be recognized from its vertex deleted subgraphs, which answers a question posed by Bondy and Hemminger in 1979. We also derive some useful counting lemmas and use them to reconstruct certain planar graphs.

4:30 pm Thursday, November 10, 2005

Colloquium: Tilings, tiling spaces and topology

by Lorenzo Sadun [mail] (University of Texas Austin) in Skile 269

Physical properties of materials modeled by aperiodic tilings (e.g., quasicrystals) are closely related to topological properties of the space of all similar tilings, especially the Cech cohomology of space. We'll go over tiling spaces, their local topology, their construction as inverse limits of branched manifolds, and their cohomology, and see how the cohomology governs deformations of the tilings.

3:05 pm Friday, November 11, 2005

Combinatorics Seminar: Extremal Functions for Graph Linkages and Rooted Minors

by Paul Wollan (School of Mathematics, Georgia Tech) in Skiles 255

Ph.D. thesis defense.

3:30 pm Monday, November 14, 2005

Algebra-Geometry-Topology Seminar: The Structure of Legendrian Invariants

by Josh Sabloff (Haverford College) in Skiles 269

The contact homology algebra is a powerful new invariant of Legendrian knots in contact manifolds. I will discuss some structure theorems for the contact homology of a Legendrian knot in the standard contact 3-space: a condition for the existence of a linearization of the algebra, a description of a fundamental class in the linearized theory, and duality in the linearized theory.

4:30 pm Monday, November 14, 2005

CDSNS Colloquium: Computing Homology

by Marian Mrozek [mail] (Jagiellonian University, Poland) in Skiles 255

Homology computations play an essential role in the evaluation of topological invariants of dynamical systems. For twelve years algorithmic homology computations have been successfully used in computer assisted rigorous analysis of dynamical systems and differential equations. An emerging area of applications of computational homology lies in computer vision and graphics. The range of usability of homology methods crucially depends on the efficiency of homology algorithms. The complexity of classical homology algorithms is cubical, which seriously limits the scope of real applications. To overcome this difficulty various preprocessing methods intended for reducing the size of the problem are applied. In the lecture I will present and compare some old and new reduction algorithms used to accelerate homology computations.

4:30 pm Tuesday, November 15, 2005

PDE Seminar: On some two-dimensional problems of conservation laws

by Dehua Wang (University of Pittsburgh) in Skiles 255

Some two-dimensional problems related to compressible Euler equations will be considered. The evolution of singularity, global structures, and wave interactions will be discussed.

2:00 pm Wednesday, November 16, 2005

Research Horizons: Using continuous and discrete SVD algorithms in dynamical systems

by Cinzia Elia (Visiting faculty from University of Bari) in Skiles 269

Stability spectra characterize the asymptotic behavior of non autonomous linear systems. The main techniques proposed to compute these quantities rely on the QR or on the SVD decomposition of the fundamental matrix solution of the system. In this talk, we first provide theoretical background on the Lyapunov and the Exponential Dichotomy spectrum on the half line. Then we describe continuous and discrete SVD techniques for the computation of these quantities. Our scope is twofold: understand under which assumption these computations are reliable and determine an efficient implementation for SVD techniques.

4:00 pm Wednesday, November 16, 2005

Analysis Seminar: Potential theory on infinite graphs

by Matt Baker (School of Mathematics, Georgia Tech) in Skiles 269

I will discuss a certain class of topological spaces (whose definition is motivated by considerations in number theory) which are inverse limits of finite metrized graphs. I'll then explain how to construct a Laplacian operator and find a fundamental solution to the Laplace equation on such spaces.

12:00 pm Thursday, November 17, 2005

Applied and Computational Mathematics Seminar: The method of topological sections in the rigorous numerics of dynamical systems

by Marian Mrozek in Skiles 269

The most challenging problem in the rigorous numerics of dynamical systems is the exponential growth of error bounds. It is fighted with many methods including special integration methods like the Lohner method and the method of intermediate sections. However, it is not difficlut to present examples of ODE's where these method cannot solve the problem. In the lecture we study one such example in which almost every trajectory escapes to infinity in very short time. We present a modified method of intermediate sections in which homology maps are composed instead of multivalued maps. We show that if the goal is rigirous computation of some topological invariants, then the method may be successfully applied in even very hard problems.

3:00 pm Thursday, November 17, 2005

Stochastic Seminar: Limiting Distributions in Some Longest Increasing or Common Subsequence Problems

by Christian Houdré (School of Mathematics, Georgia Tech ) in Skiles 269

3:05 pm Thursday, November 17, 2005

Graph Theory: The seminar has been postponed due to conflict with the Stochastics Seminar.

12:00 pm Friday, November 18, 2005

Applied and Computational Mathematics Seminar: To Be Announced

by Andrew Draganescu (Sandia Nat. Lab) in Skiles 269

3:05 pm Friday, November 18, 2005

Combinatorics Seminar: Good error control codes are good for security - coding for the wiretap channel

by Steven McLaughlin (Electrical & Computer Engineering, Georgia Tech) in Skiles 255

In this talk we introduce the idea of physical layer security where we will use conventional error-correction-based channel coding for perfect security against an eavesdropper with infinite resources. We show how perfect security is achievable under some very strict assumptions about the physical communication channels and then discuss how this can applied to more practical systems.

3:30 pm Friday, November 18, 2005

Algebra-Geometry-Topology Seminar: Yang-Baxter Equations and Their Super Solutions

by Gizem Karaali [mail] (University of California at Santa Barbara) in Skiles 269

I will start with a brief overview of the Yang-Baxter equations and their relationship to quantum groups. I will then explain the super analog of these concepts. In particular, I will discuss certain results which generalize the non-graded case and then concentrate on some examples which are peculiar to the super case. This talk should be accessible to anyone who is somewhat intrigued by this abstract but not necessarily sure of what a quantum group is.

3:30 pm Monday, November 21, 2005

Geometry/Topology Seminar: An introduction to geometric and deformation quantization

by Prof. Jorgen Ellegaard Andersen [mail] (University of Aarhus) in Skiles 269

4:30 pm Monday, November 21, 2005

CDSNS Colloquium: On Poisson-Nernst-Planck systems for ion channels

by Weishi Liu [mail] (University of Kansas) in Skiles 255

The talk is concerned with global dynamics of the Poisson-Nernst-Planck (PNP) systems for ion flows through membrane channel. As the radii of the cross-sections of three-dimensional membrane channel approaches zero, a one-dimensional limiting PNP system is derived. This one-dimensional PNP system differs from previous studied one-dimensional PNP systems on that it encodes the defining geometry of the three-dimensional membrane channels. We justify partially this limiting process by showing the upper-semi-continuity of the attractors of the three-dimensional PNP systems to that of the limiting PNP system. For large Debye numbers, the steady-state of the one-dimensional PNP system is analyzed using the geometric singular perturbation theory. Stability issue of the steady-states will also be discussed.

4:30 pm Tuesday, November 22, 2005

PDE Seminar: IBVP for multidimensional hyperbolic problems with dissipation

by Wen-Qing Xu (California State University , Long Beach) in Skiles 255

We present some results on asymptotic convergence and boundary layer behavior for IBVP of M-D hyperbolic problems under viscous or relaxation approximations. In particular, we derive a necessary and sufficient condition on the boundary conditions for a class of hyperbolic relaxation problems such that the relaxation IBVP is uniformly well-posed independent of the relaxation rate.

4:30 pm Monday, November 28, 2005

CDSNS Colloquium: Patterns Generation and Spatial Entropy in Multi-Dimensional Lattice Models

by Jung-Chao Ban (National Center for Theoretical Sciences, Taiwan) in Skile 255

Abstract

11:00 am Tuesday, November 29, 2005

Job Candidate Seminar: Can You Hear the Shape of a Potential?

by Howard Weiss (Mathematics, The Pennsylvania State University) in Skiles 255

Classical lattice spin systems provide an important and illuminating family of models in statistical physics. An interaction on a lattice determines a lattice spin system with associated potential. The pressure and free energy of the potential are fundamental characteristics of the system. However, even for the simplest systems, the information about the (microscopic) potential that the (macroscopic) free energy captures is subtle and poorly understood. We study whether, or to what extent, potentials are determined by their free energy. In particular, we show that for a one-dimensional lattice spin system, the free energy of finite range interactions typically determines the potential, up to natural equivalence, and there is always at most a finite ambiguity; we exhibit exceptional potentials where uniqueness fails; and we establish deformation rigidity for the free energy. This project was also motivated by questions in multifractal rigidity. Along the way, in the language of dynamical systems, we study whether for a subshift of finite type the unmarked Birkhoff averages for a function along the periodic orbits determine the function. We show that this rigidity problem has striking analogies to questions in spectral geometry that Kac summarized ``Can you hear the shape of a drum?".

4:30 pm Tuesday, November 29, 2005

PDE Seminar: How a shock wave cosmology describing a Big Bang of finite total mass might evolve from an inflationary spacetime.

by Blake Temple (UC Davis) in Skiles 255

I will present a computer visualization of the shock wave cosmology introduced by Smoller and Temple in Proc Natl Acad Sci, Sept 30, 2003, and then I will discuss how this model might connect up with the theory of inflation.

12:00 pm Wednesday, November 30, 2005

Ph.D. Dissertation Defense: Nodal Sets and Contact Structures

by Rafal Komendarczyk (School of Mathematics, Georgia Tech) in Skiles 255

2:00 pm Wednesday, November 30, 2005

Research Horizons: An Introduction to Longest Common and Longest Increasing Subsequence Problems

by Trevis Litherland (Georgia Tech) in Skiles 269

We will begin by introducing the classical longest common and longest increasing subsequence problems and will review some of the most important known results and open questions of the field. The main thrust of this talk will be to explore the natural ways in which typically probabilistic questions emerge as we study these problems. We will also indicate how these results are of significance to the bioinformatics community.

4:30 pm Wednesday, November 30, 2005

Applied and Computational Mathematics Seminar: Robustness of Morphogen Gradients

by Qing Nie (UCI) in Skiles 255

Many patterns of cell and tissue organization are specified during development by gradients of morphogens, substances that assign different cell fates at different concentrations. One of the central questions in cell and developmental biology is to identify mechanisms by which the morphogen gradient systems might achieve robustness to ensure reproducible embryonic patterns despite genetic or environmental fluctuations. Recently, through computations and analysis of various bio-chemical models and examination of old and new experimental data, we found a set of of new mechanisms for enhancing robustness of cell-cell signaling through non-signaling cell surface molecules (e.g., HSPG). In addition, we examined the roles of diffusive ligands (e.g., Sog) on the formation and robustness of BMP (Bone Morphogenetic Protein) gradients in the Drosophila embryo. In this talk, I shall also discuss some mathematical and computational challenges associated with such study, and present a new class of numerical algorithms for reaction-diffusion equations arising from biological models.

12:00 pm Thursday, December 1, 2005

Applied and Computational Mathematics Seminar: On the Accuracy of Homology Computations for Nodal Domains

by Thomas Wanner (George Mason University) in Skiles 269

Many partial differential equation models arising in applications generate complex patterns evolving with time which are hard to quantify due to the lack of any underlying regular structure. Such models often include some element of stochasticity which leads to variations in the detail structure of the patterns and forces one to concentrate on rougher common geometric features. From a mathematical point of view, computational algebraic topology suggests itself as a natural quantification tool and has been used in a variety of settings. In many of these instances, one is interested in the geometry of a nodal domain of a function. The nodal domain is usually approximated using an underlying discretization of the considered partial differential equation --- which immediately raises the question of the accuracy of the resulting homology computation. In this talk, I will present a probabilistic approach which gives insight into the suitability of this method in the context of random Fourier series. We will obtain explicit probability estimates, which in turn yield a-priori bounds for the suitability of certain grid sizes. Our results apply to one and two space dimensions. In addition, we address the special case of the Cahn-Hilliard models and show how the grid size has to be chosen as a function of the small model parameter in order to yield reliable results.

11:00 am Friday, December 2, 2005

Stochastic Seminar: A Coupling, and the Darling-Erdos Conjectures

by Davar Khoshnevisan (Department of Mathematics, University of Utah) in Skiles 269 (NOTE different day and time)

We present a coupling of the 1-dimensional Ornstein-Uhlenbeck process with an i.i.d. sequence. We then apply this coupling to resolve two conjectures of Darling and Erdos (1956). Interestingly enough, we prove one and disprove the other conjecture. [This is joint work with David Levin.] Time-permitting, we may use the ideas of this talk to describe precisely the rate of convergence in the classical law of the iterated logarithm of Khintchine for Brownian motion (1933). [This portion is joint work with David Levin and Zhan Shi, and has recently appeared in the Electr. Comm. of Probab. (2005)].

1:00 pm Friday, December 2, 2005

Theory Seminar: Probabilistic Inference Heuristics for Satisfiability

by Elitza Maneva (University of California, Berkeley) in CCB 102

The known NP-hardness results imply that for most combinatorial optimization problems there are no efficient algorithms that find an optimal, or even a near optimal solution, on every instance. A heuristic for an NP-hard problem is a polynomial time algorithm that produces such solutions on some input instances, but may fail on others. One of the existing methods for evaluating heuristics is to study their performance on inputs coming from a particular distribution. I will talk about heuristics based on probabilistic inference, which have recently beenapplied to Boolean satisfiability problems, and exhibit unprecedented success over the uniform distribution on formulas with fixed ratio of clauses to variables. I will show how intuition about the structure of the space of solutions of such formulas influences the design of these heuristics.

Lunch at 1pm, seminar begins shortly afterwards

2:00 pm Friday, December 2, 2005

Special Teaching Seminar: Internet-Based Multivariable Calculus and Geometry

by Thomas Banchoff (Department of Mathematics, Brown University) in Skiles 255

New Internet-based software provides accessible Java demonstrations for classroom presentation, as well as courseware for communication between teacher and student and among students. A project next semester at several universities in the greater Atlanta area will be using software developed at Brown University for teaching multivariable calculus and geometry at several levels. We are hoping to find even more participants in our NSF supported experiment.

2:00 pm Friday, December 2, 2005

Stochastic Seminar: Some Curious Results on Randomly Weighted Self-normalized Sums

by David Mason (Department of Food and Resource Ecnomics, University of Delaware) in Skiles 269 (NOTE different time and day)

We determine exactly when a certain randomly weighted self-normlized sum converges in distribution, verifying a 1965 conjecture of Leo Breiman, and then apply our results to characterize the asymptotic distribution of relative sums and to provide a short proof of a 1973 conjecture of Logan, Mallows, Rice and Shepp on the asymptotic distribution of self--normalized sums in the case of symmetry. We shall also describe the cluster sets of a special case of the these randomly weighted self--normalized sums in the domain of attraction of a stable law case. The first part of this talk is based on joint work with Joel Zinn.

4:00 pm Friday, December 2, 2005

General Colloquium: The Fourth Dimension and the Internet

by Thomas Banchoff (Department of Mathematics, Brown University) in DM Smith 105

New Internet technology affords great advances in visualization and communication of geometry for research and teaching, in particular for surfaces in the fourth dimension. The presentation will feature connections with the classic "Flatland" and the paintings of Salvador Dali.

This talk is aimed at a general audience specially the undergraduate students, whom are strongly encouraged to attend.

2:00 pm Monday, December 5, 2005

Theory Seminar: Counting Down the Tree

by Dror Weitz (Mathematics, Institute for Advanced Study) in MiRC 102

We present a novel tree representation for the hard-core lattice gas model (independent sets) on a general graph. We use this representation to show that for any graph of maximum degree \Delta, the Gibbs measure is unique (the influence of any boundary condition decays with distance) provided that the activity parameter \lambda < \lambda_c, where \lambda_c is the critical activity for the regular tree of degree \Delta. This resolves an open conjecture in statistical physics. Also, since \lambda_c is known, this extends the known uniqueness regime for many interesting graphs, including the square integer lattice Z^2. Our proof is algorithmic in nature, consisting of an elegant recursive procedure for calculating the probabilities that a given vertex is occupied. This procedure yields an efficient deterministic approximation scheme for counting independent sets (in other words, for calculating the partition function) of any graph of maximum degree \Delta in the above regime of \lambda. This extends the regime of \lambda for which an efficient approximation scheme is known to exist, and includes the interesting case of \lambda=1 (all independent sets are equally weighted) and maximum degree \Delta=5.

3:30 pm Monday, December 5, 2005

Algebra/Geometry Seminar: Homological Mirror Symmetry and Birational Geometry

by Ludmil Katzarkov (University of Miami) in Skiles 269

In this talk we will introduce homological mirror symmetry for manifolds of general type. We will discuss some applications to Birational Geomdtry and symplectic topology.

12:00 pm Wednesday, December 7, 2005

Applied and Computational Mathematics Seminar: A finite element method and solvers for a Stokes interface problem

by Maxim Olshanskii (Emory) in Skiles 269

In many numerical simulations of two-phase flows a so-called one-fluid approach is used. In such a method the two phases are modelled by a single set of conservation laws for the entire flow field; and the forces at the interface (e.g. surface tension) are treated as a part of the model. The differences in the material properties lead to jumps in the coefficients in these conservation laws. In the case of viscous incompressible fluids this leads to a Stokes type of problem with discontinuous viscosity and density coefficients. In the talk we address the stability, finite element error analysis and an iterative solver for the Stokes type problem with discontinuous viscosity and density.

12:00 pm Thursday, December 8, 2005

Applied and Computational Mathematics Seminar: New high-order, high-frequency methods in computational l electromagnetism

by Oscar Bruno (Caltech) in Skiles 269

We present a new set of algorithms and methodologies for the numerical solution of problems of scattering by complex bodies in three-dimensional space. These methods, which are based on integral equations, high-order integration, fast Fourier transforms and highly accurate high-frequency methods, can be used in the solution of problems of electromagnetic and acoustic scattering by surfaces and penetrable scatterers --- even in cases in which the scatterers contain geometric singularities such as corners and edges. In all cases the solvers exhibit high-order convergence, they run on low memories and reduced operation counts, and they result in solutions with a high degree of accuracy. In particular, our algorithms can evaluate accurately in a personal computer scattering from hundred-wavelength-long objects by direct solution of integral equations --- a goal, otherwise achievable today only by supercomputing. A new class of high-order surface representation methods will be discussed, which allows for accurate high-order description of surfaces from a given CAD representation. A class of high-order high-frequency methods which we developed recently, finally, are efficient where our direct methods become costly, thus leading to a general and accurate computational methodology which is applicable and accurate for the whole range of frequencies in the electromagnetic spectrum.

3:05 pm Thursday, December 8, 2005

Graph Theory: Remarks on Hajos' conjecture

by Csaba Biro [mail] (Math, GT) in Skiles 255

The speaker will present a recent paper by Carsten Thomassen. Hajos' conjecture says that every graph of chromatic number k contains a subdivision of the complete graph on k vertices. The conjecture is true for k<=4, open for k=5,6 and known to be false for all k>=7. We will show several classes of counterexamples that relates the conjecture to Ramsey Theory, perfect graphs and the maximum cut problem.

4:00 pm Thursday, December 8, 2005

Math Department Tea:

in Skiles 236 (lounge)

All math department staff, students, and faculty are invited. Drinks and food will be served.

3:05 pm Friday, December 9, 2005

Combinatorics Seminar: Introduction to the Caccetta-Haggkvist conjecture

by Paul Seymour (Princeton University) in Skiles 255

This conjecture is a well-known open problem dating from 1978. In its simplest form, it asserts that for any integer k>0, if G is a directed graph without parallel edges, and every vertex has outdegree at least |V(G)|/k, then there is a directed cycle of length at most k. This is easy for k = 1,2, but for k = 3 is still open. This talk will be a survey of the problem; related conjectures, partial results, approaches and counterexamples.

3:15 pm Monday, December 12, 2005

Algebra-Geometry-Topology Seminar: Horizontal open books on plumbings

by Burak Ozbagci (Georgia Tech and Koc University) in University of Georgia Math Departement, Boyd Room 303

I will start with an introduction to Giroux's correspondence between open books and contact structures on 3-manifolds, by giving some elementary examples. Then I will describe a construction of horizontal open books on oriented circle bundles (with negative Euler number) over oriented closed surfaces and on plumbings of such circle bundles. It turns out the contact structures carried by these open books are horizontal as well. Finally, I will talk about some applications of the construction. (This is a joint work with Tolga Etgu : http://front.math.ucdavis.edu/math.GT/0509611 )

4:30 pm Monday, December 12, 2005

Algebra-Geometry-Topology Seminar: Tight contact structures with trivial Ozsvath-Szabo invariants

by Paolo Ghiggini (University of Quebec, Montreal) in University of Georgia Math Departement, Boyd Room 303

Tight contact structures with trivial Ozsvath-Szabo invariants Abstract: We show that infinitely Seifert manifolds over S^2 with 4 singular fibres carry infinitely many universally tight contact structures with trivial Ozsvath-Szabo contact invariants. We obtain as a corollary that these contact structures are non weakly symplectically fillable.

3:30 pm Thursday, December 15, 2005

Job Candidate Talk: Sweeping Methods for Static Hamilton-Jacobi Equations and its Applications

by Chiu-Yen Kao (University of Minnesota, IMA) in Skiles TBA

Hamilton-Jacobi equations arise in many applications such as geometrical optics, crystal growth, path planning, seismology, and visibility problems. Viscosity solutions of these nonlinear differential equations usually develop singularities in their derivatives. In this talk, sweeping methods are introduced to approximate the viscosity solution of static Hamilton-Jacobi equations in any number of spatial dimensions. The value of a specific grid point are solved in terms of its neighbors, so that a Gauss-Seidel type nonlinear iterative method can be utilized. Furthermore, by incorporating a group-wise causality principle into the Gauss-Seidel iteration by following a finite group of characteristics, we have easy-to-implement, sweeping-type, and fast convergent numerical methods. Some convergence analysis is done for eikonal equations. Extensive 2-D and 3-D numerical examples for several applications, including crystal growth, seismology, and white matter tractography, illustrate the efficiency and accuracy of the fast sweeping methods.

11:00 am Monday, January 9, 2006

Theory Seminar: Playing the k-armed bandit with adaptive payouts

by Tom Hayes (UC Berkeley) in MiRC 102

The "k-armed bandit" is a modified slot machine, which has k arms you can pull, rather than just one. Each arm has a different payout, which can change every time you play, but is always in the range [0,1]. If we get to play this machine T times, how well can we do, compared to the best single arm in hindsight? Specifically, the goal is to minimize "regret", defined as the random variable Payout(algorithm) minus Payout(best arm). The character of this problem depends strongly on how much information is given to the algorithm. In the "full information" version of the game, the algorithm is told the payout for all k arms after each round. This version of the problem is well-understood, and randomized algorithms are known which, with high probability, guarantee regret O(\sqrt{T log k}), which is optimal. In the "bandit" version of the game, the algorithm is only told the payout for the arm it chose in the last round. This setting is harder, with a lower bound of Omega(\sqrt{T k}), and previously best upper bound O(T^{2/3}) (for k fixed).We present a new randomized algorithm for the bandit version, which achieves regret O(\sqrt{T k log k}) with high probability. This is joint work with Varsha Dani.

12:00 pm Thursday, January 12, 2006

Applied and Computational Mathematics Seminar: Recent results for the 3-body problem

by Gianni Arioli (Milan) in Skiles 269

The 3-body problem has been a subject of study for a very long time, but in recent years it has received a particular attention, thanks to new developments in the field of variational methods and computer assisted proofs. In this talk we describe some of the most recent results in this field.

4:05 pm Thursday, January 12, 2006

Combinatorics: Extensions of Graph Pebbling

by Carl Yerger (Cambridge University and Math, Georgia Tech) in Skiles 255

Mathematicians have analyzed combinatorial games on graphs such as peg solitare and checker jumping. Graph pebbling, which has applications to additive number theory, is another one of these games. In this talk, recently created extensions of pebbling, including cover pebbling, domination cover pebbling and deep graph pebbling will be discussed. Structural and probablistic aspects of these problems will be examined. Also, the complexity of the cover pebbling and domination cover pebbling decision problems will be analyzed. This talk is accessible to an undergraduate audience.

4:30 pm Tuesday, January 17, 2006

Job Candidate Talk: Hyperbolic systems of conservation laws and their approximations

by Alberto Bressan (Penn State) in Skiles 255

The talk will present a survey of basic techniques and recent results in the theory of hyperbolic systems of conservation laws. Even for smooth initial data, solutions to nonlinear conservation laws can develop shocks in finite time. This lack of regularity prevents the use of many classical tools of analysis. Recent research has shown that front tracking, vanishing viscosity and semidiscrete approximations preserve a uniform bound on the total variation of the solutions. All these approximations converge to a unique limit, depending continuously on the initial data in the L^1 norm. On the other hand, fully discrete numerical schemes can generate an arbitrary large amount of oscillations. For general hyperbolic systems, the convergence of these numerical schemes remains an open problem.

4:00 pm Wednesday, January 18, 2006

Analysis Seminar: Approximation by homogeneous polynomials

by David Benko (Western Kentucky University) in Skiles 269

Let K be a convex origin symmetric surface in R^d. Kroo conjectures that any continuous function on K can be uniformly approximated by a sum of two homogeneous polynomials. Using potential theory and weighted polynomials we resolve this problem on the plane. In higher dimensions the conjecture is still open.

12:00 pm Thursday, January 19, 2006

Applied and Computational Mathematics Seminar: Crossing bifurcations and unstable dimension variability

by Evelyn Sander (George Mason) in Skiles 269

A crisis is a global bifurcation in which a chaotic attractor has a discontinuous change in size or suddenly disappears as a scalar parameter of the system is varied. In this talk, I describe a global bifurcation in three dimensions which can result in a crisis. This bifurcation does not involve a tangency and cannot occur in maps of dimension smaller than three. The crisis produces unstable dimension variability, a type of non-hyperbolic behavior which results in a breakdown of shadowing.

3:00 pm Thursday, January 19, 2006

Stochastic Seminar: Fourier and wavelet transforms of stationary processes

by Wei Biao Wu (Statistics Department, University of Chicago) in Skiles 269

I will discuss Fourier and wavelet transforms of stationary, causal processes. Under mild conditions, Fourier transforms are shown to be asymptotically independent complex normal at different frequencies. For nonlinear time series, asymptotic properties of spectral density estimators will be discussed. I will also address the issue of what is dependence, which plays a fundamental problem in the study of random processes. By introducing a new input/output dependence structure, a strong invariance principle is established with nearly optimal rates and is applied to wavelet transforms of stationary processes.

3:00 pm Thursday, January 19, 2006

Job Candidate: Efficient Numerical Methods for Time-Harmonic Scattering Problems and Applications

by Jari Antero Toivanen (NCSU) in Skiles 255

Numerical methods are described for the solution of exterior 2D and 3D Helmholtz equations. The domains are truncated and absorbing boundary conditions are posed on the truncation boundary. A finite element discretization lead to very large systems of linear equations which are solved using the GMRES method with a domain imbedding/domain decomposition preconditioner. Numerical experiments demonstrate ability to solve resulting linear systems with billions of unknowns. As applications we consider electromagnetic backscattering by airfoils. Also the computation of acoustic scattering by an object in sediment is studied.

1:00 pm Friday, January 20, 2006

Theory Seminar: Asynchronous Pattern Matching - Metrics

by Amihood Amir (Bar-Ilan University) in MiRC 102

Traditional Approximate Pattern Matching (e.g. Hamming distance errors, edit distance errors) assumes that various types of errors may occur to the data, but an implicit assumption is that the order of the data remains unchanged. Over the years, some applications identified types of "errors" were the data remains correct but its order is compromised. The earliest example is the "swap" error motivated by a common typing error. Other widely known examples such as transpositions, reversals and interchanges are motivated by biology. We propose that it is time to formally split the concept of "errors in data" and "errors in address" since they present different algorithmic challenges solved by different techniques. The "errors in address" model, which we call asynchronous pattern matching, since the data does not arrive in a synchronous sequential manner, is rich in problems not addresses hitherto. We will consider some reasonable metrics for asynchronous pattern matching, such as the number of inversions, or the number of generalized swaps, and show some efficient algorithms for these problems. As expected, the techniques needed to solve the problems are not taken from the standard pattern matching "toolkit". (Joint work with Y. Aumann, G. Benson, A. Levy, O. Lipsky, E. Porat, S. Skienna and U. Vishne.)

3:05 pm Friday, January 20, 2006

Combinatorics: Cheeger inequalities for eigenvalues of non-reversible random walks

by Ravi Montenegro (Mathematics, UMASS, Lowell) in Skiles 255

A Cheeger inequality relates a geometric quantity to the size of eigenvalues of a transition probability matrix. Jerrum and Sinclair showed such an inequality for reversible Markov chains, while Houdre, Tetali and others refined this to a bound on the 'spectral gap'. We show that the spectral gap can be used to bound eigenvalues of non-reversible Markov chains, leading to a simple proof of a recent result of Chung, and an extension of results of Alon, Stoyanov and others to bounds on eigenvalues of non-reversible walks.

3:00 pm Monday, January 23, 2006

Job Candidate Talk: Thermally-Driven Rare Events and Action Minimization

by Maria G. Reznikoff (Princeton University) in Skiles 255

Thermal or stochastic effects are prevalent in physical, chemical, and biological systems. Particularly in small systems, noise can overpower the deterministic dynamics and lead to "rare events," events which would never be seen in the absence of noise. One example is the thermally-driven switching of the magnetization in small memory elements. Wentzell-Freidlin large deviation theory is a mathematical tool for studying rare events. It estimates their probability and also the "most likely switching pathway," which is the pathway in phase space by which rare events are most likely to occur. We explain how large deviation theory and concepts from stochastic resonance may be applied to analyze thermally-activated magnetization reversal in the context of the spatially uniform Landau-Lifschitz-Gilbert equations. The time-scales of the experiment are critical. One surprising and physically relevant result is that in multiple-pulse experiments, nonconvential "short-time switching pathways" can dominate. The effect is dramatic: the usual pathway (connected with the Arrhenius-law) underestimates the probability of switching by an exponential factor.An advantage of the method via large deviation theory is that it generalizes to systems with spatial variation. To discuss the complications and richness that emerge when spatial variation is taken into account, we consider the (simpler) Allen-Cahn equation. In this context, the rare event of interest is phase transformation from u\equiv -1 to u\equiv +1, and the most likely switching pathway is a pathway through function space. A natural reduced problem emerges in the "sharp-interface limit." We give a brief overview of some results (rigorous in d=1, heuristic in d>1). The first part of the talk is joint work with Bob Kohn and Eric Vanden-Eijnden. The second part includes work that is also joint with Felix Otto and Yoshihiro Tonegawa.

3:05 pm Monday, January 23, 2006

Combinatorics: On the Caccetta-Haggkvist Conjecture

by Jian Shen [mail] (Texas State University-San Marcos) in Skiles 149

In 1978, Caccetta and Haggkvist conjectured that any digraph with n vertices and minimum out-degree r contains a directed cycle of length at most the ceiling of n/r. An interesting special case of the conjecture claims that any digraph with n vertices and minimum out-degree at least n/3 contains a directed triangle. Even this special case of the conjecture remains open now. We survey some recent approaches and results on the Caccetta-Haggkvist conjecture. We show that, for each given r, the number of counterexamples to the conjecture, if any, is finite. If time permits, we will discuss some other related conjectures.

3:15 pm Monday, January 23, 2006

Algebra-Geometry-Topology Seminar: Existence of Engel structures

by Thomas Vogel (University of Pennsylvania and IAS) in Skiles 269

Engel structures are non-integrable plane fields on 4-manifolds who share many properties with contact structures. The existence of an Engel structure on a manifolds leads to strong restrictions on the topology of the manifold: Under certain orientation assumptions the tangent bundle of the manifold is trivial. In this talk we develop a construction which shows that the converse is also true: Every 4-manifold with trivial tangent bundle admits an Engel structure.

4:30 pm Monday, January 23, 2006

Algebra-Geometry-Topology Seminar: A survey of some smooth concordance invariants

by Matthew Hedden (Princeton) in Skiles 269

In the past few years, several powerful smooth knot concordance invariants have been discovered. Perhaps most notable are the invariants $\tau(K)$ and $s(K)$, both of whose values for the $(p,q)$ torus knots provide new proofs of Milnor's famous conjecture on the unknotting number of these knots. $\tau(K)$ was discovered by Ozsv{\'a}th-Szab{\'o}, and independently by Rasmussen, and its definition relies on the analytically defined knot Floer homology theory developed by these authors. $s(K)$, on the other hand, was discovered by Rasmussen and its definition is in terms of the combinatorial knot homology theory of Khovanov. Though quite different in their definition, the two invariants share several formal properties, and agree for many knots. Indeed, it was conjectured by Rasmussen that the two invariants are equal, up to normalization. In this talk I will survey what is known about the two invariants, and discuss some of my recent results regarding the invariant $\tau$. I will conclude by presenting the first known examples where the invariants disagree, discovered jointly with Philip Ording of Columbia University.

11:00 am Tuesday, January 24, 2006

ISyE Seminar: Natural gas value chain optimization using stochastic integer programming

by Asgeir Tomasgard (Norwegian University of Science and Technology) in Main ISyE Building, Rm. 228

We will present a value chain management model for liberalized natural gas markets. The model describes the integrated planning problem of a natural gas producer including production, transportation, storage and contract management. The decision horizon is normally between 18 months and three years. The capacities of the production fields and the transportation pipeline system limits the gas volumes delivered in the downstream markets. Demand is represented by contracts and spot markets. Contract volumes as well as spot prices are uncertain. The model is implemented as a mixed integer multistage stochastic model. Scenario trees are generated by moment matching. Preliminary results from pilot tests on Statoil's operations on the Norwegian continental shelf will be presented.

4:30 pm Tuesday, January 24, 2006

PDE Seminar: Variational Systems of Nonlinear Wave Equations

by John Hunter (UC Davis) in Skiles 255

We will consider systems of wave equations that are governed by variational principles whose Lagrangians are quadratic functions of the derivatives of the wave-field with coefficients depending on the wave-field itself. Examples include the nonlinear scalar wave equation \[ u_&ob;tt&cb; = c^2(u)\Delta u + c(u)c^\prime(u) |\nabla u|^2, \] and equations that describe the propagation of orientation waves in a massive liquid crystal director field. The vacuum Einstein equations of general relativity also belong to this class, after the imposition of a suitable gauge. The effect of nonlinearity in such hyperbolic systems of wave equations differs considerably from its effect in the much more extensively studied hyperbolic systems of conservation laws, and we will compare the two classes. In particular, we will introduce notions of genuine nonlinearity and linear degeneracy for variational wave equations that are analogous to, but different from, the corresponding notions for hyperbolic systems of conservation laws. We will derive a new weakly nonlinear asymptotic equation for waves that lose genuine nonlinearity in a system, and use it to study the propagation of `twist' waves in a massive director field. As we will show, the `twist' waves couple with genuinely nonlinear `splay' waves, whose wave-amplitude satisfies the Hunter-Saxton equation. If time permits, we will discuss the linear degeneracy of the Einstein equations, as variational wave equations, and describe some asymptotic equations for large-amplitude gravitational waves. This work is joint with Giuseppe Al\`\i.

9:00 am Wednesday, January 25, 2006

Job Candidate Talk: Optimal Sobolev regularity of a class of Fourier integral operators

3:00 pm Wednesday, January 25, 2006

Research Horizons: Algorithms and Complexity in Real Algebraic Geometry.

by Prof. Saugata Basu (GaTech) in Skiles 255

I will give a brief survey of recent developments in the area of algorithmic real algebraic geometry, and describe the main open questions. No background in real algebraic geometry will be assumed.

4:30 pm Thursday, January 26, 2006

Colloquium: Time-Frequency Representations and Statistical Models for Audio and Auditory Signal Analysis

by Patrick J. Wolfe (Division of Engineering and Applied Sciences, Harvard University) in Skiles 269

Time-frequency representations are ubiquitous in audio signal processing, their use being motivated by both auditory physiology and the mathematics of Fourier analysis. Indeed, information-carrying natural sound signals can often be meaningfully represented as a superposition of translated, modulated versions of a simple window function exhibiting good time-frequency concentration. In combination with statistical models formulated in the space of time-frequency coefficients, such an approach provides a principled way of decomposing sounds into their constituent parts, as well as an effective means of exploiting the local correlation present in the time-frequency structure of natural sound signals. Indeed, the structures of such time series lend themselves to natural prior specifications in terms of time-frequency behavior, via a decomposition according to the principles of Gabor analysis over finite cyclic groups. Several resultant applications will be addressed in this talk, including examples of speech and audio signal analysis, enhancement, and compression, as well as a brief discussion of the human peripheral auditory system from the point of view of time-frequency analysis.

12:00 pm Friday, January 27, 2006

Applied and Computational Mathematics Seminar: Augmented Techniques for Interface/Irregular Domain

by Zhilin Li (NC State U) in Skiles 269

In order to solve some interface/irregular domain problems, it is advantageous to use an augmented technique in which one or several intermediate variables called augmented variables are introduced. With these augmented variables, it is easier to discretize the partial differential equation. More important, fast solvers that are designed for regular problems or regular domains can be used. I will explain this technique with examples of elliptic interface problems with large jumps in the coefficient; fast Poisson/biharmonic solvers on irregular domains (interior or exterior); and two phase Stokes flows with large jump in the viscosity. For the last example, new jump conditions of the pressure and the velocity are coupled together. The augmented approach seems to be the only way to solve the problem accurately.

2:00 pm Friday, January 27, 2006

Geometry/Topology seminar: Quantum sl_2 invariants of rational homology spheres

by Prof. Pierre Vogel [mail] (Universite Paris VII) in Skiles 269

The quantum sl2 invariants of a homology sphere $M$ can be lifted in an invariant leaving in a suitable completion of the ring $R=[\bf Z&cb;[q^\pm]$ [Habiro]. If $M$ is a rational homology sphere the situation is completely different. I conjecture in this case that the quantum sl2 invariants of $M$ can be lifted in finitely many invariants leaving each of them in some completion of an algebraic extension of $R$. I prove this conjecture in particular in the case of lens spaces and manifold obtained by surgery on a knot.

3:05 pm Friday, January 27, 2006

Job Candidate Talk: Probabilistic reasoning and Ramsey Theory

by Benny Sudakov (Department of Mathematics, Princeton University) in Skiles 269

"Ramsey Theory" refers to a large body of deep results in mathematics concerning the partition of large collections. Its underlying philosophy is captured succinctly by the statement that "In a large system complete disorder is impossible". Since the publication of the seminal paper of Ramsey in 1930, this subject has grown with increasing vitality, and is currently among the most active areas in Combinatorics. One of the most important factors in the development of Ramsey Theory was the successful application of the so-called "Probabilistic Method". This method was initiated more than fifty years ago by Paul Erdos, and became one of the most powerful and widely used methods in Discrete Mathematics. In this talk I will describe some classical results of Ramsey Theory together with recent progress on some old questions of Erdos which was made using probabilistic arguments. I will also discuss the problem of converting existence arguments into deterministic constructions, in particular, the recent explicit constructions of Bipartite Ramsey graphs.

11:00 am Monday, January 30, 2006

Algebra-Geometry-Topology Seminar: The complexity of Positivstellensatz is elementary recursive

by Marie-Francoise Roy (Universite de Rennes 1, France) in Skiles 269 (Note different time)

The aim of the talk is to explain how to construct an algebraic certificate for the emptyness of a basic semi-algebraic set, keeping track of the degrees along the construction. The proof uses the algebraic proof by Laplace and Gauss of the fundamental theorem of algebra, and Hermite real root counting method. This is joint work with Henri Lombardi.

12:00 pm Monday, January 30, 2006

Theory Seminar: Implementation with a bounded action space

by Liad Blumrosen (Hebrew University of Jerusalem) in College of Computing, Room 109

The field of Mechanism Design considers the design of algorithms in environments where the input for the algorithms is provided by self-interested agents. Such agents may manipulate the algorithm for their own benefit. For achieving the desired system-wide outcomes, the algorithm designer should design a protocol where each agent will have the incentive to choose an action that truthfully reveals her private information (her "type"). Handling the incentive issues must be taken together with other constraints, for example, with computational or communication constraints. While traditional mechanism design typically assumes isomorphism between the agents' type- and action spaces, in many situations the agents face strict restrictions on their action space. These restrictions may be caused by, e.g., technical, behavioral or regulatory reasons. Under this restriction, a "direct revelation" of the players' private information will no longer be feasible. We devise a general framework for the study of mechanism design in single-parameter environments with restricted action spaces. We characterize settings where the information-theoretically optimal solutions are dominant-strategy implementable, and we measure the loss in these optimal mechanisms. Our results apply to various economic and computational settings, and we demonstrate their applicability to signaling games, public-good models and routing in networks. Joint work with Michal Feldman

2:00 pm Monday, January 30, 2006

Theory Seminar: Learning under the uniform distribution: Toward DNF

by Ryan O'Donnell (Microsoft Research) in MiRC 102

In this talk we will survey some recent work on the problem of learning under the uniform distribution. In this problem, there is an unknown function f : {0,1}^n -> {0,1}. The learning algorithm has access to random "examples" from the function of the form < x, f(x) >, where x is drawn uniformly at random. The algorithm's job is to output a "hypothesis" h : {0,1}^n -> {0,1} which is epsilon-close to f. Our main concern is the time-efficiency of the algorithm. The "holy grail" would be to give an algorithm taking polynomial time for any function f with a polynomial-sized DNF formula. We will discuss some algorithmic attacks on the problem, and also discuss why complete success may prove an elusive goal...

3:00 pm Monday, January 30, 2006

Physics Colloquium Job Candidate: Tilings, Dimers, and Quiver Gauge Theories

by Amihay Hanany (MIT Center for Theoretical Physics) in Howey lecture room 5

This lecture is an introduction into the recent progress in the study of the duality between string theory and gauge theory, based on the dynamics of string theory branes. We have uncovered new links between "dimers" of statistical mechanics and combinatorics, and "quiver gauge theories" which describe the dynamics of branes in string theory. The new insights have lead to a discovery of a large new class of quantum field theories, the only known examples of theories that are minimally supersymmetric, scale invariant, and 3+1 dimensionsional. Furthermore, there is an intimate connection with problems in algebraic geometry which will be touched upon during the talk. The recent progress offers new tools for analyzing and solving supersymmetric gauge theories, and a deeper understanding of their relation to string theories.

*THE RECEPTION WILL BE AT 2:30 PM IN ROOM N201*

4:30 pm Monday, January 30, 2006

CDSNS: An algorithmic approach to Conley's Decomposition Theorem

by William D. Kalies [mail] (Florida Atlantic University) in Skiles 255

In this talk, we will describe algorithms and computational issues for approximating the chain recurrent set of a dynamical system and a corresponding Lyapunov function.

9:00 am Tuesday, January 31, 2006

Research Horizons: Frames, Time-Frequency Analysis, and Wavelets

by Christopher Heill (GaTech) in Skiles 255

4:40 pm Tuesday, January 31, 2006

PDE Seminar : On Monge-Amp�re Type Equations Arising In Optimal Transportation Problems

by Truyen Nguyen (Georgia Tech) in Skiles 255

In this talk, we will discuss Monge-Amp�re type equations arising in optimal transportation problems. We prove the comparison principle, maximum principle and also a quantitative estimate of Aleksandrov type for c-convex functions. These results are in turn used to prove the solvability and uniqueness of weak solutions for the Dirichlet problems. This is a joint work with Cristian Gutierrez.

3:00 pm Wednesday, February 1, 2006

Research Horizons: Mathematical verification of Darcy law.

by Prof. Ronghua Pan (GaTech) in Skiles 255

Darcy law is very important in characterization of the motion of waves in porous medium. It has been verified by experiments in physical point of view. I will explain how to verify Darcy law in mathematical point of view. Little PDE background is needed.

4:35 pm Wednesday, February 1, 2006

Combinatorics: Counting Unions of Combinatorial Structures

by Graham Brightwell (London School of Economics, U.K.) in Skiles 255 (NOTE: Different time and day)

Given n and k, how many different graphs with vertex set [n] can be built as a union of cliques of size k? If k is small, then most graphs can be built this way; if k is large then this is a very special property and the number of such graphs will be much smaller than 2^{n(n-1)/2}, but how small? Another question of a similar flavour is this: given n and d, how many subsets of the discrete n-cube can be expressed as a union of d-dimensional subcubes? This question can be seen to be equivalent to asking about the number of Boolean functions of n variables that can be expressed using a k-SAT formula.

9:00 am Thursday, February 2, 2006

Computational Homology and Materials Science Workshop: February 2-4

by Konstantin Mischaikow in Georgia Tech Conference Center

This workshop is intended to bring together a select group of materials scientists, physicists, and applied mathematicians with an interest in quantifying the topology of microstructures and relating it to macroscopic properties of materials. In addition to providing a forum for discussing current and future applications of topological techniques, the workshop includes tutorial lectures on computational homology and the publicly available software package CHomP. Details are at http://www.math.gatech.edu/news/conferences/chomp/

3:00 pm Thursday, February 2, 2006

Stochastics Seminar: An aggregation procedure in classification. Application to adaptivity

by Guillaume Lecue (Paris VI) in Skiles 269

We consider the problem of adaptation to the margin and to complexity in binary classification. We suggest a learning method with a numerically easy aggregation step. Adaptivity both to the margin and complexity in classification, usually involves empirical risk minimization or Rademacher complexities which lead to numerical difficulties. On the other hand there exist classifiers that are easy to compute and that converge with fast rates but are not adaptive. Combining these classifiers by our aggregation procedure we get numerically realizable adaptive classifiers that converge with fast rates.

4:30 pm Thursday, February 2, 2006

Math Department Tea:

in Math Lounge (Skiles 236)

All math department staff, students, and faculty are welcome to attend. Food and beverages will be served.

3:30 pm Friday, February 3, 2006

Job Candidate Talk: Optimal Sobolev regularity of a class of Fourier integral operators

by Malabika Pramanik (CalTech) in Skiles 255

We discuss a class of Fourier integral operators that arises from averaging, and derive sharp $L^p$-Sobolev regularity properties of these operators for large values of $p$. Our results make use of a deep estimate of Wolff associated to light cones. This is joint work with Andreas Seeger.

2:00 pm Monday, February 6, 2006

Job Candidate Talk: Towards High Order Approximation to Hyperbolic Conservation Laws

by Wen Shen (Penn State) in Skiles 269

Numerical computation for hyperbolic conservation laws is an active and challenging research field. In this talk we propose a new approach, based on vanishing viscosity approximation. We add a small viscosity, and compute the solution with various viscosities, and extrapolate at the end to achieve high order. Various aspects of this approach will be discussed. One offspring from it is the optimal tracing of viscous shocks in solutions for scalar conservation laws. We introduce a nonlinear functional whose minimizers yield the viscous traveling profiles which "optimally fit" the given solution. We prove that, outside an initial time interval and away from times of shock interactions, the solution can be accurately represented by a finite number of viscous traveling waves. Further research plan within this project will be given at the end of the talk.

3:30 pm Monday, February 6, 2006

Algebra-Geometry-Topology Seminar: Higher-order Alexander invariants of plane algebraic curves

by Connie Leidy (University of Pennsylvania) in Skiles 269

We will discuss some new invariants of algebraic planar curves, called higher-order Alexander modules and their associated degrees. These are obtained from analyzing the module structure of the homology of certain solvable covers of the complement of the curve. In particular, the invariants take advantage of the non-commutativity of the fundamental group of a general curve complement. These invariants are in the spirit of those developed by T. Cochran and S. Harvey, and which were used to study knots, 3-manifolds, and finitely presented groups, respectively.

4:00 pm Monday, February 6, 2006

Job Candidate Stochastic Seminar: Malliavin calculus and gradient estimates

by Tai Melcher in Skiles 249

We will discuss the existence of "Lp-type" gradient estimates for second order differential operators. For certain values of p, these estimates imply logarithmic Sobolev and Poincare inequalities for the associated heat kernel. These inequalities have broad application and have been studied and used extensively in the fields of mathematical physics, global analysis, and geometry. "Lp-type" gradient estimates are known for elliptic operators, but the method for obtaining them fails when the operator is not elliptic. In particular, such estimates have been unknown for hypoelliptic operators. Malliavin calculus methods transfer the problem to one of determining certain infinite dimensional estimates. Malliavin calculus is a differential calculus on infinite-dimensional path space. In this talk, I will give a brief introduction to Malliavin calculus, and use it to show that "Lp-type" gradient estimates hold under some conditions.

4:30 pm Monday, February 6, 2006

CDSNS: Symmetry properties of positive solutions of parabolic equations

by Peter Polacik [mail] (University of Minnesota) in Skiles 255

In this talk, we shall examine symmetry properties of positive solutions of nonautonomous parabolic problems on bounded domains and on $R^N$. After an overview of symmetry theorems, we shall report on recent progress and discuss some new techniques.

1:30 pm Wednesday, February 8, 2006

Job Candidate: Localization and Poisson statistics in random band matrices

by Jeffrey Schenker (IAS Princeton) in Skiles 269

It is conjectured that an nxn symmetric random band matrix with independent identically distributed entries in a band of width w around the diagonal exhibits ``localization'' in the large n limit provided $w = o(\sqrt{n})$. In this talk, a proof of this phenomenon for $w = o(n^{1/5})$ will be discussed. In this context ``localization'' indicates that typical eigenfunctions have exponentially small overlap with all but a vanishing fraction of basis vectors, which is associated with Poisson statistics for local eigenvalue spacings.

3:00 pm Wednesday, February 8, 2006

Research Horizons: Time series for the environment.

by Serge Guillas (GaTech) in Skiles 255

Time series analysis can be used to address environmental issues. Several case studies are discussed in this talk: ground-level pollution (ozone and sulfur dioxide), stratospheric ozone and climate change. First, we introduce the theory of time series, including Hilbert-valued ones. Second, we detail how statistical forecasting works. Third, we present open scientific problems, such as the detection of trends, that may be solved through a careful study of uncertainties: standard regression and Bootstrap methods.

12:00 pm Thursday, February 9, 2006

Job Candidate. Applied and Computational Mathematics Seminar: Modeling crystal surface evolution: From microscopic schemes to continuum laws

by Dionisios Margetis (Department of Mathematics, M.I.T.) in Skiles 269

In traditional settings such as fluids and classical elasticity the starting point (truth) is identified with continuum equations. But in many practical cases of modeling this perspective is changed: The truth is atomistic, or takes the form of discrete schemes, by which macroscopic laws must be found. I will talk on the evolution of crystal surfaces as a prototypical case of coupling between scales, with implications in the design of novel devices. The governing, discrete equations represent the motion of interacting line defects, atomic ``steps''. In the continuum limit: (i) I derive a nonlinear PDE for the surface height; and (ii) I show analytically how microscopic details enter requisite boundary conditions, and thus affect macroscopic evolution.

3:05 pm Thursday, February 9, 2006

Graph Theory: Testing branch-width

by Sang-il Oum (Math) in Skiles 255

An integer-valued function f on the set of subsets of a finite set V is a connectivity function if it satisfies (1) f(X)+f(Y)>= f(X\cap Y)+f(X\cup Y)$ for all subsets X, Y of V, (2) f(X)=f(V-X)$ for all subsets X of V, (3) f(\emptyset)=0. Branch-width is defined for graphs, matroids, and more generally, connectivity functions. We show that for each constant k, there is a polynomial-time (in |V|) algorithm to decide whether the branch-width of a connectivity function f is at most k, if f is given by an oracle. This algorithm can be applied to branch-width, carving-width, and rank-width of graphs. In particular, we can recognize matroids M of branch-width at most k in polynomial (in |E(M)|) time if the matroid is given by an independence oracle.

3:05 pm Friday, February 10, 2006

Combinatorics Seminar: Constructing Ramsey graphs from Boolean function representations.

by Parikshit S Gopalan (College of Computing, Georgia Tech) in Skiles 255

The problem of explicit Ramsey graph construction is to construct a graph which has no large clique or independent set. We propose a simple construction based on polynomials representing Boolean functions, which are well studied in complexity theory. We show that all the best known constructions, ones due to Frankl-Wilson, Grolmusz and Alon are captured by this; they can all be derived from representations of the OR function of degree \sqrt&ob;n&cb; based on symmetric polynomials. Thus the barrier to better Ramsey constructions through current techniques appears to be the construction of lower degree representations. We show that the above Ramsey constructions cannot be improved using symmetric polynomials.

3:30 pm Monday, February 13, 2006

Topology Simiar: On the Witten-Reshetikhin-Turaev functions for integral homology spheres

by K. Habiro (RIMS, Kyoto University) in Skiles 269

The Witten-Reshetikhin-Turaev (WRT) invariant for an integral homology 3-sphere $M$ is defined for each complex root of unity $\zeta$ and takes values in the ring $Z[\zeta]$ of cyclotomic integers. Thus the WRT invariants may be defined as a function on the complex roots of unity. The behavior of this "WRT function" is controlled by a unifying invariant $J_M$ with values in the completion $R = \lim_n Z[q]/((1-q)(1-q2)...(1-q^n))$ of the polynomial ring $Z[q]$. Here the WRT invariant at $\zeta$ is recovered from $J_M$ by evaluating $J_M$ at $q=\zeta$. In this talk, I will explain some applications of this unifying invariant $J_M$. In particular, I will introduce a $p$-adic analytic version of the WRT function.

4:30 pm Monday, February 13, 2006

CDSNS Colloquium: Traveling Waves in Epidemic Models

by Shigui Ruan [mail] (University of Miami) in Skiles 255

In this talk, we first review some classical epidemic models, such as Ross-Macdonald model, Kermack-McKendrik model, Kendall model, etc. The existence of epidemic waves in some models are demonstrated. Then we propose a host-vector model for a disease without immunity in which the current density of infectious vectors is related to the number of infectious hosts at earlier times. Spatial spread in a region is modeled in the partial integro-differential equation by a diffusion term. For the general model, we first study the stability of the steady states using the contracting convex sets technique. When the spatial variable is one-dimensional and the delay kernel assumes some special form, we establish the existence of traveling wave solutions by using the linear chain trick and the geometric singular perturbation method. The results apply to several vector induced diseases such malaria, West Nile Virus, etc.

3:00 pm Tuesday, February 14, 2006

Geometry-Topology Seminar: Introduction to Convex Integration and the h-Principle, I

by David Spring (York University) in Skiles 269

We provide some historical background to the the development of the h-principle in differential topology, with special reference to the work of M. Gromov on convex integration theory. We outline the basic analytic constructions in convex integration theory, with topological examples in spaces of 1-jets.

4:40 pm Tuesday, February 14, 2006

PDE Seminar: Schr\"odinger maps and the Heisenberg model of ferromagnetism

by Chongchun Zeng (Georgia Tech) in Skiles 255

Starting with the Heisenberg model of ferromagnetic materials, we give the definition of the general Schr\"odinger maps targeted on K\"ahler manifold. Among the basic properties possibly to be discussed are the local well-posedness, gauge transformations, some special solutions, and strong potentials.

3:00 pm Wednesday, February 15, 2006

Geometry-Topology Seminar: Introduction to Convex Integration and the h-Principle, II

by David Spring (York University) in Skiles 269

We continue the Introduction, I, with an outline of convex integration theory and the h-principle in the general context of spaces of r-jets, j>= 1, with examples, including (time permitting) a discussion of solutions to systems of non-linear PDEs of a special form.

3:00 pm Wednesday, February 15, 2006

Research Horizons Seminar: Quantum group invariants of knots

by Dr. Nathan Geer (GaTech) in Skiles 255

There are deep connections between quantum group theory and knot theory. At the heart of this connection is the Jones polynomial. In this talk I will give basic definitions relating to knot theory and discuss the Jones polynomial. If there is time I will give the construction of the Reshetikhin-Turaev quantum group invariant associated to the Lie algebra sl(2) and explain how this invariant relates to the Jones polynomial.

4:00 pm Wednesday, February 15, 2006

Job Candidate: Analysis Seminar: Towards multiscale analysis on manifolds and graphs through diffusion

by Mauro Maggioni (Department of Mathematics, Yale University) in Skiles 269

The study of diffusion operators of manifolds, graphs and "data sets" is useful for the analysis of the structure of the underlying space and of functions on the space. This in turn has many and important applications to disparate fields including partial differential equations, machine learning, dynamical and control systems, data analysis. We discuss old and new ideas and algorithms for multiscale analysis associated to such diffusion operators. Given a local operator T on a manifold or a graph, with large powers of low rank, we present a general multiresolution construction for efficiently computing, representing and compressing T^t. This allows the computation, to high precision, of functions of the operator, notably the associated Green's function, in compressed form, and their fast application. The dyadic powers of T can be used to induce a multiresolution analysis, as in classical Littlewood-Paley and wavelet theory: we construct, with efficient and stable algorithms, scaling function and wavelet bases associated to this multiresolution analysis, together with the corresponding downsampling operators. This allows to extend multiscale signal processing to general spaces (such as manifolds and graphs) in a very natural way, with corresponding efficient algorithms.We will discuss motivating applications, which include function approximation, denoising, and learning on data sets, model reduction for complex stochastic dynamical systems, multiscale analysis of Markov chains and Markov decision processes, nonlinear image denoising, multiscale analysis of complex networks (e.g. corpora of documents).

12:00 pm Thursday, February 16, 2006

Applied and Computational Mathematics Seminar: ClickFox - Case Study: The Mathematical Foundations of Business Success

by Tal Cohen (ClickFox) in Skiles 269

In 2000, at the height of the bursting bubble, two Georgia Tech graduates started Clickfox. This innovative customer behavior intelligence software tracks user behavior on Internet sites and phone IVR systems, establishing a mathematical representation of user behavior and interactive applications. Clickfox currently has many Fortune 500 customers. The success of the software is based on its mathematical principles and the technological concepts that have grown around them. In this talk, we'll cover four areas: a short description of the business; a demonstration of the technology; a description of the mathematical foundations; and future challenges.

2:00 pm Thursday, February 16, 2006

Job Candidate Talk: The Entropy Condition for Hyperbolic Conservation Laws

by Michael Westdickenberg (University of Bonn) in Skiles 269

Since weak solutions of hyperbolic conservation laws may be nonunique, typically an entropy condition is imposed in order to obtain uniqueness. We discuss how the entropy condition implies regularity and structure of solutions of scalar conservation laws. For the one-dimensional system of isentropic Euler equations we show how the entropy condition gives global existence of solutions with natural bounds.

3:00 pm Thursday, February 16, 2006

Stochastic Seminar: Infinitely divisible stationary processes

by Emmanuel Roy (Ecole Normale Superieure, Paris) in Skiles 269

We will show that an infinitely divisible (ID) process without Gaussian part can be written as the independent sum of five ID stationary processes, each of them belonging to a different class characterized by its Lévy measure. The ergodic properties of each class are respectively: non ergodicity, weak mixing, mild mixing, mixing and Bernoulli. To obtain this result, we use Maruyama's representation of an ID process without Gaussian part as a stochastic integral with respect to a well chosen Poisson measure.

4:30 pm Thursday, February 16, 2006

Math Department Tea:

in Math Lounge (Skiles 236)

All math department staff, students, and faculty are welcome to attend. Food and beverages will be served.

2:00 pm Friday, February 17, 2006

Mathematical Physics Seminar: Renormalization Group Approach to the Thirring Model: Ward-Takahashi Identities, Schwinger-Dyson Equation and New Anomalies

by Pierluigi Falco (Universita' di Roma "la Sapienza") in Skiles 269

The Thirring model is a 2-dimensional, relativistic, quantum field model, "exactly" solved by Johnson in 1961. In the context of the renormalization group approach, it is possible to rigorously implement the phase and the chiral Ward Identities, which allow to construct, by convergent expansion in the coupling, the Euclidean Schwinger function and to prove that they satisfy the Osterwalder-Schrader axioms. The anomaly of the Ward Identities is not in agreement with the Adler-Bardeen theorem. The critical index of the two point Schwinger function differs from the one obtained by Johnson due to an additional contribution, which can be considered as a second (new) anomaly.

3:00 pm Friday, February 17, 2006

Geometry-Topology Seminar: The Directed Embedding Problem for Smooth Manifolds

by David Spring (York University) in Skiles 269

In 1986 Gromov used Convex Integration theory to study a new geometrical problem in differential topology, known as the directed embedding problem. In this talk we explain this problem and also our recent general solution to this problem for closed manifolds, with applications to the topological elimination of all higher order singularities of smooth projection maps from embedded submanifolds to lower dimensional manifolds.

3:05 pm Friday, February 17, 2006

Combinatorics: Traces of finite sets

by Yi Zhao (Mathematics, Georgia State University) in Skiles 255

Let G be a hypergraph (set system) on X and S be a subset of X. The trace of G on S is defined as G|_S = \&ob;E\cap S: E\in G\&cb;. We say that G contains another hypergraph H as a trace if G|_S contains a copy of H for some set S. Consider the following extremal problem: how many edges can an r-uniform hypergraph on [n] have without containing H as a trace? We give recent results when H is a power set or a complete uniform hypergraph. This is a joint work with Dhruv Mubayi.

1:00 pm Monday, February 20, 2006

Ph.D. Defense: Small Scale Stochastic Dynamics For Particle Image Velocimetry

by Christel Hohenegger (School of Mathematics) in Skiles 269

Fluid velocities and Brownian effects at nanoscales in the near-wall region can be experimentally measured in an image plane parallel to the wall, but the reconstruction of the out-of-plane dependence of the velocity profile remains extremely challenging. Particles are not only carried by the flow, but they undergo random fluctuations imposed by the proximity of the wall. We study such a system under a stochastic approach (Langevin) and a probabilistic approach (Fokker-Planck). The Langevin description leads to a coupled system of stochastic differential equations which requires the development of a numerical scheme of strong order of convergence 1. A maximum likelihood solution to the reconstruction of the out-of-plane dependence of the velocity profile is proposed based on the probability density function of mean in-plane displacements. Finally, we study, starting from the Fokker-Planck equation, the distribution of matched particles: those that starts uniformly in a measurement window and end in the same window. We quantify the bias in the mean of this distribution and discuss the relevant implications for experimentally measured velocity field.

3:30 pm Monday, February 20, 2006

Stelson Lecture: Quantum Hyperbolic Geometry

by Francis Bonahon [mail] (University of Southern California) in Clary Theatre - Bill Moore Student Success Center

This is the first of two Stelson lectures on Quantum Hyperbolic Geometry. In the past 30 years, a lot of the activity in low-dimensional topology has occurred in hyperbolic geometry and in topological quantum field theory. However, these two branches of mathematics have largely evolved in parallel, without much interaction. For instance, proofs in hyperbolic geometry tend to be analytic, whereas topological quantum field theory has a more combinatorial/algebraic flavor. The so-called Volume Conjecture now provides an exciting conjectural bridge between these two domains. Technically and conceptually, the challenge is to figure out how these two fields can fit together in a common context. We will discuss some of these issues, and briefly sketch a framework which combines hyperbolic geometry and topological quantum field theory.

4:30 pm Monday, February 20, 2006

CDSNS: Schr\"odinger maps into Hermitian symmetric spaces and their associated frame systems

by Andrea Nahmod [mail] (University of Massachusetts) in Skiles 255

In studying Schr\"odinger maps one usually reduces matters to considering an associated modified NLS system in a particular fixed gauge. One then wishes to know that solutions to the latter produce solutions to the original problem. The well-posedness results on the modified systems however, apply to a larger class of formal solutions to the equation than those which come from Schr\"odinger maps. In general, it is not a simple task to go from solutions of the modified Schr\"odinger system (MSM) to the full Schr\"odinger map system (SM) directly. The transformation formulas between a solution $u$ to the MSM system and the map $s$ are quite complex. In this talk we will describe the process to show that a fairly rough solution to the Schr\"odinger map system -- that is the weak limit of smooth solutions -- solves the associated Coulomb gauged frame system in a unique fashion and vice-versa. This is joint work with J. Shatah, L. Vega and C. Zeng.

11:00 am Tuesday, February 21, 2006

QCF Faculty Candidate Seminar: Optimal Dividend Policy with Mean-Reverting Cash Reservoir

by Abel Cadenillas (University of Alberta) in Executive Classroom, Room 228, Main Bldg., ISyE

Motivated by empirical evidence and economic arguments, we assume that the cash reservoir of a financial corporation follows a mean reverting process. The firm must decide the optimal dividend strategy, which consists of the optimal times and the optimal amounts to pay as dividends. We model this as a stochastic impulse control problem, and succeed in finding an analytical solution. We also find a formula for the expected time between dividend payments. A crucial and surprising economic result of our paper is that, as the dividend tax rate decreases, it is optimal for the shareholders to receive smaller but more frequent dividend payments. This results in a reduction of the probability of default of the firm. (Joint work with Sudipto Sarkar and Fernando Zapatero.)

11:00 am Tuesday, February 21, 2006

Stelson Lecture: Representations of the quantum Teichmuller space

by Francis Bonahon [mail] (University of Southern California) in Skiles 269

We will provide a more detailed illustration of the principles discussed in the first talk. This second talk will be focused on a punctured surface S. The quantum Teichm\uffffller space of S is a certain non-commutative deformation of the algebra of rational functions on the space of (2-dimensional) hyperbolic metrics on S. This is a purely algebraic object, closely related to the combinatorics of the Harer-Penner complex of ideal cell decompositions of the surface. It turns out that the finite-dimensional representation theory of this algebraic object is controlled by the same type of data as hyperbolic metrics on the 3-dimensional manifold product of S with the real line. We will use this correspondence to exhibit strange invariants of surface diffeomorphisms, and speculate on the relevance of this construction to the Volume Conjecture.

4:40 pm Tuesday, February 21, 2006

PDE Seminar: Normal traces and Gauss-Green formula for weakly differentiable vector fields

by Monica Torres (Purdue University) in Skiles 255

We obtain the normal trace of bounded divergence measure fields on the boundary of any set of finite perimeter $E$ as the limit of the normal traces of the vector field $F$ on smooth surfaces that approximate $\po E$ essentially from the inside of $E$ with respect to the measure $\div F$. Using this trace, we obtain the corresponding Gauss-Green formula.

3:00 pm Wednesday, February 22, 2006

Research Horizons: Frames, Time-Frequency Analysis, and Wavelets

by Christopher Heil (GaTech) in Skiles 255

We will survey frame theory and some of its uses and application. A Frame is like a basis in that every element of a given space can be written as an (infinite) linear combination of the frame vectors. The frame vectors typically serve as simple "building blocks" from which complicated signals or functions can be built. However, unlike bases, frame decompositions are not unique in general.

3:05 pm Thursday, February 23, 2006

Graph Theory/Dissertation defense: New tools and results in graph structure theory

by Rajneesh Hegde [mail] (Math, GT) in Skiles 255

We first present a ``non-embeddable extensions'' theorem for polyhedral graph embeddings. Let G be a ``weakly 4-connected'' planar graph. We describe a set of constructions that produce a finite list of non-planar graphs, each having a minor isomorphic to G, every non-planar weakly 4-connected graph H that has a minor isomorphic to G has a minor isomorphic to one of the graphs in the list. The theorem is more general and applies in particular to polyhedral embeddings in any surface. We discuss an approach to proving Jorgensen's conjecture, which states that if G is a 6-connected graph with no K6 minor, then it is apex, that is, it has a vertex v such that deleting v yields a planar graph. We relax the condition of 6-connectivity, and prove Jorgensen's conjecture for a certain sub-class of these graphs. We prove that every graph embedded in the Klein bottle with representativity at least 4 has a K6 minor. Also, we prove that every ``locally 5-connected'' triangulation of the torus, with one exception, has a K6 minor. (Local 5-connectivity is a natural notion of local connectivity for a surface embedding.) The above theorem uses a locally 5-connected version of the well-known splitter theorem for triangulations of any surface. Finally, we present a theoretically optimal algorithm for the following graph connectivity problem. A shredder in an undirected graph is a set of vertices whose removal results in at least three components. A 3-shredder is a shredder of size three. We present an algorithm that, given a 3-connected graph, finds its 3-shredders in time proportional to the number of vertices and edges, when implemented on a RAM (random access machine).

4:30 pm Thursday, February 23, 2006

Job Candidate Colloquium: Combinatorial Results Motivated by Computational Biology

by Christine Heitsch (Mathematics Department, University of Wisconsin, Madison) in Skiles 269

Under a suitable abstraction, complex biological problems can reveal surprising mathematical structure. We illustrate this phenomena with results on the combinatorics of plane trees, motivated by our work on RNA secondary structures. As will be explained, the biology inspires a new operation on plane trees, leading to a multipartite graph whose disjoint sets are enumerated by the Narayana numbers. Furthermore, the induced partial ordering gives us a lattice on the set of plane trees with n edges, which is isomorphic to the lattice of noncrossing partitions.

12:00 pm Friday, February 24, 2006

ACO Pizza Seminar: Mathematical problems in phylogeny

by Laszlo A. Szekely (University of South Carolina) in Room 228, ISyE Building (old Dupree Management building)

Biologists often face the following problem: Given a set of taxa and information about them - which can be morphological characters, corresponding segments of biomolecular sequences, gene order in their whole genomes, etc. - , find the true phylogenetic tree. Note that not every physical process allows to reconstruct its history, therefore it is good luck if phylogeny can be reconstructed at all. The choice of data and methods for phylogeny reconstruction determine the output, although reasonable choices give similar results. I will discuss the sources of disagreement about the available methods. If data type and method are agreed on, we face a large-scale computing problem that has two complexity issues: the amount of data it needs and the usual computational complexity. Analytic investigation of the methods is made possible by models of biomolecular sequence evolution. These models involve randomness. A widely-studied model for generating binary sequences is to 'evolve' them on a tree according to a symmetric Markov process. This model is known as the Cavender-Farris-Neyman model in phylogeny, and the 'symmetric binary channel' or the 'symmetric 2-state Poisson model' in other areas. The CFN model provides a simple model for the evolution of purine-pyrimidine sequences. The significance of this simple model is, that phenomena shown for the CFN model often extend to more realistic models of sequence evolution. The abstract phylogeny reconstruction problem is telling the true (model) CFN tree from the generated sequences with high probability (whp). I will discuss two recently posed decision problems related to the phylogeny reconstruction problems.

**The talk will be approximately 30 minutes, with pizza arriving around 12:30**

2:00 pm Friday, February 24, 2006

Applied and Computational Mathematics Seminar: Phase field modeling and simulation of bio-membranes

by Qiang Du (Penn State University) in Skiles 269

Lipid membranes are becoming a major focus of biological studies in recent years. In this talk, we report some recent works on the phase field modeling and simulations of the vesicle membrane deformation under elastic bending energy and the interaction with background fluid flows. Connections are made to the well-known Willmore problem in differential geometry and the Gamma convergence of nonlinear functionals in calculus of variation. We also discuss how to effectively retrieve topological information within the phase field framework which may have broad applications.

3:00 pm Friday, February 24, 2006

Topology Seminar: The Jones polynomial and its categorifications

by Anna Beliakova (University of Zurich) in Skiles 269

In this talk, we will first outline the results of Bar-Natan and Rasmussen on Khovanov homology. More precisely, after recalling the definition of the Jones polynomial, we introduce a chain complex whose graded Euler characteristic is given by this polynomial and whose homotopy type is a link invariant. We extract a new numerical invariant from this complex. With the help of this invariant, the main results of Seiberg-Witten theory (such as the Milnor conjecture or the non-equivalence of topological and smooth categories in dimension 4) can be proven purely combinatorially. Finally, we present our own results on how to extend this construction to the colored Jones polynomial.

3:05 pm Friday, February 24, 2006

ACO Colloquium: Biplanar crossing number

by Laszlo Szekely (Mathematics, University of South Carolina) in Skiles 255

The biplanar crossing number cr_2(G) of a graph G is \min cr(G_1)+cr(G_2), where cr is the planar crossing number and G_1\cup G_2=G. We show that cr_2(G)\leq (3/8)cr(G). Using this result recursively, we bound the thickness by \Theta(G)-2\leq K cr_2(G)^{.4057}\log_2 n with some constant K. A partition realizing this bound for the thickness can be obtained by a polynomial time randomized algorithm. We show that for any size exceeding a certain threshold, there exists a graph G of this size, which simultaneously has the following properties: cr(G) is roughly as large as it can be for any graph of that size, and cr_2(G) is as small as it can be for any graph of that size. The existence is shown using the probabilistic method. This is a joint work with Eva Czabarka, Ondrej Sykora, and Imrich Vrto.

4:00 pm Friday, February 24, 2006

General Colloquium: Visualization of Optimal Geometry

by John Sullivan (Technical University of Berlin) in DM Smith 105

For any topological object, we can ask for its optimal geometric shape, minimizing some geometric energy. A classical example is a soap bubble which is round because it minimizes surface area while enclosing a fixed volume. Other examples, at the frontier of current mathematical research, include knots tied tight in thick rope, which minimize their length, and surfaces which minimize elastic bending energy. The resulting shapes are not only mathematically elegant, but often exhibit striking visual beauty. We will watch two short computer-graphics videos, illustrating optimal shapes for knots and a mathematical way to turn a sphere inside out (controlled by surface bending energy), and will see other examples of mathematical visualizations arising from optimal geometry, including computer-generated sculpture.

9:30 am Saturday, February 25, 2006

Workshop on Internet-based Teaching of Multivariable Calculus: Multivariate Calculus Project

by Thomas F. Banchoff (Brown University) in Mathematics Department, Morehouse College

Tom Banchoff will visit Morehouse College on February 25 to give a workshop on his MULTIVARIABLE CALCULUS PROJECT. This is one of a series of workshops during this semester while Tom is a visiting professor at UGA. The topic of the workshop will be the online teaching resources for multivariable calculus and other courses which have been developed by Tom and his students at Brown. He will focus on those resources, such as interactive JAVA applets, which can be easily incorporated into existing courses, from precalculus to differential geometry. The workshop will be based on the MAA PREP workshop last June at Brown. See http://www.math.brown.edu/~banchoff/PREP/ for more information about this project. This will be a full day workshop, with sessions 9:30-12:30 and 1:30-4:30, in the Department of Mathematics at Morehouse College. Lunch will be provided. If you are interested in registering for the Morehouse workshop, please email Sandra Strickland (sstrickl@morehouse.edu) or Clint McCrory (clint@math.uga.edu) as soon as possible. There will be a $20 registration fee, and the number of seats is limited. Participants will receive a $50 per diem to cover the registration fee and other personal expenses. Funding to support this project has been obtained from the National Science Foundation (grant DUE-0509899). If you would like to apply for additional travel support, please include this request in your email.

12:00 pm Monday, February 27, 2006

Theory Seminar: Agnostically Learning Halfspaces

by Adam Klivans (University of Texas, Austin) in MiRC 102

We give the first algorithm that efficiently learns halfspaces (under distributional assumptions) in the notoriously difficult agnostic framework of Kearns, Schapire, and Sellie. In this model, a learner is given arbitrarily labeled examples from a fixed distribution and must output a hypothesis competitive with the optimal halfspace hypothesis. Our algorithm constructs a hypothesis whose error rate on future examples is within an additive \epsilon of the optimal halfspace in time poly(n) for any constant \epsilon > 0 under the uniform distribution over &ob;0,1&cb;^n or the unit sphere in R^n, as well as under any log-concave distribution over R^n. It also agnostically learns Boolean disjunctions in time 2^&ob;\tilde&ob;O&cb;(\sqrt&ob;n&cb;)&cb; with respect to *any* distribution. The new algorithm, essentially L_1 polynomial regression, is a noise-tolerant arbitrary-distribution generalization of the "low-degree" Fourier algorithm of Linial, Mansour, & Nisan. Joint work with A. Kalai, Y. Mansour, and R. Servedio.

3:30 pm Monday, February 27, 2006

Algebra-Geometry-Topology Seminar: Open Books On Torus Bundles Over S^1

by Jeremy Van Horn (Univeristy of Texas) in Skiles 269

For each torus bundle over S^1, there is a distinguished family of tight contact structures given by "twisting in the S^1 direction." We will provide a construction of compatible open books.

4:30 pm Monday, February 27, 2006

CDSNS Colloquium: The parameterization method for invariant manifolds

by Rafael de la Llave [mail] (University of Texas at Austin) in Skiles 255

We discuss a method to prove existence of invariant manifolds. (Precedents appear in the work of Poincare and Lyapunov) It allows to make sense of manifolds associated to non-resonant spaces, including sometimes slow manifolds. It also lends itself to numerical algorithms. (Joing work with C. Cabre, E. Fontich, A. Haro)

11:00 am Tuesday, February 28, 2006

QCF Faculty Candidate Talk: Portfolio Optimization with Drawdown Constraints

by Stan Uryasev (University of Florida, Risk Management and Financial Engineering Lab) in Executive Classroom, Room 228, Main ISyE Building

By definition, portfolio drawdown is a drop in the portfolio value compared to the previous maximum. We study a measure of risk, which depends on the portfolio drawdown curve (also called underwater curve) considered in active portfolio management. The new risk measure, Conditional Drawdown-at-Risk (CDaR), is defined as the mean of the worst x% drawdowns. This measure of risk is closely related to the Conditional Values-at-Risk risk measure. The CDaR risk measure has several important properties, which make it attractive from a practical perspective: (1) compared to variance or Value-at-Risk (VaR), it adequately reflects investors' preferences; (2) it is robust: it depends upon many significant drops in the portfolio value rather than on one extreme event; (3) information on sequence of evens is not lost (compared to approaches such as VaR or variance); (4) minimal data requirements: historical data can be directly used for path generation; (5) the technique is very stable numerically; (6) can be efficiently implemented using Linear Programming techniques. Some practical recommendations on how to use the CDaR measure for getting practically stable portfolios are provided. Using CDaR, we solved a real life allocation problem for a portfolio of derivatives.

4:40 pm Tuesday, February 28, 2006

PDE Seminar: Transport in small systems

by David Kinderlehrer (Carnegie Mellon University) in Skiles 255

Motion in small live systems has many challenges, as famously discussed in Purcell. Prominent environmental conditions are high viscosity and warmth. Not only is it difficult to move, but maintaining a course is rendered difficult by immersion in a highly fluctuating and crowded bath. We shall introduce our subject with a brief description of three examples of nanoscale transduction. We then choose one of these systems, a multiple state molecular motor, and explain how transduction and transport occur. Providing background as appropriate, we explain how Monge-Kantorovich type metrics, and hence weak topology kinetics, arise in a straightforward manner. Attention is subsequently dedicated to illustrating why there must be collaboration between these two, the chemical transduction process and the potential driven transport. Many questions arise, including some resolved by Adrian Tudorascu. This is joint work, over several years, with Michel Chipot, Stuart Hastings, Michael Kowalczyk, and Bryce McLeod.

12:00 pm Wednesday, March 1, 2006

Applied and Computational Mathematics Seminar: Invariant manifolds for quasi-periodic maps: Theory, algorithms, computations and conjectures

by Alex Haro in Skiles 269

We discuss the computation of normally hyperbolic invariant manifolds in quasi-periodic maps. We present algorithms, rigorous results that estimate the error in the numerics. Since the algorithms are implemented and error bounds are available, it is possible to explore invariant manifolds close to breakdown and conjecture mechanisms for the breakdown.

3:00 pm Wednesday, March 1, 2006

Research Horizons: The extreme sport of eigenvalue hunting.

by Evans Harrell (GaTech) in Skiles 255

Energy levels in quantum-scale devices are the eigenvalues of some differential operators that depend on the shape of the device. As a rule the energy eigenvalues depend on the geometry in a complicated way, but if we seek the shape that maximizes or minimizes the energy, there is sometimes a simple answer. I'll describe a few cases with simple extrema, including some ``isoperimetric'' results, and how to get there with a blend of geometry, operator theory, and analysis. Some of this is joint work with Michael Loss and Pavel Exner.

2:05 pm Thursday, March 2, 2006

Graph Theory: The Caccetta-Haggkvist conjecture and additive number theory

by Daniel Kral (Math, GT and Charles University) in Skiles 255

This talk is based on a lecture given by Professor Nathanson at the workshop on the Caccetta-Haggkvist conjecture held this January in Palo Alto, CA. We will present proofs of the Cauchy-Davenport theorem and the conjecture of Erdos and Heilbronn. Both proofs are algebraic and are inspired by the celebrated method of Alon and Tarsi from the area of graph choosability.

3:00 pm Thursday, March 2, 2006

Stochastic Seminar: Longest Common Subsequences, Optimal Alignments and Applications

by Heinrich Matzinger (School of Mathematics, Georgia Tech) in Skiles 269

We discuss several new results on Longest Common Subsequences (LCS) and Optimal Alignments (OA). These results concern the order of the fluctuation of the LCS and the structure of the OA-path. Furthermore, we present related problems in computational linguistics for rare semitic languages.

4:30 pm Thursday, March 2, 2006

Math Department Tea:

in Math Lounge (Skiles 236)

All math department staff, students, and faculty are welcome to attend. Food and beverages will be served.

3:05 pm Friday, March 3, 2006

Combinatorics Seminar: Tic-Tac-Toe Theory

by Jozsef Beck (Mathematics, Rutgers University) in Skiles 255

I just finished a more-than-600-page book with the same title (of course it is not published yet). As the title suggests, this is combinatorial game theory, focusing on tic-tac-toe like games. My colloquium talk is about the main results and the main techniques.

4:00 pm Friday, March 3, 2006

Geometry/Topology seminar: Resurgence of the Zagier-Kontsevich power series

by Stavros Garoufalidis [mail] (Georgia Tech) in Skiles 269

The Zagier-Kontsevich power series is a high-school power series in q, that does not converge inside or outside the unit circle, and really only makes sense at the points in the unit circle which are complex roots of unity. Needless to say, the series is in no way continuous at the complex roots of unity. Despite this naughty analytic behavior, the Zagier-Kontsevich power series is a typical example of an "arithmetic analytic function", in the sense of Habiro. Are we stuck with roots of unity? Is there a way out? In the talk, we will show that the Zagier-Kontsevich power series is resurgent, in the sense of Ecalle, which among other things it means that it does represent in a canonical and natural way an honest analytic function in the interior of the unit disc. How did Kontsevich and Zagier came up with this power series? How about other power series? And what on earth is resurgence? Are we any better replacing Habiro by Ecalle? And what does this have to do with knots? And geometry? Need help? Come and listen!

8:30 am Saturday, March 4, 2006

Southeast Geometry Conference: Talks in all areas of geometry

by students and researchers in Room 207, Tate Center, College of Charleston, Charleston, SC

The 17th annual Southeast Geometry Conference will be held at the College of Charleston, in historic Charleston, South Carolina. The talks, which will begin in the morning on Saturday, March 4th and conclude on the afternoon of Sunday, March 5th, 2006, will take place in the room 207 of the Tate Center at #9 Liberty Street (see campus map). Students and researchers from all areas of geometry (including, but not limited to, algebraic geometry, differential geometry, symplectic geometry, and applications of geometry to mathematical physics) will be presenting their research in 40-minute talks. This year the conference will have a special focus on connections between geometry and physics. The conference is free and open to the public. Feel free to show up unannounced if you would like to attend, but if possible, please let us know that you are coming so that we can be sure to have enough coffee and seating for everyone. For details see http://www.math.cofc.edu/SEGC/

2:30 pm Monday, March 6, 2006

Algebra-Geometry-Topology Seminar: Iso-length spectral arithmetic hyperbolic 3-manifolds

by Emily Hamilton (Emory University) in Boyd 222, University of Georgia

Abstract: Let M be an orientable hyperbolic 3-manifold of finite volume. The length spectrum of M is the collection of all lengths of closed geodesics in M counted with their multiplicities. The complex length spectrum of M is the collection of all complex lengths of closed geodesics in M counted with their multiplicities. Two orientable hyperbolic 3-manifolds of finite volume are called iso-length spectral (resp. complex iso-length spectral) if their length spectra (resp. complex length spectra) are identical. It is known that if M_1 and M_2 are complex iso-length spectral arithmetic hyperbolic 3-manifolds, then they are commensurable. We show that if M_1 and M_2 are iso-length spectral arithmetic hyperbolic 3-manifolds, then M_1 and M_2 are commensurable.

3:45 pm Monday, March 6, 2006

Algebra-Geometry-Topology Seminar: Pseudo-Anosov dilatations and algebra

by Chris Leininger (UIUC) in Boyd 222, University of Georgia

The dilatation of a pseudo-Anosov homeomorphism F of a closed surface S_g of genus g is a basic measure of its dynamical complexity. Penner proved that if one allows g to tend to infinity, then the logarithm of the smallest possible dilatation of such a homeomorphism tends to zero on the order of 1/g. I'll discuss joint work with Benson Farb and Dan Margalit that describes how this type of behavior is prohibited if one imposes certain algebraic restrictions. As the simplest example, we prove that if F acts trivially on homology, then the logarithm of its dilatation is bounded below by .098 (independent of g). I'll also describe how this is sharply contrasted when one considers a natural measure of topological complexity.

4:30 pm Monday, March 6, 2006

CDSNS Colloquium: Bistablility in chemical reaction and predator-prey systems

by Junping Shi [mail] (College of William and Marry) in Skiles 255

Bistability describes the phenomenon of multiple attracting regions in a dynamical system, and it has been observed in a wide range of natural phenomena. I will introduce my recent mathematical work on bistability in reaction-diffusion systems. First we consider an autocatalytic chemical reaction. If the spatial domain has dimension higher than 2 and the "order" of the reaction is high enough, then it is known that the system has a family of non-trivial steady states. We prove that each of these steady states is a "hair-trigger" for two types of long time behavior: if the initial value is below the steady state, then the solution of the system converges to a rest state of the system as time goes to infinity and so extinction occurs; if the initial value is above the steady state, then a wave front is developed and so we have the spread of "flame". This a joint work with Xuefeng Wang of Tulane University. Secondly I will consider a diffusive predator-prey system of ecology. Existence of multiple positive steady states and global bifurcation branch are examined as well as related dynamical behavior. It is found that while the predator population is not far from a constant level, the prey population could extinguish, persist or blow up depending on the initial population distributions, the various parameters in the system, and the heterogeneous environment. In particular, we examine a situation where the Allee effect is caused by the spatial heterogeneity of the environment. If time allows, a diffusive predator-prey system with a protection zone for prey will also be discussed. This is a joint work with Yihong Du of University of New England, Australia.

3:00 pm Tuesday, March 7, 2006

Algebra/Topology Seminar: Mixed Hodge polynomials of character varieties of Riemann surfaces

by Fernando Rodriguez-Villegas (UT Austin) in Skiles 269

In this talk I will describe a calculation of the number of points of the varieties of the title (parameterizing representations of the fundamental group of a Riemann surface into GL_n) over finite fields. The outcome of the calculation yields topological information on these varieties and leads to a natural conjecture about their mixed Hodge polynomials. Besides their intrinsic interest the varieties are closely related to the moduli spaces of Higgs bundles on the surface. Somewhat surprisingly the calculation also reveals a tight connection between the geometry of these character varieties and the Macdonald polynomials of combinatorics. This is joint work with T. Hausel and E. Letellier

4:40 pm Tuesday, March 7, 2006

PDE Seminar: Stable, efficient Navier-Stokes solvers and a commutator estimate

by Jie Liu (University of Maryland) in Skiles 255

For strong solutions of the incompressible Navier-Stokes equations in bounded domains with velocity specified at the boundary, we proof unconditional stability and obtain error estimates of discretization schemes that decouple the updates of pressure and velocity through explicit time-stepping for pressure. The proofs are simple, based upon a new, sharp estimate for the commutator of the Laplacian and Helmholtz projection operators. This allows us to treat an unconstrained formulation of the Navier-Stokes equations as a perturbed diffusion equation.

3:00 pm Wednesday, March 8, 2006

Research Horizons Seminar: Local rules for quasi-crystals

by Thang Le (GaTech) in Skiles 255

Quasi-periodic tilings of the Euclidean spaces can serve as models for real quasi-crystals. A local rule for a class of quasi-periodic tilings would correspond to local interactions that enforce the long-range order of the quasi-crystals. We will discuss local rules of the famous Penrose tilings, and the extistence of local rules for more general quasi-periodic tilings.

4:35 pm Wednesday, March 8, 2006

Combinatorics Seminar: Bounding the partition function of spin-systems

by David Galvin (Mathematics, University of Pennsylvania) in Skiles 243 (Note Room Change)

Let \Lambda be a system of non-negative weights on the singletons and pairs of the set [m]={1,...,m}. For any graph G, \Lambda induces a natural probability distribution on the set of functions from the vertices of G to [m] in which each such function is chosen with probability proportional to the product of the weights of the images of the vertices of G times the product of the weights of the images of the edges of G. (This framework encompasses many familiar statistical physics spin-models such as Potts, Ising and hard-core.) With P. Tetali, we considered the normalizing constant (or partition function) of the above-described distribution, in the case when all the pair-weights are either 0 or 1, and using entropy methods obtained an upper bound that is tight for the class of regular bipartite graphs. In this talk we will describe the proof of that result, and then use random graphs to extend the work to the case of general non-negative pair-weights.

12:00 pm Thursday, March 9, 2006

Applied and Computational Mathematics Seminar: Shapes of minimal aerodynamic resistance

by Alexander Plakhov in Skiles 269

A body moves through a rarefied gas; it is required to find the body�s shape minimizing the force of the medium resistance to the body�s motion. It is supposed that the gas molecules (particles) interact with the body in the absolutely elastic manner and that mutual interaction of particles can be neglected. Both conditions are rather strong: in the real world, interaction of gas molecules with surfaces is never absolutely elastic. Moreover, the interaction between the Earth atmosphere molecules can be neglected only at a height more than 150 � 200 km. The problem of resistance minimization was first posed and solved by I. Newton (1686) in the class of axially symmetric convex bodies of fixed length and width, with the symmetry axis parallel to the velocity vector. Since 1993, the problem have been studied in various classes of non-convex and/or non-symmetric bodies, under the assumption that any particle hits the body at most once (G. Buttazzo, B. Kawohl, T. Lachand-Robert et al). We consider problems of minimal (and maximal) resistance in the case where multiple collisions of particles with the body are allowed. In the talk, the following question will be addressed: 1. The problem of minimal resistance in classes of non-convex and non-symmetric bodies. Bodies of arbitrarily small resistance. 2. The problem of minimal averaged resistance for slowly rotating bodies. Rough bodies of minimal and maximal averaged resistance. Application of the Monge-Kantorovich mass transfer theory. 3. Problems of minimal resistance in media with thermal motion of particles.

3:00 pm Thursday, March 9, 2006

Stochastic Seminar: Longest Common Subsequences, Optimal Alignments and Applications, II

by Heinrich Matzinger (School of Mathematics, Georgia Tech) in Skiles 269

We discuss several new results on Longest Common Subsequences (LCS) and Optimal Alignments (OA). These results concern the order of the fluctuation of the LCS and the structure of the OA-path. Furthermore, we present related pr oblems in computational linguistics for rare semitic languages.

4:30 pm Thursday, March 9, 2006

Analysis: The Dirichlet problem in Lipschitz domains for higher order elliptic systems with rough coefficients

by Marius Mitrea (University of Missouri, Columbia) in Skiles 269

We settle the issue of well-posedness of the Dirichlet problem for a divergence form, higher order, elliptic system $L$ with complex-valued, bounded, measurable coefficients in a Lipschitz domain $\Omega$, and for boundary data in Besov spaces. The main hypothesis under which our principal result is established is in the nature of best possible and requires that, at small scales, the integral mean oscillations of the unit normal to $\partial\Omega$ and of the coefficients of the differential operator $L$ are not too large, relative to the Lipschitz constant of the domain $\Omega$, the ellipticity constant of $L$, and the indices of the corresponding Besov space. This is joint work with V. Maz'ya and T. Shaposhnikova.

3:05 pm Friday, March 10, 2006

Combinatorics Seminar: A tight bound on the diameter of random geometric graphs

by Catherine Yan (Mathematics, Texas A&M) in Skiles 255

The unit ball random geometric graph G=G^d_p(\lambda,n) has as its vertices n points distributed independently and uniformly in the unit ball, with two vertices adjacent if and only if their \ell_p-distance is at most \lambda. Like its cousin the Erdos-Renyi random graph, G has a connectivity threshold. We discuss this threshold, and determine upper and lower bounds for the graph diameter when G is connected. Specifically, we prove that almost always the diameter of G is asymptotically equal to \diam_p(\mathbf{B})/ \lambda, where \diam_p(\mathbf{B}) is the \ell_p-diameter of the unit ball \mathbf{B}.

4:00 pm Friday, March 10, 2006

Math Department Tea:

in Math Lounge (Skiles 236)

All math department staff, students, and faculty are welcome to attend. Food and beverages will be served.

3:30 pm Monday, March 13, 2006

Algebra-Geometry-Topology Seminar: Northcott's Finiteness Principle

by Matt Baker (Georgia Tech) in Skiles 269

4:30 pm Monday, March 13, 2006

CDSNS Colloquium: The Influence of Spatial/Temporal Variations on Biological

by Wenxian Shen [mail] (Auburn University) in Skiles 255

In this talk, I will first discuss the relation between biological invasions (spreading speeds) and principal eigenvalues of proper dispersal evolution equations. I will then investigate the influence of spatial/temporal variations on principal eigenvalues of general dispersal evolution equations. Following from the investigation, we will see that spatial/temporal variations of the environments favor the invading species' survival (speed up the population's spread)

11:00 am Tuesday, March 14, 2006

QCF Faculty Candidate Seminar: Unified modeling of corporate liabilities, credit derivatives and equity derivatives

by Vadim Linetsky (Northwestern University) in Executive Classroom, ISyE Main Building, Room 228

In the celebrated Black-Scholes-Merton options pricing model the firm's stock price is assumed to follow geometric Brownian motion -- a diffusion process with constant volatility and infinite lifetime (no default). To the contrary, real world firms have positive probability of default in finite time. We study the problem of developing a unified framework for modeling, valuation, and trading of corporate liabilities, equity derivatives, and credit derivatives. We propose and solve in closed form several parsimonious models of defaultable stock that enable us to price corporate debt, credit derivatives, and stock options in a unified fashion. In particular, corporate credit spreads and stock options implied volatility skews are closely linked in these models. Mathematically, our modeling framework is that of Markov processes with killing, where the killing rate (default intensity) is a function of the underlying stock price.

4:00 pm Tuesday, March 14, 2006

Spelman College Seminar: Opt Art: Using Mathematical Optimization Techniques to Construct Pictures, Portraits, and Sculpture

by Bob Bosch (Oberlin College) in NASA Auditorium, Spelman College

We will showcase the amazing utility of mathematical optimization by demonstrating its applicability in the area of art. We will begin by describing how to use integer programming to construct a portrait out of complete sets of dominoes. We will then describe how high quality solutions to certain large-scale traveling salesman problems can lead to beautiful continuous line drawings. We will conclude by presenting other examples of "Opt Art"---art constructed with the assistance of mathematical optimization techniques. In particular, we will discuss the creation of a domino image of the late Dr. Etta Zuber Falconer, longtime Spelman mathematician, done last spring as part of a scholarship fundraising effort at Spelman. The talk will be accessible to a wide audience. Note: the talk will be preceded, at 3:00pm, by a fun interactive, motivational activity for students on making domino images using only the human eye as a guide.

4:40 pm Tuesday, March 14, 2006

PDE Seminar: On the global well-posedness to the 3-D incompressible anisotropic Navier-Stokes equations

by Ping Zhang (Chinese Academy of Sciences and Courant Institute of NYU) in Skiles 255

We consider the well-posedness of anisotropic Navier-Stokes equation, where the vertical viscosity is zero, with initial data in sort of negative Besov-Soblev type spaces.

12:00 pm Wednesday, March 15, 2006

ACO Pizza Seminar: Opt Art: Using Optimization to Create Pictures, Portraits, and Sculpture

by Robert Bosch (Oberlin College) in Room 228, ISyE Building (old Dupree Management building)

Optimization deals with finding the best way to complete a task---creating a schedule for a tournament, matching professors with courses, constructing an itinerary for a traveling salesman. It has been applied successfully to such a great number of diverse disciplines that one can argue that it can be put to good use in *every* imaginable field. In this talk, we will showcase its amazing utility by describing applications in the field of art: portraits constructed out of complete sets of dominoes (via integer programming), mosaics comprised of geometric tiles (via integer programming), and continuous line drawings (via the "solution" of large-scale instances of the traveling salesman problem).

The talk will be approximately 30 minutes, with pizza arriving around 12:30.

3:00 pm Wednesday, March 15, 2006

Research Horizons Seminar: Hamiltonian ODE's in the space of probability measures

by Wilfrid Gangbo (School of Mathematics, Georgia Tech) in Skiles 255

We consider a Hamiltonian H on P2(R2d), the set of probability measures with finite quadratic moments on the phase space R2d = Rd × Rd. This is a metric space when endowed with the Wasserstein distance W2. We prove existence of solution for the initial value problem dµt/dt+\nabla \cdot (\JJd vtµt)=0, where \JJd is the canonical symplectic matrix, µ0$ is prescribed, vt is a tangent vector to P2(R2d) at µt, and belongs to \partial H(µt), the subdifferential of H at µt. When H is \lambda--convex, proper and lower semicontinuous on P2(R2d), we prove that the Hamiltonian is preserved along any solution of our evolutive system: H(µt) = H(µ0). Our study covers many systems occurring in fluid mechanics. (Joint work with L. Ambrosio)

3:05 pm Thursday, March 16, 2006

Graph Theory: Maxmin Spanning trees and the Nash-Williams and Tutte theorem

by Deeparnab Chakrabarty (CoC, GT) in Skiles 255

The Nash-Williams and Tutte's theorem relates the maximum number of edge disjoint spanning trees of a graph,k, to a quantity called strength of the graph: Given a partition of vertices, the strength of the partition is the ratio of the number of cross-edges to the number of partition sets minus one. The strength of the graph is minimum over all partitions. N-W,T. states that k exactly equals the floor of the strength. We study the maxmin spanning tree problem: Given an unweighted graph, the problem is to give weights on edges such that the total weight is unit and the cost of the minimum spanning tree is maximized. This quantity turns out to be exactly the reciprocal of the number of edge disjoint spanning trees that can be packed *fractionally*. We study the LP formulation of the maxmin problem and this also turns out to be a formulation for the reciprocal of the strength of the graph thus proving fractional packing number equals strength. Moreover, the dual of this LP has a very interesting property: if the variables are restricted to integers, the optimum can be shown, via Edmond's branching theorem, to be exactly the maximum number of spanning trees that can be packed; while removing the integer constraint only increases the optimum by atmost one. Thus, we get an LP whose integrality gap is within an additive factor one!

4:30 pm Thursday, March 16, 2006

Applied and Computational Mathematics Seminar: An integrated study of discretization, adaptivity and iterative methods for solving partial differential equations

by Jinchao Xu (PennState) in Skiles 255

This talk will touch upon three general aspects of numerical solution of partial differential equations: discretization (for transforming a continuous problem to a discrete problem), grid adaptation (for optimizing the discrete scheme) and iterative methods (for solving the resulting discrete systems). A number of recent results will be presented in all these different aspects and an integrated application of these results will be illustrated through an example from modeling of complex fluids. If time permits, applications to fuel cell simulations will also be discussed.

4:30 pm Thursday, March 16, 2006

Applied and Computational Mathematics Seminar: An integrated study of discretization, adaptivity and iterative methods for solving partial differential equations

by Jinchao Xu (PennState) in Skiles 255

This talk will touch upon three general aspects of numerical solution of partial differential equations: discretization (for transforming a continuous problem to a discrete problem), grid adaptation (for optimizing the discrete scheme) and iterative methods (for solving the resulting discrete systems). A number of recent results will be presented in all these different aspects and an integrated application of these results will be illustrated through an example from modeling of complex fluids. If time permits, applications to fuel cell simulations will also be discussed.

11:00 am Monday, March 20, 2006

Nonlinear Science: Synchronization in Addressable Excitable Media

by Jianxia Cui (Department of Chemistry, West Virginia University) in IBB 1128 (Petit Biotechnology Building Seminar Room)

The synchronization of two locally coupled excitable media is experimentally investigated with the photosensitive Belousov- Zhabotinsky (BZ) reaction. The spatial disorder of the coupled systems, with random initial configurations of spirals, gradually decreases until a final state is attained, which corresponds to a synchronized state with a single spiral in each system. The experimental observations are compared with numerical simulations of two identical Oregonator models with symmetric local coupling, and a systematic study reveals generalized synchronization of spiral waves. Spatiotemporal networks are studied in a photosensitive BZ medium that allows both local and nonlocal transmission of excitation. In addressable excitable media, local nearest-neighbor interactions occur via the spreading of reaction-diffusion waves, while nonlocal interactions take place by nondiffusive jumps to destination sites linked to excited sites. Static, dynamic, and domain link networks are experimentally and computationally studied. Transitions to synchronized behavior are exhibited with increasing link density, and power-law relations are observed for first-coverage time as a function of link probability.

3:00 pm Monday, March 27, 2006

Stochastic Seminar: On the Continuity of Local Times of Markov Processes

by Nathalie Eisenbaum (CNRS, University of Paris VI) in Skiles 269

The problem of finding a necessary and sufficient condition for the continuity of the local times for a general Markov process, is still open. Barlow and Hawks have completely treated the case of the Levy processes, Marcus and Rosen have solved the case of the strongly symmetric Markov processes. We will present an answer to that problem in the general case of Markov processes. This answer unifies the results of Barlow and Hawks, and Marcus and Rosen, by using a Gaussian process that naturally appears in a Central Limit Theorem involving the local time process. Joint work with Haya Kaspi (Technion).

4:30 pm Monday, March 27, 2006

CDSNS Colloquium: Equilibrium states and Young tower constructions in one-dimensional dynamics

by Henk Bruin [mail] (University of Surrey) in Skile 255

Young towers are being used for an increasing number of applications; recently for the construction of equilibrium states for specific potential functions. Equilibrium states are invariant measures that maximize a certain functional involving the potential function and entropy. However, when using a a Young tower construction, the resulting equilibrium state may only be unique and maximal within the class of measures that can be lifted to this specific Young tower. A priori, a different Young tower may result in different equilibrium states. In this talk I want to restrict attention to smooth maps on the interval and discuss a canonical way of constructing Young towers using the Hofbauer tower. I will show that, within the class of invariant measures with positive Lyapunov exponent, equilibrium states are indeed independent of the choice of the Young tower.

11:00 am Tuesday, March 28, 2006

CNS Seminar: Turning Colloidal Interactions with Adsorbed Functional Polymers

by Sven Behrens (Polymers Research, BASF Chemical Company) in ES&T Room L1125

Colloidal dispersions are fascinatingly versatile materials; they are ubiquitous in nature and form the basis of many products we encounter every day. The stability, transport and optical properties of colloidal dispersions hinge on the interaction between the particle surfaces. This interaction can be modified by adsorbed polymer and further tuned by the polymer's response to changes in the surrounding solution. We have studied the effects of charged and neutral polymer adsorbates on the interaction of a colloidal particle with a flat substrate, using the uncommon but particularly sensitive method of total internal reflection microscopy (TIRM). The results reveal a subtle interplay of steric, electrostatic, and van der Waals forces. The same principles govern the behavior of emulsions, another type of colloidal systems. Additional emulsion-specific properties like the instability against coalescence and ripening can also be addressed through adsorbed polymeric species. As an example, novel surfactant-free emulsions stabilized by pH and temperature switchable microgels shall be discussed, which offer unprecedented stability control.

3:00 pm Tuesday, March 28, 2006

ACO Seminar: Dispersion of Mass and the Complexity of Randomized Algorithms

by Santosh Vempala (MIT) in MiRC 102

How much can randomness help computation? Motivated by this general question and by volume computation, one of the few instances where randomness provably helps, we analyze a notion of dispersion and connect it to asymptotic convex geometry. Two of the most appealing conjectures in the latter field, called (i) slicing (or isotropic constant) and (ii) variance, together imply that for a random point X from an isotropic convex body in n-space, the variance of |X|^2 is O(n). We prove a reverse inequality: for any isotropic polytope with at most poly(n) facets, the variance of |X|^2 is AT LEAST n/ln n (up to a constant). In fact, the lower bound holds for any polytope of unit volume with poly(n) facets contained in the ball of radius poly(n). It implies that in order for most of such a convex polytope to be contained between two concentric spheres, their radii have to differ by about 1/\sqrt{\log n}; in contrast, most of a unit-volume ball lies in between two spheres whose radii differ by only about 1/\sqrt{n}. Such geometric dispersion leads to lower bounds on the randomized complexity of some basic algorithmic problems, including a quadratic lower bound for estimating the volume of a convex body. This is joint work with Luis Rademacher.

4:40 pm Tuesday, March 28, 2006

PDE Seminar: Recent progress on blowup phenomena in nonlinear Schrodinger equations

by Jim Colliander (University of Toronto.) in Skiles 255

This talk will survey recent progress and open questions concerning the blowup phenomena in nonlinear Schrodinger equations. Aspects of the problems related to initial data having low or minimal regularity and distinctions between the mass critical and mass supercritical problems will be highlighted.

3:00 pm Wednesday, March 29, 2006

Research Horizons Seminar: Knitting as a means of Visualizing One-Sided Surfaces

by Nathanael Berglund (School of Mathematics, Georgia Tech) in Skiles 255

What do mathematicians do for a little leisurely fun? Probably things like this! In this talk I consider knitting as a medium for visualizing one-sided (i.e. non-orientable) surfaces, particularly the Klein Bottle and Projective Plane. I discuss the various ways of representing these surfaces in R^3, along with their pros and cons from a visualization standpoint. Since we cannot hope to embed a non-orientable surface in R^3, I discuss why an immersion is the "next best" thing, and why knitting is especially well-suited to immersed surfaces! I will begin the talk by giving some of the basic topological theory of compact surfaces, and how thanks to a couple of theorems from differential topology (the Whitney embedding and immersion theorems) we may embed any smooth m-dimensional manifold in R^{2m} and immerse it in R^{2m-1}. I will then talk about the actual process of knitting such a surface, and the difficulties/challenges I encountered in my own attempt to produce a pattern for, and then actually knit, an immersion of P^2 known as "Boy's Surface". I will give some examples of how knitting and crocheting have been used by others to visualize one-sided and other surfaces, and as a way to illustrate the metric properties of certain surfaces. Finally, I will discuss directions for potential research into knitting of surfaces, such as automatic computer generation of stitch patterns for surfaces, which likely has connections with the problem of "ideal" texture mapping in computer graphics. This talk will be light on the mathematical side, being geared towards a general audience (and in particular anyone who likes to knit!).

4:30 am Thursday, March 30, 2006

Colloquium : A geometric problem and the Hopf Lemma

by Yanyan Li [mail] (Rutgers University) in Skiles 269

Abstract: A classical result of A.D. Alexandrov states that a connected compact smooth $n-$dimensional manifold without boundary, embedded in $\Bbb R^{n+1}$, and such that its mean curvature is constant, is a sphere. Here we study the problem of symmetry of $M$ in a hyperplane $X_{n+1}=$constant in case $M$ satisfies: for any two points $(X', X_{n+1})$, $(X', \widehat X_{n+1})$ on $M$, with $X_{n+1}>\widehat X_{n+1}$, the mean curvature at the first is not greater than that at the second. Symmetry need not always hold, but we establish it under some additional conditions. Some variations of the Hopf Lemma will also be presented. This is a joint work with Louis Nirenberg.

12:00 pm Thursday, March 30, 2006

Applied and Computational Mathematics Seminar: To Be Announced

by Andrei Draganescu (Sandia Nat. Lab) in Skiles 269

2:00 pm Thursday, March 30, 2006

International Loemker Conference on Leibniz: The Metaphysical and Mathematical Discussion of the Status of Infinitesimals in Leibniz's Time

by Conference Participants in Emory University Conference Center

On occasion of the 50th anniversary of graduate studies at the Department of Philosophy at Emory University we will have an International Loemker Conference on Leibniz. On March 30th, 2006 at 7pm at the Emory Conference Center there will be a public lecture by Robert Mulvaney entitled "Loemker, Leibniz and Philosophy at Emory" dedicated to Leroy Loemker, Leibniz Scholar and former Chair of the Department of Philosophy at Emory University. There will be a reception following the lecture.

10:05 am Friday, March 31, 2006

Graph Theory: Locally constrained graph homomorphisms

by Jan Kratochvil (Charles University, Prague) in Skiles 255 ***Please note different time and day***

Graph homomorphisms (edge-preserving vertex-mappings between graphs) are intensively studied as a generalization of graph coloring. A homomorphism is called locally bijective (injective, surjective, respectively) if the neighborood of every source vertex is mapped bijectively (injectively, surjectively, respectively) into (onto) the neighborhood of the target vertex. We will show that this view unifies previously studied classes of problems, such as graph covers, role assignment graphs and distance constrained graph labelings. We will also discuss relations to the Constrained Satisfaction Problem. The main question in this area is the P/NPC dichotomy of the decision problem of the existence of a locally constrained homomorphism to a parameter graph H. Among other results we show that imposing lists yields dichotomy in the case of locally injective homomorphisms.

3:05 pm Friday, March 31, 2006

Combinatorics Seminar: Hereditary properties of ordered and directed graphs

by Rob Morris (Mathematics, University of Memphis) in Skiles 255

A hereditary property of structures (e.g., graphs, permutations, posets), {\cal P}, is a collection of such structures which is closed under renaming vertices, and under taking induced sub-structures. For each n \in {\mathbb N}, let {\cal P}_n = {G \in {\cal P}: V(G) = {1,...,n}}$, the 'nth slice' of {\cal P}. The speed of {\cal P}, introduced by Erdos in 1964 for properties of labelled graphs, is the function n -> |{\cal P}_n|. We are interested in what can be said about the speed of a hereditary property. Properties of labelled graphs have been well studied, for example by Erdos, Frankl and Rödl, Alekseev, Scheinerman and Zito, Bollobás and Thomason, and Balogh, Bollobás and Weinreich, who showed that the speeds of such a property are far from arbitrary. More precisely, there often exists a family {\cal F} of functions f: {\mathbb N} -> {\mathbb N} and another function F: {\mathbb N} -> {\mathbb N} with F(n) much larger than f(n) for every f \in {\cal F}, such that if for each f \in {\cal F} the speed is infinitely often larger than f(n), then it is also larger than F(n) for every n \in {\mathbb N}. Putting it concisely: the speed jumps from {\cal F} to F. The possible speeds for graphs are now known quite precisely, so a natural next step is to look at objects with more structure, and ask whether similar results still hold. For example, we might wish to orient the edges of our graphs, or to put a linear order on the vertices. The problem for labelled directed graphs turns out to be no more difficult than for graphs, however for ordered graphs, things get somewhat more complicated. In this talk I shall present some of our recent work on the possible speeds of these structures, including three generalizations of the Stanley-Wilf Conjecture, which was recently proved by the combined results of Klazar, and Marcus and Tardos. All the work in the talk is joint with József Balogh and Béla Bollobás.

4:00 pm Friday, March 31, 2006

Math Department Tea:

in Math Lounge (Skiles 236)

All math department staff, students, and faculty are welcome to attend. Food and beverages will be served.

2:30 pm Monday, April 3, 2006

Number Theory Seminar: Multiple Dirichlet series and Gauss sums

by Solomon Friedberg (Department of Mathematics, Boston College) in Skiles 255

In this lecture I will discuss a family of multiple dirichlet series that are built out of sums of n-th order Gauss sums and a given root system of rank r. The combinatorics of the root system plays a key role in the definition. If n is sufficiently large, we show that these multiple Dirichlet series, originally defined in a product of r right half planes, have meromorphic continuation to C^r and satisfy a group of functional equations isomorphic to the Weyl group of the root system. If n is smaller, there are intriguing connections to combinatorics and representation theory in one case; these suggest a larger picture, that is also connected to the theory of metaplectic Eisenstein series. This work is all joint with Brubaker and Bump, and parts are also joint with Chinta and Hoffstein.

Reception at 3:30 pm in Room 236

3:30 pm Monday, April 3, 2006

Geometry Seminar: The ascent of liquid on a needle

by Erich Miersemann (University of Leipzig) in Skiles 269

If one dips a vertical cylinder of radius r into a bath of liquid, the surface of the bath remains a graph over a region exterior to the vertical projection of the cylinder. The speaker has obtained and will describe an asymptotic expansion for the shape of this graph in the singular limit as r tends to zero.

4:30 pm Monday, April 3, 2006

CDSNS Colloquium: Solving polynomial system

by Tien-Yien Li (Michigan State University) in Skile 255

Abstract

11:00 am Tuesday, April 4, 2006

Teaching Mathematics Workshop: Case Study Workshop

by Solomon Friedberg (Department of Mathematics, Boston College) in Skiles 269

A project to develop new training materials -- Case Studies -- for use in TA-development programs for mathematics graduate students. Over the past few years numerous workshops and talks on these materials have been given. http://www.bc.edu/bc_org/avp/cas/math/publicprojectPI/

4:40 pm Tuesday, April 4, 2006

PDE Seminar: Boltzmann Diffusive Expansion beyond Navier-Stokes

by Yan Guo (Brown University) in Skiles 255

We establish the global in time validity of a diffusive expanison to the rescaled Boltzmann equation (diffusive scaling) inside a periodic box. In particluar, our reuslts lead to error estimate for the well-known Navier-Stokes-Fourier approximation and beyond.

3:00 pm Wednesday, April 5, 2006

Research Horizons Seminar: The variation of area, mean curvature, and open problems

by John McCuan (Georgia Institute of Technology) in Skiles 255

I will describe the general formula for the first variation of area for surfaces and its relation to capillary surfaces and soap films. I will also describe several problems associated with capillary surfaces and soap films that are easy to state; some have been recently solved and some are still open.

12:15 pm Thursday, April 6, 2006

ACO Pizza Seminar: Prediction of Protein Structure, Function and Druggability on a Proteomic Scale

by Jeffrey Skolnick (Center for the Study of Systems Biology) in Room 228, ISyE Main Building (old Dupree Management building)

A novel method for the prediction of protein function and druggability based on the sequence-to-structure-to-function paradigm has been developed. We first present recent results from the application of our structure prediction algorithm, TASSER, in CASP6 and then describe the structure prediction of all putative GPCRs in the human genome. Based on confidence criteria, 90% should have correct structures, and clustering shows that structurally similar GPCRs have similar function even when their sequences are diverse. We then describe our multimeric threading algorithm, MULTIPROSPECTOR, and its application to the prediction of protein-protein interactions. Next, we describe newly developed methods for the accurate inference of protein biochemical function and present results of the comprehensive analysis of all sequenced genomes and the automated assignment of proteins to metabolic pathways. Finally, we combine all these approaches into a pathway based method for the prediction of druggable protein targets and apply the resulting methodology to the human genome.

The talk will be approximately 30 minutes, with pizza arriving around 12:30.

3:00 pm Thursday, April 6, 2006

Stochastic Seminar: Mirror averaging, aggregation and model selection

by Philippe Rigollet (Paris VI) in Skiles 269

Given a collection of M different estimators or classifiers, we study the problem of model selection aggregation, i.e., we construct a new estimator or classifier, called aggregate, which is nearly as good as the best among them with respect to a given risk criterion. We define our aggregate by a simple recursive procedure which solves an auxiliary stochastic linear programming problem related to the original non-linear one and constitutes a special case of the mirror averaging algorithm. We show that the aggregate satisfies sharp oracle inequalities under some general assumptions. The results allow one to construct in an easy way sharp adaptive nonparametric estimators for several problems including regression, classification and density estimation.

4:30 pm Thursday, April 6, 2006

Colloquium: Integrable models and operator algebras

by Detlev Buchholz (Universit\"at G\"ottingen) in Skiles 269

Recently, it has been possible to establish rigorously the existence of an abundance of 1+1-dimensional relativistic quantum field theories with factorizing scattering matrices by operator-algebraic means. This novel approach, which is complementary to the advanced methods of constructive quantum field theory, settles some longstanding questions in the context of integrable models (form-factor program) and sheds new light on the problem of constructing quantum field theories. In this talk, a survey is given on the basic ideas, results and perspectives of this approach.

9:00 am Friday, April 7, 2006

Bursting Workshop: Origin and Regulation of Bursting activity of Neurons

by Workshop Participants (Georgia State University) in Loudermilk Conference Center, Georgia State University

Andrey Shilnikov (GSU), Ronald Calabrese (Emory) and Gennady Cymbalyuk (GSU) are organizing a two day meeting "Origin and Regulation of Bursting activity of Neurons", here in Atlanta on April 7-8, 2006. At this meeting we intend to discuss bifurcation routes to bursting activity, mechanisms of its regulation and synchronization. Please, find more details at http://www.mathstat.gsu.edu/~meetings/

11:00 am Friday, April 7, 2006

Combinatorics and Math Physics Seminar: Kinetically constrained spin models: rigorous results

by Fabio Martinelli (University of Rome III) in Skiles 255

Kinetically constrained spin models are lattice 0-1 spins evolving under a Glauber (or Metropolis) type of dynamics, usually reversible w.r.t. trivial Bernoulli product measure, in which a single updating at a given vertex can occur only if the current configuration around that vertex satisfies certain specific constraints. They are intensively studied in the physics literature in connection with "glass and jamming transitions" and are closely related to bootstrap percolation models. The simplest example is the so called Fredrickon-Andersen (FA-f) in which the spin at site x can flip only if "f" among its neighbors have value zero. Due to the degenaracy of the jump rates the configuration space can be broken into different irreducible components and the invariant measure is not unique. Such non uniqueness may lead to dynamical phase transition in the thermodynamic limit. The only rigorous result available so far has been obtained some years ago by Aldous and Diaconis for the East model in one dimension. In this talk I will report on a series of new results for a wide class of models in dimension greater than one obtained in collaboration with N. Cancrini (Rome), C. Roberto and C. Toninelli (Paris). The main achievements are upper bounds on the relaxation time up to the critical point, exponential decay of the so called "persistence function" and sharp asympotics near the critical point. Some of our findings contradict some previous conjectures based on numerical simulations. Our technique is based on a novel block dynamics which takes into account the kinetical constraints on larger and larger scales.

2:00 pm Friday, April 7, 2006

Applied and Computational Mathematics Seminar: An Inverse Problem for a Reaction Diffusion Equation: A Model Problem for Program Validation

by Todd Dupont (U Chicago) in Skiles 269

This talk addresses some issues of optimal control of systems governed by partial differential equations. The work was motivated, at least in the first instance, by the difficulty of comparing experiments and simulations. Physical experiments are often used as tools in gaining confidence in our ability to model the phenomena. Using information from experiments can be difficult, often requiring close collaboration between the experimentalists and the simulators. The difficulties come from many sources, such as the incompleteness of the experimental data and the need to invoke mathematical models of unresolved scales. The large goal of our research program is to use the context of PDE constrained optimization to determine if experimental and computational results are in contradiction. The model problems addressed are simple equations that we have chosen in trying to determine the computational feasibility of this process. The optimization problems that we are confronted with involve millions of parameters, but can be solved in work that is equivalent to a few function evaluations. The work described here is joint with Andrei Draganescu, Sandia National Laboratories, Albuquerque, soon to be at University of Maryland Baltimore County.

3:30 pm Monday, April 10, 2006

Geometry-Topology Seminar: Mahler measure of multivariable polynomials and polylogarithms

by Matilde Lalin [mail] (IAS, Princeton) in Skiles 269

The Mahler measure of an n-variable polynomial P is the integral of log|P| over the n-dimensional unit torus T^n with the Haar measure. For one-variable polynomials, this is a natural quantity that appears in different problems such as Lehmer's question. While the algebraic nature of the values of the Mahler measure for one-variable polynomials with integral coefficients is well understood, the knowledge of the several-variable case is reduced to a collection of examples, some of which can be understood as special values of polylogarithms. It turns out that the polylogarithms appear as a consequence of evaluating regulators.

4:30 pm Monday, April 10, 2006

CDSNS Colloquium: Energy estimates of free boundary problems of the Euler equation

by Jalal Shatah [mail] (New York University) in Skiles 255

In this talk we consider free boundary problems for Euler equation with and without surface tension. We will derive a priori estimates that ensure the regularity of the velocity field and the free boundary. These estimates will also show that as the surface tension goes to zero, solutions of thel surface tension problem converges to solutions of the zero surface tension problem considered by S. Wu and by D. Christodoulou and H. Lindblad. This is a joint work with Chongchun Zeng.

11:00 am Tuesday, April 11, 2006

PDE Seminar: Sobolev--type metrics in the space of curves

by Andrea Mennucci (Scuola Normale Pisa) in Skiles 255

The space M of immersed curves in the plane is a fundamental model for Shape Theory, a branch of Computer Vision. For example, the approach to Image Segmentation via Active Contours is to define an Energy Functional E on M, and utilize the Calculus of Variations to derive a curve evolution to minimize E. These evolutions are referred to as Gradient Flows: this implies a certain Riemannian metric on the space of curves, that we call $H0$; but this fact has been largely overlooked. Surprisingly, $H0$ does not yield a well define metric structure, since the associated distance is identically zero; moreover, some simple Shape Optimization tasks are ill-defined when using the $H0$ metric. For this and other reasons, a regularization term is often added to the energy E: this remedy, though, does change the energy, and ends up solving a different problem. In this talk we introduce the Computer Vision problems; we recall basilar definitions from Riemannian Geometry; we discuss some metrics that have been proposed in the recent literature, and in particular present a family of Sobolev-type metrics; we show the substantial improvements gained both in theory and in practical segmentation and tracking applications.

3:00 pm Tuesday, April 11, 2006

Analysis-Mathematical Physics Seminar: New results on unstable system dynamics

by Pavel Exner (Czech Academy of Sciences) in Skiles 269

We discuss dynamics of an unstable system characterized by a Hamiltonian H and projection P on the state Hilbert space \mathcal&ob;H&cb;, in particular, from the viewpoint of relations between the subspace P\mathcal&ob;H&cb; of decaying states and the domain of the Hamiltonian. An important question concerns the behavior of such a system in the limit of permanent observation. If H\ge 0 and H^&ob;1/2&cb;P is densely defined, the decay is prevented by the monitoring and the Zeno dynamics is generated by the self-adjoint operator associated with the form \psi \mapsto \|H^&ob;1/2&cb;P\psi\|^2. Comparison of the Zeno dynamics with the stable one corresponding to "switching off" the interaction in H shows that very different situations can occur. On the other hand, if \psi\in P\mathcal&ob;H&cb; is "far enough" from D(H) the anti-Zeno effect occurs. What is more, even the decay law undisturbed by measurements can exhibit unusual features as we shall illustrate on the example of Winter model.

3:00 pm Tuesday, April 11, 2006

Stochastic Seminar: Analytic definitions of free convolutions. CLT for additive free

by Gennadiy Chystyakov (Universität Bielefeld and Institute for Low Temperature Physics (Kharkov, Ukraine)) in Skiles 255

We give a new approach to the definition of additive and multiplicative free convolutions which is based on the theory of Nevanlinna and of Schur functions. Based on an analytical approach to the definition of additive free convolution on probability measures on the real line we prove free analogs of limit theorems for sums for non-identically distributed random variables in classical Probability Theory.

3:00 pm Tuesday, April 11, 2006

CNS Seminar: Dielectric Microcavity Lasers: A System with Classical and Quantum Chaos

by Douglas Stone (Applied Physics, Yale University) in Howey room N110 (Physics)

Resonant electromagnetic states in a passive dielectric cavity represent a realization of quantum/wave chaos, that is, a system described by a wave equation with a chaotic classical dynamics associated with its classical limit. We have shown that important properties of such states, such as their directional emission pattern, can be understood by analyzing this limiting classical dynamics. When such a cavity contains a gain medium and is pumped it will emit laser radiation and the resulting system of radiation plus gain medium is described by three coupled non-linear (Maxwell-Bloch) equations, which can exhibit various kinds of non-linear behavior such as mode competition, cooperative frequency locking and even chaotic time-dependence of the laser emission. A number of experiments have been done on this type of laser and interesting results have been obtained; for example a three order of magnitude variation of the output intensity with cavity shape. As conventional laser theory has little to say about such complex cavity lasers, we have developed a generalized formalism for their treatment and will report initial results on their directional emission patterns and output power as a function of shape.

4:40 pm Tuesday, April 11, 2006

PDE Seminar: Some properties of the ground states of the infinity Laplacian

by Yifeng Yu (UT Austin) in Skiles 255

We will talk about several properties of infinity ground states which are ground states of the infinity Laplacian in the sense of Juutinen-Lindqvist-Manfredi [J-L-M1]. We will give a sufficient condition of the domain such that the distance function is the unique infinity ground state up to some constant factor. Those sufficient domains include the annulus, the ball, the stadium, etc. Also, we show that if the domain is convex, then a variational infinity ground state is a viscosity solution of the infinity Laplacian equation in the subdomain where it is $C^1$.

2:00 pm Wednesday, April 12, 2006

Analysis Seminar: Schur flows and orthogonal polynomials on the unit circle

by Leonid Golinski (Columbus, Ohio) in Skiles 269

The relation between the Toda lattices and similar nonlinear chains and orthogonal polynomials on the real line has been elaborated immensely for the last decades. We examine another system of differential-difference equations known as the Schur flow, within the framework of the theory of orthogonal polynomials on the unit circle. This system can be presented in equivalent form as the Lax equation, and the corresponding spectral measure undergoes a simple transformation. The general results are illustrated on the modified Bessel measures on the unit circle and the long time behavior of their Verblunsky coefficients.

3:00 pm Wednesday, April 12, 2006

Annual Joseph Ford Commemorative Lecture: Einstein's unknown insight and the problem of quantizing chaotic motion

by A. Douglas Stone (Department of Applied Physics, Yale University) in Howey Lecture Rm. 3 (Physics)

In 1917 Einstein authored a little-known paper on the problem of generalizing the old quantum theory to problems with several degrees of freedom that are not separable. This paper was his only published work on the correct quantization rule for matter, which was of course not known at that time. His work laid the foundation for a method which is completely correct (within its sphere of applicability), now known as Einstein-Brillouin-Keller quantization, a multi-dimensional generalization of the WKB approximation. However he pointed out that the method fails if there do not exist a number of integrals of motion equal to the number of degrees of freedom, i.e. unless the system is integrable. He suggested that non-integrable classical dynamics is typical and presents an open problem for quantum theory. This brilliant insight was ignored until the late sixties when it became well-known to physicists that partially chaotic motion is indeed generic in classical mechanical systems. The problem noted by Einstein is fundamental and has never been fully overcome; but alternative semiclassical approaches to the quantum mechanics of classically chaotic systems have been developed and applied to interesting problems in atomic, condensed matter and optical physics. I will review Einstein's arguments and place them in a modern context. Then I will mention a few experimental systems to which "quantum chaos theory" can be applied, focusing on the topic of chaotic dielectric microlasers studied at Yale and elsewhere.

Refreshments will be served in Rm. N201 at 2:00 PM

3:00 pm Wednesday, April 12, 2006

Research Horizons Seminar: Uniform Approximation in Laws of Large Numbers

by Vladimir Koltchinskii (Georgia Institute of Technology) in Skiles 255

Let $X, X_1, \dots, X_n$ be independent random points in a space $S$ with common distribution $P$ and let ${\cal F}$ be a class of real valued functions on $S.$ For $f\in {\cal F},$ denote by $Pf$ the expectation of $f(X)$ and denote by $P_nf$ the following average: $$ P_nf:=n^{-1}\sum_{i=1}^n f(X_i). $$ What is the size of the random variable $$ \xi_n({\cal F}):=\sup_{f\in {\cal F}}|P_nf-Pf|? $$ This question is very basic in many areas of Probability and Statistics. In the case when ${\cal F}$ consists of a single function $f,$ the answer is given by the classical Law of Large Numbers, Central Limit Theorem, Law of Iterated Logarithm and related inequalities. In the case of infinite ${\cal F},$ the question is central in Probability in Banach Spaces. In the recent years, it has become clear that many mathematical problems in Theoretical Computer Science (namely, in Learning Theory) are also related to this question. We will discuss several situations in which the answer is known as well as some open problems.

4:00 pm Wednesday, April 12, 2006

Analysis: Greedy approximations with regard to bases

by Vladimir Temlyakov (University of South Carolin) in Skiles 259

Theory of greedy approximations is a part of nonlinear approximations. The standard problem in this regard is the problem of $m$-term approximation where one fixes a basis and looks to approximate a target function by a linear combination of $m$ terms of the basis. The primary object of our discussion is the Thresholding Greedy Algorithm (TGA) with regard to a given basis. The TGA, applied to a function $f$, picks at $m$th step an element with the $m$th biggest in absolute value coefficient of the expansion of $f$ in the series with respect to the basis. We show that this algorithm is very good for a wavelet basis and is not that good for the trigonometric system. We discuss in detail the behavior of the TGA with regard to the trigonometric system.

11:00 am Thursday, April 13, 2006

Applied and Computational Mathematics Seminar: Nonlinear filters and mathematics

by Frederick E Daum (Raytheon Wireless Services) in Skiles 269

A nonlinear filter is an algorithm that predicts the state of a dynamical system given sparse noisy measurements. For example, weather forecasting is an important application. The key problem in nonlinear filtering is the �curse of dimensionality,� which means that the computational complexity grows exponentially with the dimension of the state. A common misconception is that particle filters beat the curse of dimensionality. Theoretical and numerical results show that particle filters depend crucially on a good proposal density, without which particle filters suffer from the curse of dimensionality. We will describe a new algorithm that exploits the smoothness of the Fokker-Planck equation and uses the meshfree adjoint method to compute an optimal density of points to represent the solution. More generally, we connect nonlinear filters with recent progress in quasi-Monte Carlo methods which are many orders of magnitude faster than Monte Carlo methods for certain high dimensional problems.

3:00 pm Thursday, April 13, 2006

Stochastic Seminar: Fast learning rates for plug-in classifiers

by Alexandre Tsybakov (Universite Pierre et Marie Curie) in Skiles 269

It has been recently shown that, under the margin (or low noise) assumption, there exist classifiers attaining fast rates of convergence of the excess Bayes risk, i.e., the rates faster than n-1/2. The works on this subject suggested the following two conjectures: (i) the best achievable fast rate is of the order n-1, and (ii) the plug-in classifiers generally converge slower than the classifiers based on empirical risk minimization. We show that both conjectures are not correct. In particular, we construct plug-in classifiers that can achieve not only the fast, but also the super-fast rates, i.e., the rates faster than n-1. We establish minimax lower bounds showing that the obtained rates cannot be improved. This is a joint work with J.-Y.Audibert.

4:30 pm Thursday, April 13, 2006

Colloquium: Hilbert's 3rd problem and invariants of 3-manifolds

by Walter Neumann (Columbia University) in Skiles 269

This talk will describe the status of Hilbert's third problem on equidecomposability of polytopes, whose solution by Dehn in 1900 led to refinements that remain unsolved and that have interesting links with invariants of 3-manifolds.

3:00 pm Friday, April 14, 2006

ACO Distinguished Lecture Series: New Market Models and Algorithms

by Vijay V. Vazirani (College of Computing, Georgia Tech) in CCB Lecture Hall 16

The notion of a "market" has undergone a paradigm shift with the Internet -- totally new and highly successful markets have been defined and launched by companies such as Google, Yahoo!, Amazon, MSN and Ebay. Another major change is the availability of massive computational power for running these markets in a centralized or distributed manner. In view of these new realities, the study of market equilibria, an important, though essentially non-algorithmic, theory within Mathematical Economics, needs to be revived and rejuvenated with new models, ideas, and an inherently algorithmic approach. The last five years has seen a surge of activity, within Algorithmic Game Theory, on obtaining algorithms for market equilibria. Interestingly enough, some of this work has already contributed handsomely to the theory of algorithms as well. In this two lecture series I will provide a historical perspective on the study of markets as well as give an in-depth feel for the exciting work going on within Algorithmic Game Theory.

3:00 pm Monday, April 17, 2006

Physics Faculty Candidate Seminar: Nonlinear Dynamics and Arrhythmias of the Heart

by Stefan Luther (Laboratory of Atomic and Solid State Physics, Cornell University) in Physics, Howey Lecture Rm. 5

Heart disease is one of the most prevalent diseases in the world and is the leading cause of death in industrialized countries. In the United States alone, heart rhythm disorders cause approximately 300,000 sudden deaths annually. Sudden cardiac death occurs unpredictably as a result of fast-developing electromechanical malfunctions of the heart. During normal functioning, the contraction of the heart is periodically triggered by planar electrical activation waves propagating across the heart followed by a refractory period. Numerical simulations suggest that during ventricular tachycardia (VT) the plane wave undergoes a transition to spiral waves (which are analogous to vortices in the Ginzburg-Landau model). Subsequently, spiral wave breakup results in a chaotic state corresponding to ventricular fibrillation (VF). Ventricular fibrillation essentially inhibits the contraction of the heart due to lack of spatial coherence. This becomes lethal to the organism if not stopped within a very short time. Some of the questions are: How are vortical structures created? How do they evolve and finally break up? Key to answering these questions is the development of a measurement system capable of imaging the spatio-temporal excitation patterns in cardiac tissues. Combining experimental data and complex nonlinear computer models of the heart, we are seeking strategies to understand and ultimately prevent sudden cardiac death. This understanding is key to the development of effective therapies, intelligent pacemakers and implantable micro defibrillators.

3:15 pm Monday, April 17, 2006

Algebra-Geometry-Topology Seminar: Predicting DNA Knot/Link Type after Protein Action

by Dorothy Buck (Imperial College London) in Skiles 269, Ga Tech

DNA molecules often have a circular, or topologically constrained, central axis. The topology of this axis can influence which, and how, proteins interact with the underlying DNA. Subsequently, there are protein families (e.g. recombinases) that change the DNA axis topology, for example converting an unknot into a torus knot. Experimentally determining the minimal crossing number (MCN) of the newly formed knots or links is tractable, but determining the exact knot/link type is very difficult and expensive. Unfortunately there are there are 1,701,936 knots with MCN \leq 16, so a finer sieve for predicting the knots/links that arise from protein action is needed. In this talk, I'll describe recent joint work with Erica Flapan, where we topologically prove that all knots/links formed during DNA recombination must fall within a few well-defined families. This substantially narrows the experiments needed to pinpoint the exact knot/link type.

4:30 pm Monday, April 17, 2006

Algebra-Geometry-Topology Seminar: A gluing construction of higher genus special Lagrangian cones

by Mark Haskins (Imperial College London) in Skiles 269, Ga Tech

We describe recent joint work with Nicos Kapouleas in which we construct infinitely many new singularity types for special Lagrangian submanifolds in dimension three. More specifically, for every odd natural number n we construct a countably infinite family of special Lagrangian cones whose link is a surface of genus n. The method we use is a 'gluing' argument or singular perturbation result.

4:30 pm Monday, April 17, 2006

CDSNS Colloquium: Existence and nonexistence of nontrivial patterns of a reaction-diffusion system

by Huiqiang Jiang [mail] (University of Minnesota) in Skiles 255

In this talk, we will discuss steady states of a reaction-diffusion system named after Gierer and Meinhardt which is used to model pattern formations in morphogenesis. Various existence and non-existence results will be presented. Especially, we will show that the smallness of ratio of two diffusion coefficients alone can prevent the formation of nontrivial patterns.

4:30 pm Tuesday, April 18, 2006

Colloquium: Concentration inequalities for functions of independent random variables with applications in statistical learning theory

by Gabor Lugosi (Pompeu Fabra University, Barcelona) in Skiles 269

A general method for obtaining moment inequalities for functions of independent random variables is presented. It is a generalization of the entropy method which has been used to derive concentration inequalities for such functions. The new inequalities prove to be a versatile tool in a wide range of applications. We illustrate the power of the method by showing how it can be used to effortlessly re-derive some classical inequalities for sums of independent random variables, moment inequalities for suprema of empirical processes, and moment inequalities for Rademacher chaos and $U$-processes. Some of these corollaries are apparently new. We also discuss applications for other complex functions of independent random variables, such as suprema of boolean polynomials which include, as special cases, subgraph counting problems in random graphs. A special attention will be paid to applications in statistical learning theory such as classification and ranking problems.

4:40 pm Tuesday, April 18, 2006

PDE Seminar: Diffusion vs. Advection

by Yuan Lou (Ohio State University.) in Skiles 255

We study a Lotka-Volterra reaction-diffusion-advection model for two competing species in a heterogeneous environment. The species are assumed to be identical except their diffusion strategies: one disperses by random diffusion only, the other by both random diffusion and advection along environmental gradient. When the two competitors have the same diffusion rates and the strength of the advection is relatively weak in comparision to that of the random diffusion, we show that the "smart" competitor wins provided that the underlying spatial domain is convex, and the competitive advantage can be reversed for certain non-convex habitats. When the advection is strong relative to random diffusion, we show that both species can can coexist stably.

3:00 pm Wednesday, April 19, 2006

Research Horizons Seminar: Weak KAM theorem on Lagrangian dynamics

by Yongfeng Li (Georgia Institute of Technology) in Skiles 255

In the Hamiltonian systems, finding an invariant set by the Hamiltonian flow is the same as finding global solutions of the associated Hamilton-Jacobi equation. Such solution do not exist in general. KAM theory asserts that a Cantor set of global solutions exists when the Hamiltonian is a small C^k perturbation of a completely integrable Hamiltonian. While the weak KAM theorem of Albert Fathi asserts that such solutions always exist in a weak sense if the Hamiltonian or the associated Lagrangian is convex and superlinear. In this talk, a brief introduction to weak KAM theorem will be presented based on Fathi's notes about Weak KAM Theorem on Lagrangian Dynamics.

4:00 pm Wednesday, April 19, 2006

Analysis Seminar: Interval exchange maps, renormalisation and continued fractions

by S. Marmi (Scuola Normale Superiore and Princeton University) in Skiles 269

Interval exchange maps are characterized by combinatorial and metric data. They arise naturally as generalizations of rotations and as first return maps of linear flows on translation surfaces. The analysis of first return times on an interval (renormalisation) leads to several generalisations of the classical continued fraction algorithm (Rauzy, Veech, Zorich). A further acceleration of these schemes can be used to characterise a class of interval exchange maps of Roth type for which the cohomological equation can be solved provided the datum is sufficiently regular. (The seminar is based joint work with Pierre Moussa and Jean-Christophe Yoccoz: see Journal of the A.M.S. 18 (2005) 823-872.)

4:30 pm Wednesday, April 19, 2006

Special Seminar: Interpreting Eigenmaps

by Sharad Goel (Stanford University) in Skiles 255

The eigenmap algorithm was recently introduced by Belkin and Niyogi as a method for visualizing high-dimensional point clouds by mapping to low-dimensional Euclidean space. This mapping is defined in terms of eigenfunctions of a random walk on the data points with transitions that come from a similarity measure on the points. In this talk I'll give a detailed analysis of the eigenmap algorithm applied to a specific dataset: the 2005 U.S. House of Representatives roll call votes. In this case, eigenmap outputs 'horseshoes' that are characteristic of dimensionality reduction techniques. We show that in general, a latent ordering of the data gives rise to these patterns, and our results appear to be the first rigorous work in this direction. This work is joint with Persi Diaconis.

1:30 pm Thursday, April 20, 2006

Graph Theory: Adjacency Lemmas and their applications

by Yue Zhao (University of Central Florida) in Skiles 255

In this talk, first we will talk about Vizing's Adjacency Lemma and its applications, then we will introduce some other adjacency lemmas obtained after Vizing's Adjacency Lemma and talk about their applications.

3:05 pm Friday, April 21, 2006

Combinatorics Seminar: Random Graphs, Random Regular Graphs and Couplings

by JeongHan Kim (Microsoft Research) in Skiles 255

The study of random regular graphs, started in late 70's, has recently attracted much attention. Main questions in this area have been whether the random regular graph contains a perfect matching, a Hamilton cycle, and a Hamilton decomposition. These properties are closely related to the contiguity of random models. Roughly speaking, two models are contiguous if they are essentially the same. For example, one may consider the uniform random 3-regular graph and the union of three independent random perfect matchings, and ask whether the two models are essentially the same or not. We will discuss contiguity of various random regular graph models. We will also introduce some attempts to study random (hyper)graphs by means of random regular (hyper)graphs. In particular, we will discuss recent improved bounds for Shamir's problem regarding when the random uniform hypergraph contains a perfect matching.

3:30 pm Friday, April 21, 2006

Algebra-Geometry-Topology Seminar: Kaehler decomposition of 4-manifolds

by Inanc Baykur (Michigan State University) in Skiles 269

Given any closed oriented smooth 4-manifold X, we have shown that it can be decomposed into two compact exact Kaehler manifolds with strictly pseudoconvex boundaries, up to orientation, such that contact structures on the common boundary induced by the maximal complex distributions are isotopic. The decomposition gives rise to a folded Kaehler structure on X, a globally defined 2-form which is a particular generalization of a symplectic form. Moreover, folded Lefschetz fibrations, a certain analogue of Lefschetz fibrations, are seen to be the geometric counterpart of these structures. In this talk we would like to outline these existence results.

3:30 pm Monday, April 24, 2006

Algebra-Geometry-Topology Seminar: Non-archimedean analogues of the Ahlfors Islands Theorem

by Robert Benedetto (Amherst College) in Skiles 269

The Ahlfors Islands Theorem states that for any five simply connected domains V_1,...,V_5 in the Riemann sphere with disjoint closures, there is a simple geometric condition (C) such that any meromorphic f from the disk to the sphere satisfying (C) must map some domain bijectively onto one of the ``islands'' V_i. If one replaces ``meromorphic'' by ``holomorphic,'' then only three domains in the complex plane are required. We will present corresponding $p$-adic (and more generally, non-archimedean) statements. In particular, only two islands are required in the non-archimedean holomorphic case, and four in the meromorphic case.

4:30 pm Monday, April 24, 2006

CDSNS Colloquium: Regularity of maps for the optimal transport problem

by Mar�a del Mar Gonz�lez Nogueras [mail] (University of Texas, Austin) in Skiles 255

The optimal transport problem is formulated as follows: given two distributions with equal masses (an embankment and an excavation), find a transrport map T which carries the first distribution into the second and minimizes the transport cost. Existence of such a map for a convex cost is well known, but there are many open questions on its regularity. Here we deal with a particular case, and prove interior second derivative estimates. The methods involve a perturbation argument from a standard Monge-Ampere equation.

3:00 pm Wednesday, April 26, 2006

Research Horizons Seminar: Are Polynomials Lightweight?

by Doron Lubinsky (Georgia Institute of Technology) in Skiles 255

We discuss a number of problems involving polynomials: how fast can we approximate by them, or by rational functions? When can they approximate "all" functions? What about approximating by Muntz polynomials, or on unbounded intervals? Finally, why is potential theory so useful in analyzing them?

4:00 pm Wednesday, April 26, 2006

Analysis: Resurgence of the Kontsevich-Zagier series

by Stavros Garoufalidis (Gatech) in Skiles 269

4:00 pm Wednesday, April 26, 2006

Analysis: Resurgence of the Kontsevich-Zagier series

by Stavros Garoufalidis [mail] (Gatech) in Skiles 269

I plan to prove that the Kontsevich-Zagier power series is resurgent. This involves explicit and conceptual calculations from analytic number theory and asymptotic analysis.

12:00 pm Thursday, April 27, 2006

Applied and Computational Mathematics Seminar: Fixed point indices of iterations

by Grzegorz Graff (School of Mathematics, Georgia Tech and Gdansk University of Technology) in Skiles 269

One of the most powerful instrument to study the dynamics of a map f near a fixed point p is the sequence of fixed point indices of its iterations ind(f^n,p). It is known that for a continuous map this sequence can not take arbitrary integer values but must satisfy some conditions called Dold congruences. Additional assumptions put on f, such as smoothness, may give stronger bounds on the form of indices of iterations. In this talk we discuss different types of restrictions and pay a special attention to the case when p is an isolated invariant set in the plane, which enables us to apply the Conley index methods.

2:00 pm Thursday, April 27, 2006

Analysis: Stability properties of phase field equations with memory

by Amy Novick-Cohen (Technion) in Skiles 255

3:00 pm Thursday, April 27, 2006

Mathematical Physics Seminar: Variational Methods in the Study of Imaging, Foams, Quantum Dots ... and More

by Irene Fonseca (Department of Mathematical Sciences, Carnegie Mellon University) in Skiles 255

Several questions in applied analysis motivated by issues in computer vision, physics, materials sciences and other areas of engineering may be treated variationally leading to higher order variational problems and to models involving lower order density measures. Their study often requires state-of-the-art techniques, new ideas, and the introduction of innovative tools in partial differential equations, geometric measure theory, and calculus of variations. In this talk it will be shown how some of these questions may be reduced to well understood first order problems, while in others the higher order plays a fundamental role. Applications to phase transitions, to the equilibrium of foams under the action of surfactants, imaging, micromagnetics and thin films will be addressed.

3:05 pm Thursday, April 27, 2006

Graph Theory: Towards a Grid Theorem for Rank-width and Clique-width

by Sang-il Oum (Math, GT) in Skiles 255

Robertson and Seymour proved that if a graph G does not contain a fixed planar graph, then the tree-width of G is bounded. This theorem has been generalized to binary matroids by Geelen, Gerards, and Whittle. It would be interesting to know whether there is an analogous theorem for rank-width (equivalently, for clique-width). A conjecture is that if a graph G does not contain a fixed bipartite circle graph as a pivot-minor, then the rank-width of G is bounded. Interestingly, this conjecture would, if true, imply the previous two theorems. We survey the known special cases of this conjecture and discuss a new result (joint work with Jim Geelen).

4:30 pm Thursday, April 27, 2006

Colloquium: A Report on the Group of Invertibles of a Banach Algebra

by Claude Schochet (Wayne State University) in Skiles 269

We report progress on the rational homotopy type of $GL(A)$ when $A$ is a homogeneous continuous trace $C^*$-algebra such as those that arise in the definition of twisted $K$-theory. The core examples are when $A = C(X)\otimes M_n({\mathbb C})$ and, more generally, when $A$ is the algebra of continuous sections of a bundle over $X$ with fibre $M_n(\mathbb C )$ or the compact operators. Such a bundle of algebras is classified by its Dixmier-Douady invariant in $H^3(X;{\mathbb Z})$. To what extent does the twisting of the bundle affect the rational homotopy of $GL(A)$?

9:00 am Friday, April 28, 2006

Mathematics Curriculum Conference: Rethinking the Mathematics Curriculum for Engineering and Science Students

by Various engineers, computer scientists and mathematicians (Georgia Tech, Univ of Illinois-Urbana, Virginia Tech and others) in Lecture Hall 203 Management Building

This conference will bring together engineers, computer scientists and mathematicians to discuss the way that recent developments in science and engineering are changing the kind of mathematics that is being used in upper level science and engineering courses and in research, the ways mathematics is used there, and the depth of understanding that is required for effective use. See http://www.math.gatech.edu/~carlen/CurConf/ for more details.

3:00 pm Friday, April 28, 2006

Combinatorics Seminar: Graph Partitioning and Embeddability of Negative Type Metrics into L_1

by Nisheeth Vishnoi (College of Computing, Georgia Tech and IBM-India) in Skiles 255

An important problem in CS/VLSI is to partition a graph into two "roughly" equal parts so as to minimize the number of edges crossing the partition. A classic result of Leighton and Rao ('88) gave a Linear Programming based O(log n) factor approximation algorithm for this problem. Soon, a Semidefinite Programming (SDP) based approach was suggested and it was conjectured that it would lead to an O(1) factor approximation algorithm for this problem. In 2004, the breakthrough result of Arora, Rao and Vazirani made progress towards this by showing that this SDP based approach gives an O(sqrt(log n)) factor approximation algorithm. There is an intimate connection between this SDP based approach and the theory of metric embeddings. Feasible solutions to the SDP for this problem correspond to Negative Type (or L_2^2) metrics, and the worst distortion needed by such an n-point metric to embed into a metric in L_1 upper bounds the approximation factor. This led Goemans and Linial to make a (stronger) conjecture that every negative type metric embeds into L_1 with O(1) distortion. We disprove both these conjectures by constructing an \Omega(log log n) integrality gap instance for this SDP. This implies an n-point negative type metric which requires distortion at-least \Omega(log log n) to embed into L_1. The focus of this talk will be metric embeddings and I will describe the construction and outline the proof. The talk is based on two papers: 1) with Subhash Khot in FOCS '05 and 2) with Nikhil Devanur, Subhash Khot and Rishi Saket in STOC '06.

4:00 pm Monday, May 1, 2006

ACO Distinguished Lecture Series: The Abelian Sandpile Model

by Laszlo Babai (The University of Chicago) in Emory University, MSC E208

http://www.mathcs.emory.edu/News/Seminar/200605012359.Babai.pdf

Evans/Hall award reception to follow MSC atrium

11:05 am Thursday, May 4, 2006

ACO Colloquium: Many Hamiltonian Cycles

by Jeff Kahn (Mathematics, Rutgers University) in Skiles 255

We'll begin with the following theorem, which proves a conjecture of Sarkozy, Selkow and Szemeredi, and try to use it as an excuse to talk about other things (e.g., Bregman's Theorem, entropy, the "incremental random method," statistics physics ...).

Theorem Any n-vertex Dirac graph (i.e., graph of minimum degree at least n/2) contains at least (2-o(1))^{-n}n! Hamiltonian cycles.

Joint with Bill Cuckler.

3:05 pm Friday, May 5, 2006

Combinatorics seminar: The hardness of approximating the dimension of a poset

by Rajneesh Hegde (Math, GT) in Skiles 255

The dimension of a partially ordered set (poset) is the minimum integer k such that the partial order can be expressed as the intersection of k total orders. We prove that unless P = NP there exists no polynomial-time algorithm to approximate the dimension of a poset on $n$ elements within a factor of sqrt(n). The same hardness of approximation holds for the fractional version of poset dimension.

8:05 am Wednesday, May 10, 2006

IEEE International Conference: Data and Learning (Keynote Speech)

by Stephen Smale (Department of Mathematics, University of California, Berkeley) in Georgia State University

IEEE-GrC2006 and GSU have jointly organized a distinguished lecture series that features two "Nobel" Laureates: Stephen Smale (winner of the Fields Medal, often known as the "Nobel Prize" for Mathematics), Lotfi Zadeh (winner of the IEEE-Medal of Honors, often known as the "Nobel Prize" for Electrical Engineering) and two pioneer experts: T. Y. Lin in Granular Computing and V. Vapnik in Support Vector Machines. The three morning keynote/panel events are free for Georgia State University faculty and students (see more detailed information at http://www.cs.sjsu.edu/~grc/program.html . If you'd like to attend, please send an email to Tammie Dudley (tdudley@cs.gsu.edu). The registration fee is $40 for each morning session. Please note that seating is limited and priority will be given to registered participants.

11:00 am Thursday, May 25, 2006

Combinatorics Seminar: Coloring Non-Uniform Hypergraphs red and blue

by Lincoln Lu (Department of Mathematics, University of South Carolina) in Skiles 255

Let f(r)=\min_H\sum_{F\in E(H)}\frac{1}{2^{|F|}}, where H ranges over all 3-chromatic hypergraphs with minimum edge cardinality r. Erdos-Lovasz (1975) conjectured f(r)\to\infty as r\to \infty. This conjecture was proved by Beck in 1978. Here we show a new proof for this conjecture with a better lower bound: f(r)\geq (\frac{1}{4}-o(1))\frac{\ln r}{\ln\ln r}.

3:05 pm Monday, June 5, 2006

Combinatorics Seminar: Robust Mixing Time

by Murali Krishnan Ganapathy (University of Chicago) in Skiles 255

We develop a new notion of "robust mixing time" of a Markov chain, which is the mixing time that results when steps of the chain are interleaved with steps of a stochastic matrix chosen by an (oblivious) adversary. We use this framework to estimate the mixing times for certain non-Markovian processes and for reversible liftings of Markov Chains. Non-Markovian card shuffling processes: The random-to-cyclic transposition process is a non-Markovian card shuffling process, which at time t, exchanges the card at position t (mod n) with a random card. Mossel, Peres and Sinclair (2004) showed that the mixing time of this process lies between (0.0345 + o(1))n log n and 4x10^5 n log n + O(n). We reduce the constant C to 1 by showing that the random-to-top transposition chain (a Markov Chain) has robust mixing time n log n + O(n) when the adversarial strategies are limited to strategies which preserve the symmetry of the underlying Markov Chain. Reversible liftings: Chen, Lovasz and Pak showed that for a reversible ergodic Markov Chain P with stationary distribution \pi, any reversible lifting Q of P must cannot mix faster than P by more than a \log (1/\pi_*) factor, where \pi_* = \min_x \pi(x). Looking at a specific adversarial strategy allows us to show that the mixing time of Q is at least the relaxation time of P. This helps identify cases where reversible liftings cannot improve the mixing time by more than a constant factor.

1:00 pm Thursday, July 27, 2006

REU MiniConference Presentation: The Four Vertex Theorem and the Extension of its Converse to the Sphere

by Bobby DeMarco (University of Delaware) in Skiles 269

Adolf Knesser proved in 1912 that any simple closed planar curve has at least four vertices, or points of maximum or minimum curvature. Over the past almost hundred years many versions and extensions of this theorem have been explored. Notably, in 1971 Herman Gluck proved a type of converse to the four vertex theorem. Gluck showed that given a strictly positive curvature function with at least four vertices, it is possible to construct a simple closed planar curve as a function of time, t, such that the curvature at any point on this curve corresponds to the given curvature function. This presentation will discuss the continuation of Gluck's work and examine how Professor Ghomi and I hope to extend it to the sphere.

1:30 pm Thursday, July 27, 2006

REU MiniConference Presentation: Alexandrov's Conjecture: Intrinsic diameter and area of convex surfaces

by Brian Nakamura (School of Mathematics, Georgia Tech) in Skiles 269

This 50 year old conjecture by Alexandrov has commonly been cited, yet there are very few known results. An introduction to the conjecture and some recent progress will be presented. A few observations and partial results produced during the summer REU will also be discussed.

2:15 pm Thursday, July 27, 2006

REU MiniConference Presentation: The Homology of 17-Crossing Knots: A Computational Approach to Knot Theory

by Darshan Bryner (School of Mathematics, Georgia Tech) in Skiles 269

Knot Theory is a relatively new field in Mathematics, and hence many mathematical properties of knots remain unfounded. To date, a comprehensive list of all knots without repetitions of up to only 16 crossings exists. In this research project, Lew, Stavros, and I have created a database to store knots of up to 17 crossings and implemented a program to hopefully form the most accurate list in existence of such knots without repetitions. My talk will address knot theory terminology and definitions, the software tools used in our computation, the structure of our database, and the use of parallel processors to achieve the necessary computational power.

2:45 pm Thursday, July 27, 2006

REU MiniConference Presentation: Time-Optimal Control of Bioterror Response Logistics: The Case of Anthrax

by Andrew Brown (Biology, Georgia Tech) in Skiles 269

This project begins the analysis of optimal control of bioterror response logistics. The necessary conditions for time-optimal control are given, based on the calculus of variations and dynamic programming. A gradient algorithm is implemented and used to test the queueing network model as well as the algorithm's own reliability. Experimental simulation results are presented and discussed.

3:30 pm Thursday, July 27, 2006

REU MiniConference Presentation: Discrete Biorthogonal Polynomials

by Beth Hart (School of Mathematics, Georgia Tech) in Skiles 269

We have been analyzing discrete biorthogonal polynomials, which generalize the notion of discrete orthogonal polynomials. Our polynomials P_n of degree n satisfy a biorthogonality relation such as \sum_&ob;j=0&cb;^\infty&ob;P_n(x_j)(\psi (x_j))^k w(x_j)&cb; = 0, for k = 0,1,...,n-1. Here x_0,x_1,... is a sequence of points, the weight w is positive on all these points, and \psi is an increasing function on an interval containing all x_j. The special case \psi (x) = x generates discrete orthogonal polynomials. We investigate expicit formulae for P_n for various choices of \psi, x_j, and w.

4:00 pm Thursday, July 27, 2006

REU MiniConference Presentation: An Analysis of the Three Hat Problem

by Brian Benson (School of Mathematics, Georgia Tech) in Skiles 269

In 2003, the Three Hat Problem written by Donald Aucamp appeared in MIT's Technology Review. The puzzle gives a scenario in which three people wearing hats are sitting together and each hat can be seen by everyone except the person that is wearing that hat. Each person is told that all of the hats contain a positive integer and that two of the integers add to the third. In an ordered, turn-wise, modular fashion, each person truthfully states whether or not he knows his number. We present a mathematical method by which to analyze all positive integer cases of the Three Hat Problem puzzle. Using this analysis, we show how and why it is always the case that one of the three people can figure out the integer that is on his hat. Further, we give a simple algorithm to derive the dialogue for any case and ordering of three integers that might occur in the puzzle.

4:30 pm Thursday, July 27, 2006

REU MiniConference Presentation: Searching for Periodicity in the Game of Officers

by Garret Thompson (Computer Science, Georgia Tech) in Skiles 269

A taking-and-breaking game is played by removing beans from a heap and then splitting what remains into new heaps, where the number of beans removed and the number of heaps depends on the rules of the game. According to the Sprague-Grundy theorem, each position in such a game is equivalent to a position in a simpler game called Nim, and if you can determine a function mapping heap sizes to "Nim-values," you can play the game optimally. It is conjectured that this function is periodic for most taking-and-breaking games, and we explore techniques that have been used to solve other games, applying them to a specific game called Officers.

1:00 pm Friday, July 28, 2006

REU MiniConference Presentation: An algebraic approach to a graph choosability problem

by Masanori Koyama (Harvey Mudd College) in Skiles 269

A graph is called n-list colorable if for every family of sets S_v (v in G) with |S_v| =k for all v, there exists a proper coloring of the graph such that each v is colored from the colors in S_v. The smallest n such that the graph G is n-list colorable is called the choosability of G and is denoted ch(G). It is known that \chi(L(G)) = ch(L(G)) =3 for 2-connected cubic planar graphs. It is not known if this holds for cubic non-planar graphs, although it is known for to hold for K_&ob;3,3&cb;. Using an algebraic method, attempt was made to show the 3 edge-choosability of K_&ob;3,3&cb; with vertices replaced with triangles.

1:30 pm Friday, July 28, 2006

REU MiniConference Presentation: On the Abel-Jacobi Map from a Graph to its Jacobian

by Dragos Ilas (Physics, Georgia Tech) in Skiles 269

There is a natural way to attach to any graph a finite Abelian group called the Jacobian group, or Picard group, of the graph. There is also a natural map from the graph to its Jacobian group called the Abel-Jacobi map. A program was written to calculate the Jacobian group and the Abel-Jacobi map using the Smith Normal Form. We discuss some conjectures and theorems about the injectivity and surjectivity of powers of the Abel-Jacobi map. The latter is related to the number of linearly independent cycles which the graph possesses. Finally, we investigate the relationship of these questions to a certain chip-firing game on graphs.

2:15 pm Friday, July 28, 2006

REU MiniConference Presentation: Markov Chains and Traffic Analysis

by Emanuel Indrei (School of Mathematics, Georgia Tech) in Skiles 269

In this study, we use Markov chains to construct a theoretical traffic system. The presentation is organized into three major parts: The first two deal with the construction of two spaces in which objects may interact. The third part analyzes the evolution of one particular object. Using the central limit theorem and bounds given by the law of iterated logarithm, we prove that after a large number of time steps, the probability of locating this object in the traffic network diminishes to zero. We conclude by offering methods of analyzing the evolution and interaction of multiple objects and accounting for accidents.

2:45 pm Friday, July 28, 2006

REU MiniConference Presentation: GOEDEL

by Lee Martie (Computer Science, Georgia Tech) in Skiles 269

The GOEDEL program, Belinfante's computer implementation of Kurt Godel's algorithm for class formation in Mathematica, was used to formulate definitions and theorems in the theory of relations, abstract algebra and arithmetic. The ultimate goal would be to use the results derived to obtain automated proofs using McCune's automated reasoning program Otter. The specific focus of this research was to translate into the language of the GOEDEL program the formulation of Peano arithmetic in terms of unary algebras as expounded by Birkhoff and Bartee in their book on Modern Applied Algebra. Several Mathematica notebooks were prepared and posted on the web, detailing some of the more interesting results obtained in the course of this work. The first notebook dealt with the class UNOPS of all unary operations. A unary operation is a function whose range is contained in its domain. General properties of this class were derived, and examples of unary operations were provided. The second notebook posted was about partial orders and equivalence relations. In this notebook, direct proofs were obtained for results that had initially been found using reification, a procedure for associating relations to constructors. The third notebook was about a theorem concerning unions of commuting transitive relations. Two notebooks were posted about powers of unary operations. A formula for the class of mappings with one-point domains was derived, as a first step toward proving theorems about mappings using induction on the size of their domains. The final topic studied concerned cyclic unary algebras, and the clock algebra in particular.

3:30 pm Friday, July 28, 2006

REU MiniConference Presentation: Distinguishing Legendrian Knots

by Gokhan Civan (School of Math and Aerospace Engineering, Georgia Tech) in Skiles 269

Legendrian knots are knots that satisfy certain geometric properties. There are various algebraic invariants that are used to differentiate between Legendrian knots. Recently a very complicated invariant in the form of a differential graded algebra was discovered. There are also invariants that are derived from this more general invariant through a process of linearization. There are open questions about the equivalence or relative strengths of these derived invariants. This research consisted of investigating examples in hopes of shedding light onto some of these questions. The effort has been assisted by a Mathematica code which has been developed along the way and could culminate in a general tool to compute some of these invariants for Legendrian knots.

4:00 pm Friday, July 28, 2006

REU MiniConference Presentation: A Mathematical Framework for Paper Folding

by Steven Britt and Laura Stiltz (School of Mathematics, Georgia Tech) in Skiles 269

Our research has focused on utilizing the standard ideas of differential geometry to mathematically describe the notions associated with paper folding. We define folding functions both along lines, as is traditionally associated with folding, as well as more generally along planar curves. We then prove the intuitively necessary condition that any folding as we have defined is isometric to a plane. Further questions arise upon examination of the processes by which the paper is folded as a function of time. We also present a few examples, conjectures, and proofs on various topics in paper folding.

4:00 pm Thursday, August 10, 2006

Theory Seminar: Heat Flow and a Faster Algorithm to Compute the Surface Area of a Convex Body

by Hariharan Narayanan (University of Chicago) in CoC 102

We draw on the observation that the amount of heat diffusing outside of a heated body in a short period of time is proportional to its surface area, to design a simple algorithm for estimating the surface area of a convex bodies given by a membership oracle. Our method has a complexity of O^*(n^4), where $n$ is the dimension, compared to O^*(n^{8.5}) for the previous best algorithm. We show that this estimate cannot be improved given the current state-of-the-art in volume computation.

4:30 pm Monday, August 21, 2006

CDSNS Colloquium: Computing Arnol'd tongue scenarios

by Frank Schilder [mail] (University of Bristol) in Skiles 269

A famous phenomenon in circle-maps and synchronisation problems leads to a two-parameter bifurcation diagram commonly referred to as the Arnol'd tongue scenario. One considers a perturbation of a rigid rotation of a circle, or a system of coupled oscillators. In both cases we have two natural parameters, the coupling strength and a detuning parameter that controls the rotation number/frequency ratio. The typical parameter plane of such systems has Arnol'd tongues with their tips on the decoupling line, opening up into the region where coupling is enabled, and in between these Arnol'd tongues, quasi-periodic arcs. In this talk we present a unified framework for computing these objects for both, maps and vector fields. We illustrate our methods by numerically investigating the Arnold tongue scenario for a generic caricature map example.

4:30 pm Monday, August 21, 2006

Combinatorics seminar: Spending Constraint Utilities, with Applications to the Adwords Market

by Vijay V. Vazirani (CoC, GT) in Skiles 255

The notion of a "market" has undergone a paradigm shift with the Internet -- totally new and highly successful markets have been defined and launched by companies such as Google, Yahoo!, Amazon, MSN and Ebay. Another major change is the availability of massive computational power for running these markets in a centralized or distributed manner. In view of these new realities, the study of market equilibria, an important, though essentially non-algorithmic, theory within Mathematical Economics, needs to be revived and rejuvenated via an inherently algorithmic approach. Such a theory should not only address traditional market models but also define new models for some of the new markets. We present a new, natural class of utility functions which allow buyers to explicitly provide information on their relative preferences as a function of the amount of money spent on each good. These utility functions offer considerable expressivity, especially in Google's Adwords market. In addition, they lend themselves to efficient computation, while still possessing some of the nice properties of traditional models. This talk is based on the following paper: http://www-static.cc.gatech.edu/fac/Vijay.Vazirani/spending.pdf

11:00 am Thursday, August 24, 2006

Nonlinear Science Seminar: Density Dependent, Age Structured Population Models

by Howard Weiss (Georgia Tech) in Howey N110

We discuss our long-term program to understand the dynamics of general classes of density dependent, age structured (Leslie) population models. For some families we find a plethora of extremely complicated dynamical behaviors, several of which have not been previously observed in age structured population models, and which may give rise to new paradigms in population biology. We attempt to provide a rigorous foundation for the population biologist's notion of "ergodicity" by constructing SRB or natural measures for some of these models. Using 20 years of population data from a local research stream, my Penn State fisheries colleague Robert Carline has discovered density (and seasonal stream flow) dependence of some survival probabilities for brown trout. I will discuss our preliminary progress in modeling this trout population using a density dependent, age structured model.

2:00 pm Thursday, August 24, 2006

WebCT : Setting up WebCT

by Klara Grodzinsky (School of Mathematics, Georgia Tech) in Skiles 255

Learn the basics about WebCT. This will be a very informal introduction of the system. For example, learn how to set up a grade book, so your TA can record grades. I'll create a standard WebCT shell that can be amended for your course.

2:30 pm Monday, August 28, 2006

Geometry-Topology Seminar: Contact structures and foliations

by John Etnyre (GaTech) in Boyd 222, University of Georgia

In the min 1990's Eliashberg and Thurston proved that any foliation (apart from the foliation of S^2 X S^1 by S^2's) can be perturbed into a contact structure. This has had major implications in contact geometry and low dimensional topology over the last few years. After surveying some of the past work in this area I will discuss a recent observation that all contact structures are perturbations (even deformations) of foliation. This result follows form Giroux's connection between open book decompositions of 3-manifolds and contact structures and some simple constructions and computations.

4:00 pm Monday, August 28, 2006

Geometry-Topology Seminar: Laminations, branched surfaces and contact structures.

by Skander Zannad in Boyd 222, University of Georgia

Laminations are an intermediate between surfaces and foliations. In particular, essential laminations generalize both incompressible surfaces and tight foliations. The main topological result about essential laminations, is that the universal cover of a manifold of dimension 3 carrying an essential lamination is $\mathbb{R}^3$. To study laminations, branched surfaces appear to be useful objects. In this talk, we will give a sufficient condition for a branched surface to fully carry a lamination, giving a piece of answer to a problem D. Gabai. In an attempt to link the theory of contact structures to the theory of branched surfaces and laminations, we give a sufficient condition so that two contact structures forming a pair are carried by the same branched surface. We will see how this result may be used to get topological results.

4:30 pm Monday, August 28, 2006

CDSNS Colloquium: Some phenomena in networks modeled by generalized Lotka-Volterra systems

by Valentin Afraimovich (San Luis Potosi University, Mexico ) in Skiles 255

A model of sequential activity in the networks of active elements is proposed. It is based on the notion of stable heteroclinic sequence. The following phenomena are supposed to be discussed: reproducibility of sequential dynamics; dynamics of sequential decision making as a form of information dependent transient activity; learning of sequences. Some theoretical and numerical results will be presented.

3:00 pm Tuesday, August 29, 2006

PDE Seminar: Solutions of the isentropic Euler equations with Finite Energy

by Michael Westdickenberg (School of Mathematics, Georgia Tech) in Skiles 255

We consider the isentropic Euler equations and establish the global existence of entropy solutions to the Cauchy problem. We only assume that the initial data have finite total mass and energy. Our proof relies on a new higher integrability property for the density and on the propagation of higher integrability for the total energy. To obtain strong convergence of sequences of approximate solutions, we use compensated compactness and Young measures based in L^p-type uniform bounds. (Joint work with Philippe G. LeFloch)

4:00 pm Wednesday, August 30, 2006

Analysis Seminar: Small Ball Inequality in 3 dimensions

by Michael Lacey (GT) in Skiles 255

Let $h_R$ be Haar functions adapted to a dyadic rectangle in three dimensions, taking the values $-1,0,1$. We prove a non trivial lower bound on the $L^\infty$ bound of hyperbolic sums of these functions. This bound is a substantial improvment, and simplification of a famous, and famously difficult result of Jozef Beck. It is related to questions in Irregularities in Distribution, Approximation Theory, and Probability Theory.

4:30 pm Wednesday, August 30, 2006

Applied and Computational Mathematics Seminar: Lanczos Methods for Ill-Posed Problems in Image Processing

by James Nagy (Emory University) in Skiles 269

Ill-posed problems arise in many image processing applications, including microscopy, medicine and astronomy. Iterative methods are typically recommended for these large scale problems, but they can be difficult to use in practice. For example, it may be difficult to determine an appropriate stopping criteria for fast algorithms, such as the conjugate gradient method; noise contaminates the iterates very quickly, so an imprecise stopping criteria can lead to poor reconstructions. Lanczos based hybrid methods have been proposed to slow the introduction of noise in the iterates. These methods require choosing a regularization parameter for a small subproblem at each iteration. It has been shown that if these parameters are chosen optimally, then the Lanczos based methods can be very effective. In this talk we illustrate difficulties that can arise in practice when attempting to choose the regularization parameters automatically, and consider a modification of the generalized cross validation method for this purpose. Image processing examples are used to illustrate concepts and to test and compare algorithms.

10:00 am Thursday, August 31, 2006

QCF Seminar: A horse-race among competing option pricing models using S&P 500 index options

by Minqiang Li (College of Management, Georgia Tech) in Skiles 269

The last three decades have witnessed a whole array of option pricing models. We compare the predictive performance of a selection of models by carrying out a "horse race" on S&P 500 index options along the lines of Jackwerth and Rubinstein (2001). Trader rules still dominate mathematically more sophisticated models, and the performance of the trader rules is further improved by incorporating the stable index skew pattern documented in Li and Pearson (2005). Mathematically more sophisticated models vary in their overall performance and their relative accuracy in forecasting future volatility levels and future volatility skew shapes.

3:05 pm Thursday, August 31, 2006

Stochastic Seminar: Concentration and Fluctuations for Lévy Processes

by Christian Houdré (School of Mathematics, Georgia Tech) in Skiles 269

We estimate a median of f(Xt) where f is a Lipschitz function, X is a Lévy process and t is an arbitrary time. This leads to concentration inequalities for f(Xt). In turn, corresponding fluctuation estimates are obtained under assumptions typically satisfied if the process has a regular behavior in small time and a, possibly different, regular behavior in large time. Joint work with P. Marchal.

3:30 pm Friday, September 1, 2006

Geometry-Topology Reading Seminar: Introduction to Knot Heegaard Floer Theory

by John Etnyre (GaTech) in Skiles 269

I will being discussing Heegaard decompositions associated to knots and links and use this to define the Heegaard Floer Knot groups. This might take a couple of lectures. After this we will try to understand recent work that provides an algorithm for computing these groups.

2:00 pm Wednesday, September 6, 2006

Research Horizons Seminar: Some of my favorite coloring problems

by Prof. Robin Thomas (Georgia Institute of Technology) in Skiles 255

I will discuss several problems in the area of graph coloring, especially coloring graphs on surfaces and problems that could be attacked using algebraic methods. Some are undoubtedly hard, but I will focus on those that are of current interest and where I think decent progress can be made.

4:30 pm Wednesday, September 6, 2006

Analysis : Asymptotic results for the minimum energy and best-packing problems

by S.V. Borodachov (GTech) in Skiles 255

We consider a generalization of the classical Thomson problem dealing with the final (equilibrium) positions of N electrons repelling each other on the surface of a sphere. The potential of the repelling force in our consideration is assumed, more generally, to be proportional to the reciprocal of the power s>0 of the distance between points and the points (electrons) are constrained to a compact rectifiable set A embedded in Euclidean space of a higher dimension. We study this problem for s being greater than or equal to the Hausdorff dimension of the set (for the rest of the values of s the solution follows from the potential theory). When N is fixed and s gets large, the minimum energy problem becomes the problem of maximizing the separation distance between N points on A (also known as the packing radius). Using concepts from the geometric measure theory we obtain asymptotic behavior as N gets large, of the minimal energy and the packing radius on a general rectifiable set, as well as the limit distribution of the optimal configurations. Possible applications of this problem are also considered. Joint work with E. Saff and D. Hardin, both of Vanderbilt.

3:00 pm Thursday, September 7, 2006

PDE Seminar: Eigenvalues, the ABP inequality and the Dirichlet Problem for non-proper Hamilton-Jacobi-Bellman and Isaacs operators

by Boyan Sivakov (Universite de Paris 10, France) in Skiles 255 (Note change in day)

We study uniformly elliptic fully nonlinear equations of the type F(D^2u, Du, u, x)=f(x). We show that convex positively 1-homogeneous operators possess two principal eigenvalues and eigenfunctions, and study these objects; we prove one-sided ABP inequalities; we obtain existence and uniqueness results for non-proper operators whose principal eigenvalues are positive; non-uniqueness and multiplicity results if only one eigenvalue is positive; finally, we obtain an existence result for non-proper Isaac's equations.

3:05 pm Thursday, September 7, 2006

Stochastic Seminar: Some Recent Results of Sniady Regarding Random Permutations with Constraints

by T. Litherland (School of Mathematics) in Skiles 269

In this talk we present some recent work of Piotr Sniady concerning the limiting shape of the Young diagrams associated with random permutations whose longest decreasing subsequences are no greater than some fixed d in length. It is shown that, up to a rescaling, the limiting law is that of the joint distribution of the eigenvalues of the d x d traceless Gaussian Unitary Ensemble.

4:15 pm Thursday, September 7, 2006

Math Department Tea:

in Skiles 236 (lounge)

This will be our first math department tea. Refreshements (tea, coffee, and sodas) and food will be served. All math faculty, students, and staff are welcome to attend.

10:00 am Friday, September 8, 2006

Graph theory seminar: Spanning 3-colourable subgraphs of small bandwidth in dense graphs

by Mathias Schacht (Humboldt Universitat, Berlin) in Skiles 255

Finding sufficient degree conditions, on a given graph G, which imply that G contains a particular spanning subgraph H is a typical question in extremal graph theory. A well known example of such a result is Dirac's theorem. It asserts that any n-vertex graph G on with minimum degree at least n/2 contains a spanning, so called Hamiltonian, cycle H. We discuss a related result for 3-chromatic graphs H of bounded maximum degree and small bandwidth, which settles a conjecture of Bollobas and Komlos. The proof is based on Szemeredi's regularity lemma and the so called blow-up lemma. This is joint work with Julia Boettcher and Anusch Taraz.

3:30 pm Friday, September 8, 2006

Geometry-Topology Reading Seminar: More on knot Heegaard-Floer theory

by John Etnyre (GaTech) in Skiles 269

2:00 pm Monday, September 11, 2006

Theory of Computation Colloquium: New Results for Learning Noisy Parities and Halfspaces

by Subhash Khot (College of Computing, Georgia Tech) in MiRC 102A

I will present some new results on the learnability of parities and halfspaces in the presence of noise. The results can be informally stated as: (1) In uniform distribution setting, learning juntas on k variables reduces to learning noisy parities on k variables. Learning DNFs reduces to learning noisy parities on O(log n) variables. (2) Learning halfspaces: Given a set of labeled examples in R^n such that there is a halfspace that correctly classifies 1-\eps fraction of examples, it is NP-hard to find a halfspace that is correct on 1/2+\eps fraction of examples for any \eps > 0. This is optimal. (3) It is hard to PAC-learn majorities of halfspaces assuming that the Ajtai-Dwork lattice-based cryptosystem is secure (which amounts to assuming that the shortest vector problem for n-dimensional lattices is hard to approximate within factor n^&ob;O(1)&cb;). Joint work with Vitaly Feldman, Parikshit Gopalan, and Ashok Kumar Ponnuswami. To appear in FOCS'06.

3:30 pm Monday, September 11, 2006

Geometry-Topology Seminar: Construction of new symplectic cohomology S^2 X S^2

by Anar Ahmadov (GaTech) in Skiles 269

There has been some recent activity in the discovery of new simply-connected 4-manifolds with small Euler characteristics. Motivated by these results, we will present new symplectic and infinite family of non-symplectic 4-manifolds with same cohomology as S^2 x S^2. Generalization of this construction will be given as well, i.e an infinite family of symplectic(non-symplectic) 4-manifolds cohomology equivalent to connected sums of S^2 x S^2.

4:30 pm Monday, September 11, 2006

CDSNS Colloquium : Gearhart-Pr\"uss Theorem and linear stability for Riemann solutions of conservation laws

by Xiaobiao Lin [mail] (North Carolina State University ) in Skiles 255

In this talk, we will review the Hille-Yosida Theory, Paley-Wiener Theorem and Gearhart-Pr\"uss Theorem on the asymptotic behavior of semigroups. We consider the spectral and linear stability of of the Riemann solutions with multiple Lax shocks for systems of conservation laws $u_\tau + f(u)_\xi = 0$. Using the self-similar change of variables $x = \xi/\tau, t= ln(\tau)$, Riemann solutions become stationary to the system $u_t + (Df(u) - x I) u_x = 0$. In the space of $O((1+|x|)^&ob;-\eta&cb;)$ functions, we show that if $\Re \lambda > -\eta$, then $\lambda$ is either an eigenvalue or resolvent point. Eigenvalues of the linearized system are zeros of the determinant of a transcendental matrix. On some vertical lines in the complex plane, there are &ob;\em resonance values&cb; where the determinant can be arbitrarily small but nonzero. A $C^0$ semigroup is construced and using the Gearhart-Pr\"uss Theorem, we show that the solutions are of $O(e^&ob;\gamma t&cb;)$ if $\gamma$ is greater than the largest real parts of the eigenvalues and the resonance values. We study examples where Riemann solutions have two or three Lax-shocks.

3:00 pm Tuesday, September 12, 2006

PDE Seminar: Slow Motion in Gradient Flows

by Maria Reznikoff (School of Mathematics, Georgia Tech) in Skiles 255

Sometimes physical systems exhibit "metastability," in the sense that states get drawn toward so-called metastable states and are trapped near them for a very long time. A familiar example is the one-dimensional Allen Cahn equation: initial data is drawn quickly to a "multi--kink" state and the subsequent evolution is exponentially slow. The slow coarsening has been analyzed by Carr & Pego, Fusco & Hale, Bronsard & Kohn, and X. Chen. In general, what causes metastability? Our main idea is to convert information about the energy landscape (statics) into information about the coarsening rate (dynamics). We give sufficient conditions for a gradient flow system to exhibit metastability. We then apply this abstract framework to give a new analysis of the 1-d Allen Cahn equation. The central ingredient is to establish a certain nonlinear energy-energy-dissipation relationship. One benefit of the method is that it gives a natural proof of the fact that exponential closeness to the multi-kink state is not only propagated, but also generated. (This work is joint with Felix Otto, University of Bonn).

4:30 pm Tuesday, September 12, 2006

ACO seminar: Design is as Easy as Optimization

by Aranyak Mehta (IBM Almaden Research Center) in MiRC 102A

Over the last four decades, theoreticians have identified and studied several fundamental genres of algorithmic problems. These include decision, search, optimization, counting, enumeration, random generation, and approximate counting problems. We identify a new genre of algorithmic problems -- design problems. Every optimization problem leads to a natural design problem. For instance, the sparsest cut problem leads to: given an undirected graph $G = (V,E)$ and a bound $B$, find a way of distributing weight $B$ on the edges of $G$ so that the sparsity of the sparsest cut is maximized. Practitioners have always been faced with design problems and such problems have been studied individually by theoreticians as well. However, this genre has not been formally defined before and subjected to a systematic complexity-theoretic study. Using techniques of Freund-Schapire and its follow-up works in learning theory, we show that for a large class of problems, the design version is as easy as the optimization version. Joint work with Deeparnab Chakrabarty and Vijay V. Vazirani. Paper available at: http://www-static.cc.gatech.edu/~aranyak/design-fullversion.pdf

Refreshments will be served at 4:00PM.

5:00 pm Tuesday, September 12, 2006

Undergraduate: Applying to Grad School

by Michael Lacey (GT) in Skiles 269

A guided tour through the steps one has to go through to apply for graduate school, and a few Fellowships. While mainly directed to GT Math Majors, it would also be useful to any GT undergrad considering graduate school in a science or engineering program. After all, many GT Math undergrads go into other programs of study.

2:00 pm Wednesday, September 13, 2006

Research Horizons Seminar: Orthogonal Polynomials and their Relatives

by Prof. Doron Lubinsky (Georgia Institute of Technology) in Skiles 255

We discuss some features of orthogonal polynomials and their cousins (such as biorthogonal and Muntz orthogonal polynomials). Some applications will also be briefly outlined.

4:00 pm Wednesday, September 13, 2006

Analysis Seminar: A new approach to universality limits involving orthogonal

by Doron Lubinsky (GT) in Skiles 269

Universality limits arise in the theory of random matrices and many questions in mathematical physics. We present a new way to prove them for a fixed weight. This method works in a uniform sense when the weight satisfies a local Lipshitz condition of order 1/2. Without any smoothness, it works in an L1 sense. Up to now, the universality limit has been proved only for piecewise analytic weights.

3:05 pm Thursday, September 14, 2006

Stochastic Seminar: Localization and skew-invariant solutions of parabolic models

by Yuri Bakhtin (School of Mathematics, Georgia Tech) in Skiles 269

I will discuss directed polymers in random potential and associated Anderson-type parabolic models. I will consider one very specific model and show how localization for directed polymers is connected with existence of an attracting skew-invariant positive solution for that model. If I have time I will talk about possible generalizations of this result leading to a Perron-Frobenius theory for non-compact linear skew-products. Joint work with K. Khanin (University of Toronto).

4:05 pm Friday, September 15, 2006

Combinatorics: Some analogies between graphs and Riemann surfaces

by Matt Baker (Gerogia Tech) in Skiles 255

I will discuss some analogies between finite graphs and Riemann surfaces, including a Riemann-Roch formula for graphs and some new results concerning the Abel-Jacobi map from a graph to its Jacobian. This is joint work with Serguei Norine.

2:00 pm Monday, September 18, 2006

Theory of Computation Colloquium : An Approach to Bounded Rationality

by Ehud Kalai (Northwestern University) in MiRC 102A

A central question in game theory, learning, and other fields is how a rational intelligent agent should behave in a complex environment, given that it cannot perform unbounded computations. We study strategic aspects of this question by formulating a simple model of a game with additional costs (computational or otherwise) for each strategy. While a zero-sum game with strategy costs is no longer zero-sum, we show that its Nash equilibria have an interesting structure and the game has a new type of "value." We also show that potential games with strategy costs remain potential games. Both zero-sum and potential games with strategy costs maintain a very appealing property: simple learning dynamics converge to Nash equilibrium.

3:30 pm Monday, September 18, 2006

Geometry-Topology Seminar: Tunnel number versus Heegaard genus

by Ken Baker (GaTech) in Skiles 269

Any 3-manifold obtained by Dehn surgery on a tunnel number t knot in S^3 has Heegaard genus at most t+1 -- the Heegaard genus is bounded above by a function of t. One may then wonder if something similar works the other way around: If p/q Dehn surgery on a knot K in S^3 produces a 3-manifold of Heegaard genus g, then is the tunnel number of K bounded above by a function of g? We will discuss the relevant concepts, hypotheses, and approach that allow us to address this question. This is a joint work (in progress) with Cameron Gordon and John Luecke.

4:30 pm Monday, September 18, 2006

CDSNS Colloquium: Dynamics of Polarized Light in Resonant Optical Media

by Gregor Kovacic [mail] (Rensselaer Polytechnic Institute ) in Skiles 255

In a resonant interaction, light of specific wavelengths excites electron transitions between atomic energy levels or energy bands in an active optical medium such as gas aor crystal. In the lambda-configuration, light interacts with a medium via a pair of electron transitions between an energetically higher and two energetically lower atomic levels, which involve light of two different colors and/or opposite circular polarizations. We have identified a switching mechanism in this interaction: The color/polarization of the light will switch so that it will interact with the medium only through the transition between the higher level and the lower level less populated with electrons. If the initial occupation of the two lower levels varies randomly, an optical pulse passing through this material will switch randomly between two colors/polarizations. Mathematically, this phenomenon is described by exact, stochastically varying solutions of a completely integrable random partial differential equation, thus combining the opposing concepts of integrability and disorder. This work was done in collaboration with Ethan Atkins, Courant Institute; Julie Byrne, Siena College; Ildar Gabitov, University of Arizona; Peter Kramer, Rensselaer Polytechnic Institute.

4:00 pm Tuesday, September 19, 2006

ACO Seminar: Large Anonymous Games

by Ehud Kalai (Northwestern University) in Skiles 269

In strategic games with many semi anonymous players, continuous payoff functions and independent types all the equilibria are (asymptotically) structurally robust. This robustness property has important implications in overcoming modeling difficulties. Some examples are: modeling partially-specified games (like games played on the web), games embedded in bigger worlds, and stability of Nash prices in market games. The lecture elaborates on the implications of structural robustness and on sufficient-conditions to attain it.

Refreshments will be served at 4:00PM -- Talk begins at 4:30PM

2:00 pm Wednesday, September 20, 2006

Research Horizons Seminar: Proving and Disproving

by Prof. Tom Trotter (Georgia Institute of Technology) in Skiles 255

The famous "1/3 - 2/3" conjecture of Kislytsin asserts that if P is a poset which is not a chain, then there is an incomparable pair (x,y) for which the fraction of linear extensions in which x is less than y is at least 1/3 and at most 2/3 of the total number of linear extensions. This conjecture was first posed in 1966 and remains open today. In this talk, we sketch proofs of three interesting partial results on this conjecture: 1. The conjecture holds for any poset which has width 2 (size of largest antichain is two; alternatively, union of two chains). 2. The conjecture holds for any poset which is a semiorder (unit length intervals with [a, a+1] < [ b, b+1] when a+1 < b in R). 3. There is a width 2 semiorder for which the conjecture fails.

4:30 pm Wednesday, September 20, 2006

ACM: High-Order Surface Relaxation vs. the Ehrlich-Schwoebel Effect in Thin-film Growth

by Prof. Bo Li (UC San Diego) in Skiles 269

The surface of an epitaxially growing thin film often exhibits a mound-like structure with its characteristic lateral size increasing in time. In this talk, we consider two competing mechanisms for such a coarsening process: (1) surface relaxation described by high-order gradients of the surface profile; and (2) the Ehrlich-Schwoebel (ES) effect which is the upper-lower terrace asymmetry in the adatom attachment and detachment to and from atomic steps. We present a theory based on a class of continuum models that are mathematically gradient-flows of some effective free-energy functionals describing these mechanisms. This theory consists of two parts: (1) variational properties of the energies, such as ``ground states'' and their large-system-size asymptotics, showing the unboundedness of surface slope and revealing the relation between some of the models; (2) rigorous bounds for the scaling law of the roughness, the rate of increase of surface slope, and the rate of energy dissipation, all of which characterize the coarsening process. Predictions on scaling laws made by our theory agree well with experiments.

4:30 pm Wednesday, September 20, 2006

Analysis Seminar: Estimates for Jacobi polynomials and the spectral gap of the physical Kac model

by Eric Carlen (School of Mathematics, Georgia Tech) in Skiles 255

In joint work with Carvalho and Loss, the Kac conjecture for physical, momentum conserving three dimensional collision was established. A key part of the proof is a correlation inequality that relies on some rather subtle and interesting estimates for Jacobi polynomials. This will be explained in the lecture in the hope that experts on orthogonal polynomials can shed further light on the matter.

3:05 pm Thursday, September 21, 2006

Stochastic Seminar: Metastability in irreversible diffusion processes and stochastic resonance

by Barbara Gentz (WIAS Berlin, Germany) in Skiles 269

During the last years, significant progress has been made in the quantitative description of metastability in reversible diffusion processes. In particular, recent works by Bovier, Eckhoff, Gayrard and Klein have established precise relations between metastable lifetimes, the potential landscape, and the small eigenvalues of the diffusion's generator. Much less is known for irreversible diffusions. For a class of irreversible, two-dimensional diffusions, we shall discuss the problem of diffusion exit from domains whose boundary is an unstable periodic orbit. In this situation, the theory of large deviations yields no information on the distribution of exit locations. Studying the first-passage time through the unstable periodic orbit and going beyond exponential asymptotics, we derive an explicit expression for the density of the first-passage time. The density is found to be close to a periodically modulated exponential one, where the periodic modulation is governed by a universal function, depending on a single parameter related to the period of the periodic orbit. As an application, we shall discuss the phenomenon of stochastic resonance, for which our result on the first-passage time distribution allows to determine a precise expression for the residence-time distribution. This is joint work with Nils Berglund (CPT-CNRS Marseille, France).

4:00 pm Thursday, September 21, 2006

Department Tea:

in Skiles 236

This will be our next math department tea, which will be in conjunction with the social event for the ACO colloquium. All are welcome.

4:30 pm Thursday, September 21, 2006

Joint ACO/Theory of Computation Colloquium: Line of Sight Networks

by Alan Frieze (Carnegie Mellon University) in Skiles 269

Random geometric graphs have been one of the fundamental models for reasoning about wireless networks: one places n points at random in a region of the plane (typically a square or circle), and then connects pairs of points by an edge if they are within a fixed distance of one another. In addition to giving rise to a range of basic theoretical questions, this class of random graphs has been a central analytical tool in the wireless networking community. For many of the primary applications of wireless networks, however, the underlying environment has a large number of obstacles, and communication can only take place among nodes when they are close in space and when they have line-of-sight access to one another --- consider, for example, urban settings or large indoor environments. In such domains, the standard model of random geometric graphs is not a good approximation of the true constraints, since it is not designed to capture the line-of-sight restrictions. Here we propose a random-graph model incorporating both range limitations and line-of-sight constraints, and we prove asymptotically tight results for k-connectivity. Specifically, we consider points placed randomly on a grid (or torus), such that each node can see up to a fixed distance along the row and column it belongs to. (We think of the rows and columns as "streets" and "avenues" among a regularly spaced array of obstructions.) Further, we show that when the probability of node placement is a constant factor larger than the threshold for connectivity, near-shortest paths between pairs of nodes can be found, with high probability, by an algorithm using only local information. In addition to analyzing connectivity and k-connectivity, we also study the emergence of a giant component, as well an approximation question, in which we seek to connect a set of given nodes in such an environment by adding a small set of additional "relay" nodes. Joint work with Jon Kleinberg, R. Ravi and Warren Debaney.

Refreshments will be served at 4:00 PM in Skiles 236

3:30 pm Friday, September 22, 2006

Geometry-Topology Reading Seminar: Intorductions to Khovanov/Rasmussen Theory

by Shea Vick (Univeristy of Pennsylvania, visiting Ga Tech) in Skiles 269

3:30 pm Friday, September 22, 2006

Geometry-Topology Reading Seminar: Intorductions to Khovanov/Rasmussen Theory

by Shea Vick (Univeristy of Pennsylvania, visiting Ga Tech) in Skiles 269

3:30 pm Monday, September 25, 2006

Geometry-Topology Seminar: Representation theory of hyperbolic groups

by Igor Belegradek (Georgia Tech) in Skiles 269

In the last 20 years the theory of infinite groups has been dominated by geometric methods, and in particular, there has been an extensive study of hyperbolic groups (that can be characterized as groups satisfying linear isoperimetric inequality). In the talk I shall explain that hyperbolicity of a group cannot be detected by the representation theory, more precisely, for every finitely generated group G and any positive integer n, there exists a hyperbolic group that has the same complex n-dimensional representations as G. The proof involves small cancellation theory, harmonic map superrigidity, and algebraic geometry of representation varieties. (This is joint work with Denis Osin).

4:30 pm Monday, September 25, 2006

CDSNS Colloquium: Patterns and Inhomogeneities

by Arnd Scheel [mail] (University of Minnesota) in Skiles 255

Sand ripples, animal coat patterns, or surface waves in fluid convection --- the regularity of spatio-temporally periodic patterns strikes us when the complexity of the system seems to predict incoherence. Motivated by chemical and biological patterns, we study oscillatory patterns in reaction-diffusion systems. The talk will lead us through old and recent results, from existence and bifurcations of wave trains with periodic boundary conditions, modulations of wave numbers and pattern selection in large domains, to the role of impurities and boundary conditions. Most of the results strive for a unified view on generic behavior, which would give us insight into the dynamics of complex systems when little is known about the underlying equations.

3:00 pm Tuesday, September 26, 2006

PDE Seminar: A class of self-dual partial differential equations and its variational principles

by Nassif Ghoussoub (UBC, Vancouver and PIMS) in Skiles 255

Motivated in part by the basic equations of quantum field theory (e.g. Yang-Mills, Ginzburg-Landau, etc...), we introduce and analyse a general -and remarkably encompassing- class of self (and antiself) dual partial differential equations. The class contains many of the basic families of linear and nonlinear, stationary and evolutionary partial differential equations: Transport equations, Nonlinear Laplace equations, Cauchy-Riemann systems, Navier-Stokes equations, Shrodinger equations, but also -infinite dimensional- gradient flows of convex potentials (e.g., heat equations), Hamiltonian systems, and many other parabolic-elliptic equations. We then proceed to develop appropriate variational principles for a systematic resolution of such equations. In both stationary and dynamic cases, the equations associated to the proposed variational principles are not derived from the fact they are critical points of the action functional, but because they are also zeroes of certain derived non-negative Lagrangians.

4:30 pm Tuesday, September 26, 2006

ACM: Laplacian eigenfunctions as a tool for general-shape image analysis

by Naoki Saito (UC Davis) in Skiles 269

I present a method to analyze images recorded on a domain of general shape by expanding them into the eigenfunctions of Laplacian defined there. To compute these eigenfunctions, we diagonalize the integral operator commuting with the Laplacian using a fast algorithm based on Alpert wavelets, instead of directly solving the Helmholtz equation on the domain (which can be quite complicated and costly). We also present its application to image approximation and statistical image analysis and compare the performance with the standard methods such as wavelets and Pricipal Component Analysis/Karhunen-Loeve Transform.

3:00 pm Wednesday, September 27, 2006

Research Horizons Seminar: Sharp constants in conformaly invariant inequalities

by Michael Loss (Georgia Institute of Technology) in Skiles 255

I will talk about a integral inequality that goes back to Hardy, Littlewood and Sobolev. The sharp constant for this inequality was calculated by Lieb in 1983 overcoming substantial difficulties. In this talk I'll give a simple geometric prove due to Carlen and myself that yields also the sharp constant.

10:00 am Thursday, September 28, 2006

QCF Seminar: Mean-Variance Convergence around the World

by Cheol Eun (School of Management, Georgia Tech) in Skiles 269

In this paper, we show (i) that the risk-return characteristics of our sample of 17 developed stock markets of the world have converged significantly toward each other during our study period 1974-2004, (ii) that the speed of convergence, however, varies greatly across individual markets, largely reflecting the "initial position" of each market relative to the rest of markets in the risk-return space, and that (iii) the documented international convergence in risk-return characteristics is driven mainly by the declining "country effect", rather than the rising "industry effecct", suggesting that the convergence is associated with international market integration. Specifically, we first compute the risk-return distance among international stock markets based on the Euclidean distance and find that the distance thus computed has been deceasing significantly over time, implying a mean-variance convergence. In particular, the average risk-return distance has decreased by about 43% over our sample period. Lastly, we document that the risk-return characteristics of our sample of 14 emerging markets have been converging rapidly toward those of developed markets in recent years. This development notwithstanding, emerging markets still remain as a distinct asset class.

3:05 pm Thursday, September 28, 2006

Graph Theory seminar: Riemann-Roch formula for graphs

by Serguei Norine (Math, GT) in Skiles 255

We present proofs of graph-theoretic analogues of the classical Riemann-Roch theorem and Jacobi's inversion theorem. The arguments used are purely combinatorial. As a consequence we characterize the existence or non-existence of a winning strategy for a certain chip-firing game on graphs. This is a follow-up to the talk "Some analogies between graphs and Riemann surfaces" by Matt Baker and is based on joint work with him. Our presentation will not assume familiarity with Matt's talk.

4:15 pm Thursday, September 28, 2006

Department Tea:

in Skiles 236

Food and beverages will be served. All math faculty, students, and staff are welcome.

3:30 pm Friday, September 29, 2006

Geometry-Topology Reading Seminar: More on Khovanov/Rasmussen Theory

by Shea Vick (Univeristy of Pennsylvania, visiting Ga Tech) in Skiles 269

9:00 am Saturday, September 30, 2006

ECOAS Conference: Fourth East Coast Operator Algebras Symposium

by Various Invited Speakers in Skiles 249

This is the fourth meeting of a conference series on Operator Algebras, Noncommutative Geometry, with applications to a wide range of areas in Mathematics and Quantum Physics. For details, see http://www.math.gatech.edu/ecoas06

3:30 pm Monday, October 2, 2006

Geometry-Topology Seminar: Multivariable Link Invariants

by Nathan Geer (GaTech) in Skiles 269

In this talk I will discuss a quantum group type invariant of links arising from finite dimensional modules of a Lie superalgebra type I. This gives rise to a set of multivariable link invariants. I will explain how these invariants are related to other well known invariants.

4:30 pm Monday, October 2, 2006

CDSNS Colloquium: Large Prandtl Number Behavior of the Boussinesq System

by Xiaoming Wang [mail] (Florida State University) in Skiles 255

We consider large Prandtl number behavior of the Boussinesq system for Rayleigh-B\'enard convection at large time. We first show that the global attractors of the Boussinesq system converge to that of the infinite Prandtl number model. This is accomplished via a generalization of upper semi-continuity property with respect to parameters of dissipative dynamical systems (due to Hale) to the case of singular limit of two time scale problems of relaxation type. We then show that stationary statistical properties (in terms of invariant measures) of the Boussinesq system converge to that of the infinite Prandtl number model. In particular, we derive a new upper bound on heat transport in the vertical direction (the Nusselt number) for the Boussinesq system. The new upper bound agrees with the recent physically optimal upper bound on the infinite Prandtl number model at large Prandtl number (due to Constantin, Doering, Otto and Reznikoff).

3:00 pm Tuesday, October 3, 2006

PDE Seminar: Free-Surface Problems in Irrotational 3D Fluids

by David Ambrose (Clemson University) in Skiles 255

In this talk, I will describe a framework for studying free-surface problems in irrotational 3D fluids. The corresponding method for 2D fluids has been used to prove well-posedness of vortex sheets with surface tension, water waves, and Hele-Shaw flows in two dimensions. The method is to first make a careful choice of dependent variables and parameterization of the free surface; this allows the equations of motion to be rewritten in a convenient way. Then, to prove well-posedness, energy estimates are performed.

2:00 pm Wednesday, October 4, 2006

Research Horizons: Arithmetic Progressions in Subsets of Finite Abelian Groups

by Ernie Croot [mail] (Georgia Institute of Technology) in Skiles 255

Suppose that G is a finite abelian group. Denote by r_3(G) the size of the largest subset of G having no three-term arithmetic progressions, which are triples n,n+d,n+2d. What can one say about r_3(G) ? In this talk I will survey some of the known and unknown problems related to this basic question.

4:00 pm Wednesday, October 4, 2006

ACO Distinguished Lecture Series: Advances in Convex Optimization: Conic Programming

by Arkadi Nemirovski (ISyE, GT) in Instructional Center 205

During the last two decades, major developments in Convex Optimization were focusing on Conic Programming, primarily, on Linear, Conic Quadratic and Semidefinite optimization. Conic Programming allows us to reveal rich structure which usually is possessed by a convex program and to exploit this structure in order to process the program efficiently. In the talk, we overview the major components of the resulting theory (conic duality and primal-dual interior point polynomial time algorithms), outline the extremely rich ``expressive abilities'' of Conic Quadratic and Semidefinite Programming and discuss a number of instructive applications.

Reception to follow.

3:05 pm Thursday, October 5, 2006

Graph Theory seminar: Finite subgraphs of uncountably chromatic graphs

by Peter Komjath (Eotvos University, Budapest) in Skiles 255

We survey some recent results on the classes of finite graphs that occur in graphs with uncountable chromatic number

4:30 pm Thursday, October 5, 2006

ACM (Special time for this week): A shallow water model: derivation of a nonlinear Saint-Venant system and its numerical analysis using the pair $P_1$ nonconform - $P_1$ ($P_1^&ob;NC&cb;-P_1$) to compute fluid velocity and pressure.

by Abdou Sene (Universit\'e Gaston Berger, St. Louis, Senegal) in Skiles 269

A serious issue in numerical analysis of Saint-Venant equations is the creation of spurious modes by the most classical numerical schemes such as the pair $P_1 - P_1$. To solve this problem, instead of adding viscosity as usual, we use the pair $P_1^&ob;NC&cb;-P_1$ which does not produce spurious modes. Moreover, the $P_1^&ob;NC&cb;$ elements being orthogonal, the mass matrix related to velocity is diagonal. The latter characteristic reduces the number of operations in solving the system. This model involves streaming waters, inundations and ground percolation.

3:00 pm Friday, October 6, 2006

Geometry/Topology Seminar: Refined analytic torsion

by Maxim Braverman (Northeastern University) in Skiles 269

For a representation of the fundamental group of a compact oriented odd-dimensional manifold we define a refinement of the Ray-Singer torsion associated to this representation. If the representation is acyclic then our new invariant is a non-zero complex number, which can be viewed as an analytic counterpart of the refined combinatorial torsion introduced by Turaev. The refined analytic torsion is a holomorphic function of the space of acyclic representation of the fundamental group. When the representation is unitary, the absolute value of the refined analytic torsion is equal to the Ray-Singer torsion, while its phase is determined by the eta-invariant. The fact that the Ray-Singer torsion and the eta-invariant can be combined into one holomorphic function allows to use methods of complex analysis to study both invariants. I will present several applications of this method. In particular, I will calculate the ration of the refined analytic torsion and the Turaev torsion. (Joint work with Thomas Kappeler)

4:00 pm Friday, October 6, 2006

Combinatorics: Arithmetic Progressions and Fourier Transforms

by Ernie Croot (Georgia Tech) in Skiles 255

In this talk I will prove a certain result which says that if the Fourier transform of a set in the vector space F_p^n, decays sufficiently rapidly, then the underlying set must have lots of three-term arithmetic progressions. This result in particular shows that certain discrete analogues of ``smooth functions'' f : F_p^n -> [0,1] have the property that the support of f contains many three-term arithmetic progressions.

4:15 pm Friday, October 6, 2006

Geometry-Topology Seminar: Integration and symplectic reduction for Teichmueller space

by Scott A. Wolpert (Univeristy of Maryland) in Skiles 269

The Teichmueller space $\mathcal T$ is the space of homotopy conformal structures for a surface . Recent progress on the Weil-Petersson (WP) geometry of $\mathcal T$ includes understanding: the $CAT(0)$ WP geometry, harmonic maps to , the quasi isometric equivalence of $\mathcal T$ to the Thurston-Hatcher pants complex, the existence of designer metrics, and the work of Maryam Mirzakhani on integration and symplectic reduction. We will present an overview of Mirzakhani's integration scheme for and connections to applications. The considerations include McShane's identity, and for the moduli space of Riemann surfaces recursion formulas for WP volumes and intersection numbers of tautological bundles. The recursion formulas satisfy the string equation and the dilaton equation. Greg McShane found for the lengths of simple closed geodesics on a once punctured torus with hyperbolic metric the universal identity \[ \sum_{scg's\ \gamma}\frac{1}{1+e^{\ell_{\gamma}}}=\frac12 \] for {\em simple closed curves}. Mirzakhani generalized the identity and developed a scheme for calculating WP volumes. An example is provided by the volume of the moduli space of hyperbolic metric tori with geodesic boundaries of prescribed lengths $L_1,\ L_2$ \[ V_{1,2}=\frac{1}{192}(4\pi^2+L^2_1+L^2_2)(12\pi^2+L^2_1+L^2_2). \]

2:00 pm Monday, October 9, 2006

Theory of Computation Colloquium: Beyond Hirsch Conjecture: on Smoothed Complexity of the Simplex Algorithm

by Roman Vershynin (UC Davis) in MiRC 102A

Smoothed analysis of algorithms is concerned with the expected running time of an algorithm under slight random perturbations of arbitrary inputs. Spielman and Teng proved that the (shadow-vertex) simplex method had polynomial smoothed complexity. On a slight random perturbation of arbitrary linear program, the simplex method finds the solution after a walk on polytope(s) with expected length polynomial in the number of constraints n, the number of variables d and the inverse standard deviation of the perturbation 1/sigma. Our main result is that the length of walk in the simplex method is actually polylogarithmic, rather than polynomial, in the number of constraints n. This shows that the tight Hirsch conjecture n-d on the length of walk on polytopes is not a limitation for the smoothed Linear Programming. Random perturbations create short paths between vertices. Smoothed analysis is related to theoretical open problems of independent interest, such as estimating the condition number of a random matrix, and the problem of "decoupling" phase-I from phase-II in linear programming. We will discuss some approaches to these problems based on concentration inequalities.

4:30 pm Monday, October 9, 2006

CDSNS Colloquium: The Ohta-Kawasaki equation of diblock copolymers

by Xiaofeng Ren [mail] (Utah State University) in Skiles 255

The Ohta-Kawasaki equation of diblock copolymers is an integral differential equation that arises from a density functional theory of diblock copolymer melts. The equation can also be written as an elliptic system of a Cahn-Hilliard like equation coupled to a linear equation. We will discuss the existence and stability of solutions that exihibit lamellar patterns. Wide stripe patterns are found by the Gamma-convergence method and narrow stripe patterns are found by the Lyapunov-Schmidt redution method. A stability analysis also yieds wiggled lamellar patterns.

3:00 pm Tuesday, October 10, 2006

PDE Seminar: Pressure estimate for the Incompressible Navier-Stokes Equation in a Bounded Domain

by Jian-Guo Liu (University of Maryland) in Skiles 255

Based upon a new, sharp estimate for the commutator of the Laplacian and Helmholtz projection operators, we show that the pressure gradient is bounded in L2 norm by the viscosity term times a constant less than one, up to lower order terms. By consequence, NSE can be regarded as a perturbed diffusion equation, rather than a perturbed Stokes system. This leads to stability results for discretization schemes that (a) provide simple proofs of existence and uniqueness of local strong solutions, and (b) help explain the success of recently developed numerical methods that are fast, accurate near boundaries, and simple and flexible in structure. (This is joint work with Bob Pego, CMU and Jie Liu, U. Maryland.)

10:00 am Wednesday, October 11, 2006

QCF Seminar: A statistical inquiry into the plausibility of Epstein-Zin-Weil utility

by Han Hong (Duke University) in Skiles 269

See http://www.econ.duke.edu/~hanhong/papers/bq.pdf

2:00 pm Wednesday, October 11, 2006

Research Horizons Seminar: The Weierstrass Uniform Approximation Theorem and its History

by William L. Green (Georgia Institute of Technology) in Skiles 255

In 1885 Weierstrass proved that every continuous function on a closed bounded interval is uniformly approximable there by polynomials. Over the next few decades, many other mathematicians gave additional illuminating proofs of this useful result. In the late 1930's, Marshall Stone discovered an important extension of Weierstrass's result to continuous functions on compact Hausdorff spaces, and by 1914 C. Muntz had already published a generalization that took the result in yet a different direction. We give a brief introduction to these results and to the kinds of arguments that have been used to establish them.

3:00 pm Wednesday, October 11, 2006

Mathematical Biology & Ecology Seminar: Mathematical Needs in Systems Biology

by Eberhard Voit [mail] (Department of Biomedical Engineering at Georgia Tech and Emory University) in Skiles 255

The combination of high-throughput methods of molecular biology with advanced mathematical and computational techniques has propelled the emergent field of systems biology into a position of prominence. Unthinkable only a decade ago, it has become possible to screen and analyze the expression of entire genomes, simultaneously assess large numbers of proteins and their prevalence, and characterize in detail the metabolic state of a cell population. While very important, the focus on comprehensive networks of biological components is only one side of systems biology. Complementing large-scale assessments are more subtle analyses that rationalize the design and functioning of biological modules in exquisite detail. This intricate side of systems biology aims at identifying the specific roles of processes and signals in smaller, fully regulated systems by computing what would happen if these signals were lacking or organized in a different fashion. I will exemplify this type of approach with a detailed analysis of an intriguing control system regulating glucose uptake and utilization in the bacterium Lactococcus lactis. The seminar will conclude with a system biologist�s wish list of important topics requiring rigorous mathematics.

4:30 pm Wednesday, October 11, 2006

Analysis : On the Holonomy of the Coulomb Connection over 3-manifolds with Boundary

by William Gryc (Georgia State ) in Skiles 255

The Coulomb gauge (or Coulomb connection) of Yang-Mills theory was introduced by physicists as a way to choose a unique representative connection from the equivalence class of connections related by gauge transformations. Unfortunately, Gribov showed that with reasonable conditions at infinity, there cannot be a unique choice in the Coulomb gauge. This is the so-called Gribov ambiguity. Narasimhan and Ramadas showed that this ambiguity is "maximal" in the following sense: the restricted holonomy group of the Coulomb connection is dense in the connected component of the identity of the gauge group when one considers the product principal bundle S^3 x SU(2) -> S^3. We consider whether this "maximal ambiguity" exists in a different situation. Instead of a base manifold S^3, we consider a base manifold with a boundary and use Dirichlet boundary conditions on connections as defined by Marini. In contrast to an intermediate result of Narasimhan and Ramadas, the image of the curvature form of the Coulomb connection at one fixed point is never dense in the gauge algebra in this with-boundary case. However, if the base manifold is an open subset O of R^3 with smooth boundary then the restricted holonomy group of the Coulomb connection on the product principal bundle O x K -> O is again a dense subset of the connected component of the identity of the gauge group. As unfriendly as this all may sound, this talk will introduce all relevant differential geometric and Yang-Mills concepts and should be accessible to anyone interested in analysis.

4:30 pm Wednesday, October 11, 2006

ACM: Fast Solvers for $C^0$ Interior Penalty Methods

by Susanne Brenner (LSU) in Skiles 269

$C^0$ interior penalty methods are discontinuous Galerkin methods for fourth order elliptic boundary value problems that have many advantages. In this talk we will first give a brief introduction to these methods and then discuss multigrid and domain decomposition methods for solving the resulting systems.

1:30 pm Thursday, October 12, 2006

Clemson Mini-Conference: Discrete Mathematics and Algorithms

in Student Senate Chambers, Clemson University

Continuing a tradition of many years, the Clemson mini-Conference will be held Thursday afternoon and all day Friday. It consists of about a dozen 40-minute talks from invited speakers talking on any topic they like. Registration starts at 1pm on Thursday, with the first talk at 1:30pm. Friday talks are from 9am to 5pm (http://www.cs.clemson.edu/~goddard/MINI/)

3:05 pm Thursday, October 12, 2006

Graph Theory seminar: List-coloring the Square of a Subcubic Graph

by Dan Cranston (University of Illinois, Urbana) in Skiles 255

The square G^2 of a graph G is the graph with the same vertex set as G and with two vertices adjacent if their distance in G is at most 2. Thomassen showed that for a planar graph G with maximum degree Delta(G)=3 we have chi(G^2)<= 7. Kostochka and Woodall conjectured that for every graph, the list-chromatic number of G^2 equals the chromatic number of G^2, that is chi_l(G^2)=chi(G^2) for all G. If true, this conjecture (together with Thomassen's result) implies that every planar graph G with Delta(G)=3 satisfies chi_l(G^2)<= 7. We prove that every graph (not necessarily planar) with Delta(G)=3 other than the Petersen graph satisfies chi_l(G^2)<= 8 (and this is best possible). In addition, we show that if G is a planar graph with Delta(G)=3 and girth g(G)<= 7, then chi_l(G^2)<= 7. Dvorak, Skrekovski, and Tancer showed that if G is a planar graph with Delta(G) = 3 and girth g(G) <= 10, then chi_l(G^2)<= 6. We improve the girth bound to show that: if G is a planar graph with Delta(G)=3 and g(G) <= 9, then chi_l(G^2) <= 6. All of our proofs can be easily translated into linear-time coloring algorithms.

3:05 pm Thursday, October 12, 2006

Stochastic Seminar: Estimation of the Frequency of Sinusoids from Noisy Observations

by Ta-Hsin Li (Department of Mathematical Sciences, IBM T.J. Watson Research Center) in Skiles 269

Estimation of the frequency of sinusoidal signals is an important problem in many applications such as radar, sonar, and telecommunications. Statistical efficiency and computational simplicity are among the desired properties of estimation methods. The Gaussian maximum likelihood (GML) method has the first property but suffers from some serious computational problems. This talk discusses an alternative method, based on iterative filtering, which overcomes the computational problems and retains the statistical efficiency of the GML method.

4:30 pm Thursday, October 12, 2006

School of Mathematics Colloquium: A phase transition in a model for the spread of an epidemic

by Harry Kesten (Cornell) in Skiles 269

Abstract: http://www.math.gatech.edu/~bakhtin/phasetrn4.pdf

2:00 pm Friday, October 13, 2006

Stochastic Seminar: Empirical Graph Laplacian Approximation of Laplace-Beltrami Operators: Large Sample Results

by Evarist Gine (Department of Mathematics, University of Connecticut) in Skiles 255 (Note: Day, Time, Room changes)

See http://www.math.gatech.edu/news/seminars/abstracts/gine.pdf

3:00 pm Friday, October 13, 2006

Mathematical Physics Seminar: Finite speed of propagation in quantum lattice systems and applications

by Bruno Nachtergaele (UC Davis) in Skiles 255

We give a short proof of the Lieb-Robinson bound for a general class of quantum lattice systems and discuss several applications: an upper bound on the speed of information transmission in quantum channels, an upper bound on the propagation of correlations and entanglement, an upper bound on the correlation length in the ground state of quantum spin models in terms of the spectral gap above the ground state, and a Lieb-Schultz-Mattis Theorem in arbitrary dimensions.

2:00 pm Wednesday, October 18, 2006

Research Horizons Seminar: Basic Fourier Analysis in Additive Combinatorics

by William McCLain (Georgia Institute of Technology) in Skiles 255

I'll show some problem solving techniques in Additive Combinatorics that are derived from Fourier Analysis. I'll offer different perspectives on interpreting the problem solving techniques and present them in the most elementary way possible. First we'll see a basic proof of Roth's theorem addressing 3-term Arithmetic Progressions in subsets of the integers. Then we'll see the complications and adaptations of the problem solving techniques used to tackle problems concerning 4-term arithmetic progressions in subsets of the integers.

4:30 pm Wednesday, October 18, 2006

ACM: Numerical Solution of the Nonlinear Helmholtz Equation

by Semyon Tsynkov (NC State) in Skiles 269

The nonlinear Helmholtz equation (NLH) models the propagation of electromagnetic waves in Kerr-type media and describes an array of important phenomena in optics and in other areas. We will discuss the main difficulties that it presents for analysis, and introduce a new fourth-order method for its numerical solution. A key element of the method is a special nonlocal two-way artificial boundary condition. It facilitates reflectionless propagation of all the outgoing waves while also fully transmitting the given incoming field at the boundaries of the computational domain. For the first time in the literature, the method enables direct simulation of nonlinear self-focusing in the nonparaxial regime, and gives a quantitative prediction of the important phenomenon of backscattering. We will present numerical results for a variety of settings, including critical self-focusing and subcritical propagation of the solitary waves with collisions; the latter computed using expansions with respect to non-orthogonal systems of eigenfunctions that satisfy local Sommerfeld radiation boundary conditions in the transverse direction. Recently, this work has been taken to a new dimension by introducing compact high-order discretizations and a Newton-based nonlinear solver. The compact schemes are of a finite volume type and allow us to analyze material discontinuities. Newton's linearization for the problem is nontrivial because the Kerr nonlinearity contains absolute values of the field and becomes non-differentiable in the sense of Frechet for complex-valued time-harmonic solutions. Thus, the NLH has to be recast as a system of two equations in the real form, in which case Newton's method guarantees rapid convergence of nonlinear iterations and enables computations for very high levels of nonlinearity that could not be achieved previously. Joint work with G. Baruch and G. Fibich, Tel Aviv University.

4:30 pm Wednesday, October 18, 2006

Analysis Seminar: Matrix valued orthogonal polynomials and differential equations

by Alberto Grunbaum (UCBerkeley) in Skiles 255

The problem of determining families of matrix valued orthogonal polynomials (a subject that goes back to M G Krein) with extra properties, such as satisfying differentail equations (with matrix coefficients) has received certain attention recently. I review some progress in the area, in particular the study of the noncommutative algebra of differential operators that has a given family of orthogonal polynomials as its common eigenfunctions (with a scalar valued eigenvalue parameter).

3:05 pm Thursday, October 19, 2006

Graph theory seminar: Some results from Graph Minors XI

by Torsten Inkmann (Math, GT) in Skiles 255

It is a well known result that the branchwidth of a graph can be expressed as the maximum order of a tangle in the graph. In general, tangles are not easily characterized, but in Graph Minors XI, Robertson & Seymour discuss how a particular type of tangles can be described in a geometric way, if the graph is embedded on some surface. These results also form the basis of an algorithm to compute the branchwidth for planar graphs in polynomial time. We will discuss some of the results of Graph Minors XI. No familiarity with branchwidth or tangles will be assumed.

3:05 pm Thursday, October 19, 2006

Stochastic Seminar: Fence Methods for Mixed Model Selection

by Jiming Jiang (UC Davis) in Skiles 269

Many model search strategies involve trading off model fit with model complexity in a penalized goodness of fit measure. Asymptotic properties for these types of procedures in settings like linear regression and ARMA time series have been studied, but these do not naturally extend to more mixed model scenarios where simple definitions of sample size and variance components are harder to come by. This paper introduces a new class of strategies, known as fence methods, for mixed model selection. The general idea involves a procedure to isolate a subgroup of what are known as correct models (of which the optimal model is a member). This is accomplished by constructing a statistical fence, or barrier, to carefully eliminate incorrect models. Once the fence is constructed, the optimal model is selected from amongst those within the fence according to model simplicity. We describe a variety of fence methods, based on the same principle but applied to different situations, including clustered and non-clustered data, linear or generalized linear mixed models, and Gaussian or non-Gaussian random effects. We show the broad applicability and study the performance of fence methods by giving a number of examples, each supported by simulation results or applied data analysis. We also give sufficient conditions for consistency of fence, a desirable property for a good model selection procedure. Finally, extension of the fence as a general principle for model selection is discussed. This work is joint with J. Sunil Rao of Case Western Reserve University and Zhonghua Gu and Thuan Nguyen of the University of California, Davis.

4:30 pm Thursday, October 19, 2006

SoM Colloquium: Dynamic Depletion of Vortex Stretching and Nonlinear Stability of 3D Incompressible Flow

by Thomas Y. Hou [mail] (California Institute of Technology) in Skiles 269

Whether the 3D incompressible Euler or Navier-Stokes equations can develop a finite time singularity from smooth initial data has been an outstanding open problem. Here we review some existing computational and theoretical work on possible finite blow-up of the 3D Euler equations. We show that the local geometric properties of vortex filaments can lead to dynamic depletion of vortex stretching, thus avoid finite time blowup of the 3D Euler equations. Further, we perform large scale computations of the 3D Euler equations to re-examine the two slightly perturbed anti-parallel vortex tubes which is considered as one of the most attractive candidates for finite time blowup of the 3D Euler equations. We find that there is tremendous dynamic depletion of vortex stretching and the maximum vorticity does not grow faster than double exponential in time. Finally, we present a new class of solutions for the 3D Euler and Navier-Stokes equations, which exhibit very interesting dynamic growth property. By exploiting the special nonlinear structure of the equations, we can prove nonlinear stability and the global regularity of this class of solutions.

3:30 pm Friday, October 20, 2006

Geometry-Topology Reading Seminar: Heegaard Floer Homology in Branched Covers

by Ken Baker (GaTech) in Skiles 269

4:30 pm Friday, October 20, 2006

Mathematical Physics: Anderson versus displacements: Two very different random Schr\"odinger

by Gunther Stolz [mail] (University of Alabama Birmingham) in Skiles 255

The most studied random Schr\"odinger operators are Anderson models, which describe alloy-type materials by placing a fixed potential at each lattice site and multiplying it with a random coupling constant. Another physically motivated way to model a disordered medium is to randomly displace the single site potential from each point of the lattice. The mathematical investigation of these two models turns out to be quite different. This is mostly due to the fact that the displacement model lacks monotonicity properties of the Anderson model. We will discuss this mainly in the context of determining the almost sure spectral minimum of the displacement model. Here we can handle dimension one, but have mostly open conjectures to offer in higher dimension.

3:30 pm Monday, October 23, 2006

Geometry-Topology Seminar: Plane fields on 3-manifolds

by John Etnyre (GaTech) in Skiles 269

4:30 pm Monday, October 23, 2006

CDSNS Colloquium: Coexistence in chemostats with state-dependent feedback

by Sergei S. Pilyugin [mail] (University of Florida) in Skiles 255

Under normal operating procedure, the chemostat (also known as the continuous bioreactor) is run with fixed operating parameters. If a single growth-limiting nutrient is supplied, at most one microbial species can survive in the long run. This principle of competitive exclusion can be violated if the operating parameters vary in time. Recently, the idea of designing a chemostat with state-dependent feedback has been actively discussed. It is known, for instance, how to design the feedback that warrants the coexistence of two microbial species. The results that I will present address the question of stable coexistence of three or more species. Specifically, I will talk about unfolding a codimension two bifurcation (Hopf + transcritical) and prove the existence of stable periodic solutions where three species persist.

3:00 pm Tuesday, October 24, 2006

PDE Seminar: BV estimates on damped p-system

by Ronghua Pan (School of Mathematics, Georgia Tech) in Skiles 255

In contrast to great success in hyperbolic conservation laws, BV theory for hyperbolic balance laws are wildely open except for strictly dissipative class. Damped p-system severs as a simplest example with weak dissipation. Previous results are valid for isothermal gas dynamics and for perturbations near constant equilibrium. Recently, we proved the uniform BV estimates for generic small BV data, and thus establish the global BV theory to the problem. One of the by product is the large time behavior of the weak solutions with sharp decay rates. This is joint with C. M. Dafermos.

2:00 pm Wednesday, October 25, 2006

Research Horizons Seminar: Small Ball Inequalities

by Michael Lacey (Georgia Institute of Technology) in Skiles 255

This talk will combine some of our favorite topics: combinatorics, orthogonality, analysis, probabilistic reasoning. We will work with Haar functions in dimensions 2 and 3, Such functions are supported on dyadic rectangles. The objects of interest are sums of such functions where the volume of the rectangles is held fixed---the `hyperbolic' assumption. And we seek a lower bound on the sup norm of the such sums. It is easy to establish such a lower bound on average, and the conjecture is that the average case estimate by a factor that is the square root log of the volume. This is true in dimension 2--a theorem of Talagrand--we will recall a short mysterious proof due to Temlyakov. The case of three dimensions is much harder, as will be explained. There are partial results due to Jozef Beck and the speaker and Dmitry Bilyk. These questions arise in the setting of probability, approximation theory, and number theory.

3:00 pm Wednesday, October 25, 2006

Mathematical Biology & Ecology Seminar: Selective advantage for sexual reproduction

by Emmanuel Tannenbaum [mail] (School of Biology, Georgia Tech) in Skiles 255

This talk develops a simplified model for sexual reproduction within the quasispecies formalism. The model assumes a diploid genome consisting of two chromosomes, where the fitness is determined by the number of chromosomes that are identical to a given master sequence. We also assume that there is a cost to sexual reproduction, given by a characteristic time tau_*seek* during which haploid cells seek out a mate with which to recombine. If the mating strategy is such that only viable haploids can mate, then when tau_*seek*=0, it is possible to show that sexual reproduction will always out compete asexual reproduction. However, as tau_*seek* increases, sexual reproduction only becomes advantageous at progressively higher mutation rates. Once the time cost for sex reaches a critical threshold, the selective advantage for sexual reproduction disappears entirely. The results of this paper suggest that sexual reproduction is not advantageous in small populations per se, but rather in populations with low replication rates. In this regime, the cost for sex is sufficiently low that the selective advantage obtained through recombination leads to the dominance of the strategy. In fact, at a given replication rate and for a fixed environment volume, sexual reproduction is selected for in high populations because of the reduced time spent finding a reproductive partner.

4:30 pm Wednesday, October 25, 2006

Analysis Seminar: Resurgence of a power series invariant of the simplest hyperbolic knot

by Stavros Garoufalidis (School of Mathematics, Georgia Tech) in Skiles 255

Perturbative quantum field theory associates formal power series to knotted 3-dimensional objects. These series are factorially divergent (of class Gevrey-1), and its has been conjectured that they have resurgent borel transform with a geometrically interesting set of singularities. We will write down explicitly the series associated to the simplest hyperbolic knot, and we will prove its resurgence. The nearest singularities are at I*2.02...This is joint work with Ovidiu Costin.

4:30 pm Wednesday, October 25, 2006

ACM: Charged Bubbles

by John Pelesko (University of Delaware) in Skiles 269

In the mid-1960's G.I. Taylor launched the field of electrohydrodynamics through a series of seminal papers on electrified drops, films, and jets. This work led to the discovery of the Taylor Cone and Taylor Jet; both of which have since found significant industrial applications. In one paper, as an approximation to a liquid drop, Taylor studied a planar soap film in an electrostatic field. This work, and the mathematical model of the experiment of Ackerberg, turned out much later to be important in the study of micro- and nanoelectromechanical devices. In previous work, we continued the Taylor soap film experiments and extended the theory to a variety of situations. Recently, in our lab, we have begun studying a new non-planar geometry, linking the subject of electrohydrodynamics with that of liquid bridges. In particular, we study deformations of a catenoid, made from soap film, subjected to an electric field. In this talk, we'll outline the theory for this system and discuss recent experimental results.

3:05 pm Thursday, October 26, 2006

Stochastic Seminar: Metastability for complex systems with small noise

by Weinan E (Department of Mathematics, Princeton University) in Skiles 269

Many important processes in nature are rare events. Familiar examples include conformational changes of bio-molecules, nucleation events and chemical reactions. From an abstract viewpoint, this can be formulated as the problem of navigating a system over its energy landscape. For simple systems, the classical transition state theory and large deviation theory provide an effective description for the transition rates of such rare events. For systems with complex energy landscapes, however, the transition state theory is not the most efficient tool. We will discuss a new theory, the transition path theory, that is more suited for describing transitions in complex systems. We will also discuss the string method for finding transition pathways and transition rates, that has proven to be quite effective for a wide variety of problems. Applications will be presented for simple examples of phase transformation in solids, conformational changes of bio-molecules with explicit solvent and thermal noise induced switching of magnetic thin films.

4:30 pm Friday, October 27, 2006

Analysis : Counting polynomial configurations on dense subsets of the integers

by Nikos Frantzikinakis (University of Memphis) in Skiles 255

The polynomial Szemeredi theorem of Bergelson and Leibman states that every integer subset with positive density contains infinitely many configurations of the form x,x+p_1(n),...x+p_k(n), where p_1,...,p_k is any fixed family of integer polynomials with zero constant term. Unlike Szemeredi's theorem on arithmetic progressions the only known proof of this result uses ergodic theory. Recent developments in ergodic theory have enabled us to show that integer subsets with positive density in fact contain "many" polynomial patterns, where "many" means at least as many as in the random case. For example Bergelson, Host and Kra showed that we can always find "many" arithmetic progressions of length 4 with fixed step (suprisingly this fails for length 5 progressions). Jointly with Kra we showed that a similar result holds for all polynomial patterns arising from linearly independent polynomials. Recently, I showed that the same holds for all polynomial patterns that arise from two polynomials and in "most" cases for three polynomials. I am going to discuss these and several other recent results in this direction.

2:00 pm Monday, October 30, 2006

Theory of Computation Colloquium: The Challenge of Coding Biological Information in Nucleotide Sequences

by Christine Heitsch (School of Mathematics, Georgia Tech) in MiRC 102A

In the nucleus, lengthy DNA molecules have a canonical double-stranded helix structure well-adapted for information storage and retrieval. In the laboratory, short single-stranded DNA sequences have a variety of possible applications, ranging from microarrays to nanomolecular structures and DNA computation. Furthermore, ongoing discoveries highlight the many vital regulatory and catalytic functions performed by different RNA molecules, other than mediating the production of proteins from DNA. Thus, increasing interest is focused on the self-bonding of RNA molecules and on the design, analysis, and prediction of RNA secondary structures.

3:00 pm Monday, October 30, 2006

Geometry-Topology Seminar (joint Emory): Toric geometry on near-symplectic manifolds

by Margaret Symington (Mercer University) in MSC E408 (at Emory)

4:30 pm Monday, October 30, 2006

CDSNS Colloquium: One model, one regime, and many phenomena

by David (Shen-Ou) Cai [mail] (Courant Institute, New York University) in Skile 255

We will present our modeling and large-scale computational work of the primary visual cortex (V1). In particular, we will discuss network mechanisms underlying spatiotemporal dynamics associated with spontaneous on-going activity of the V1 and the line-motion illusion.

3:00 pm Tuesday, October 31, 2006

PDE Seminar: A gradient approach to an evolution problem arising in superconductivity

by Sylvia Serfaty (Courant Institute, NYU) in Skiles 255

In a joint work with Luigi Ambrosio, we study a PDE arising in superconductivity and describing the evolution of the vortex-density. We interpret it as a gradient-flow (in the formal framework of Otto and rigorous framework of Ambrosio-Gigli-Savare) for the Wasserstein structure on probability measures, and derive existence results with energy-dissipation estimates.

2:00 pm Wednesday, November 1, 2006

Stochastic Seminar: Empirical Likelihood Based Confidence Intervals for the Sensitivity at a Fixed Level of the False Positive Rate

by Jeff Qin (Georgia State University) in Skiles 269

When the response of a diagnostic test is continuous, it is of interest to find the range of the sensitivity of the diagnostic test at a fixed level of its false positive rate. In this paper, we first define an empirical likelihood ratio for the sensitivity of a continuous-scale diagnostic test and show that its limiting distribution is a scaled chi-square distribution. We then propose three new empirical likelihood based confidence intervals for the sensitivity of the test by using the scaled chi-square distribution. A simulation study is conducted to evaluate the small sample performance of the newly proposed intervals. The simulation results show that the newly proposed hybrid bootstrap and empirical likelihood intervals have good coverage accuracy even when the false positive rate is high.

2:00 pm Wednesday, November 1, 2006

Research Horizons Seminar: The exotic world of 4-manifolds

by John Etnyre (Georgia Institute of Technology) in Skiles 255

Four dimensions is unique in many ways. For example n-dimensional Euclidean space has a unique smooth structure if and only if n is not equal to four. In other words, there is only one way to understand smooth functions on R^n if and only if n is not 4. There are many other way that smooth structures on 4-dimensional manifolds behave in surprising ways. In this talk I will discuss this and I will sketch the beautiful interplay of ideas (you got algebra, analysis and topology, a little something for everyone!) that go into proving R^4 has more that one smooth structure (actually it has uncountably many different smooth structures but that that would take longer to explain).

3:00 pm Wednesday, November 1, 2006

Mathematical Biology & Ecology Seminar: Do Plankton wander aimlessly or is their Walk far from Random?

by Jeannette Yen (School of Biology, Georgia Tech) in Skiles 255

Plankton are aquatic organisms that form the base of the aquatic food web and therefore, the balance in aquatic ecosystems depends on their survival. The term plankton is derived from the Greek word "planktos", meaning "wanderer" or "drifter". To determine when their movement patterns depart from random, we perform kinematic analyses of their swimming behavior. From quantitative analyses of their three-dimensional trajectories, propulsion and morphology, and small-scale turbulence, we learn that plankton often do not go with the flow. At the small-scale where biologically-generated behavior differs from physically-derived flow, we find plankton self-propel themselves, are aware of each other, and evolve in response to the fluid environment in surprising ways. We present results of two modeling efforts of plankton behavior. First, we evaluate the strength of biological forces that maintain swarms against physical dispersion where we regard not the interaction between animals but the interaction between the animal and its local, stimulus field. Whatever the driving mechanism, the tendency that counters dispersion in a swarm is not just a collective, statistical property, but rather must be observable in each swarmer's individual motion as a hidden regularity. This notion suggests that we may be able to apprehend the dynamics of a large observationally unwieldy aggregation by studying the behavior of a few typical individuals. Ongoing efforts simulate the aggregative behavior of krill schools using a self-organizing model.

4:30 pm Wednesday, November 1, 2006

Applied Mathematics & Computational Seminar: Multigrid methods as efficient solvers for ill-posed problems

by Andrei Draganescu (University of Maryland at Baltimore County) in Skiles 269

An air contamination event takes place in a heavily populated area. A chemical agent is being diffused in the air and moved by the winds. Sensors monitoring air quality detect increased concentrations of the pollutant. At what location was the pollutant released in the air? What areas will be affected over the next few minutes, hours, days, and and to what degree? Fast answers to these questions are critical for hazard containment and assessment, and for evacuation strategies. An efficient response to the above scenario requires the backward solution of a time-dependent advection-reaction-diffusion equation, a problem that is ill-posed. In this talk I will present a method for efficiently solving the regularized inverse problem of identifying initial conditions given the end-time state for an equation of parabolic type. The method is a new embodiment of the well-known multigrid paradigm, which consists in taking advantage of several discretization levels for the same continuous problem in order to speed up the solution process. I will discuss mathematical results for both the linear and semilinear cases, and I will present supporting numerical results. In the end I will show the result of applying this algorithm to the scenario above, which proves that the associated inverse problem can be resolved in a reasonable time-frame.

4:30 pm Wednesday, November 1, 2006

Analysis Seminar: The Pick Property for Planar Domains

by Richard Rochberg (Washington U., St. Louis) in Skiles 255

The Pick property is a useful property enjoyed by some but not all Hilbert spaces of holomorphic functions. The Hardy and Dirichlet spaces for the disk have the property; the Bergman space of the disk and the Hardy space of the annulus do not. I will describe a renormings of the Hardy spaces of multiply connected plane domains which automatically have the Pick property. The construction uses the Drury-Arveson-Hardy space of holomorphic functions on the ball. Joint work with: Nicola Arcozzi, U. Bologna, Eric Sawyer, McMaster U.

3:05 pm Thursday, November 2, 2006

Graph theory seminar: Triangle-free digraphs

by Paul Seymour (Princeton University) in Skiles 255

A digraph is k-free if it has no directed cycle of length at most k. A special case of the Caccetta-Haggkvist conjecture asserts that every 3-free digraph G has a vertex of outdegree less than |V(G)|/3. This is still open, but more generally, what can we say about 3-free digraphs? In joint work with Blair Sullivan and partly with Maria Chudnovsky, we have studied several related questions. For instance:

# A 3-free tournament is acyclic; what can one say about a 3-free digraph that is almost a tournament?

# It is not true that a 3-free digraph G must have average outdegree less than |V(G)|/3 (the transitive tournament is a counterexample); but what if we average the outdegree of v over all EDGES uv? This ``edge-average'' for the transitive tournament is less than |V(G)|/3...

# How many three-edge directed paths can there be in a 3-free digraph? There is a pretty argument that gets close to the correct bound.

We survey these and several other questions.

3:00 pm Friday, November 3, 2006

Analysis: Weyl-Titchmarsh type formulas for the spectral density for two classes of Jacobi matrices

by Sergey Simonov (St Petersburg St) in Skiles 255

We consider small perturbations of two models of absolutely continuous Jacobi matrices: discrete Schrodinger operator and Hermite operator. Considering the asymptotics of the orthogonal polynomials associated with the matrix, we introduce the coefficients in the matrix, in terms of which the spectral density of the perturbed operator can be expressed. These formulas are analogous to the Weyl-Titchmarsh formulas for the Schrodinger operator in continuous case.

3:30 pm Friday, November 3, 2006

Geometry-Topology Reading Seminar: Heegaard-Floer Theory in Lens spaces

by Ken Baker (GaTech) in Skiles 269

4:30 pm Friday, November 3, 2006

Mathematical Physics Seminar: Lieb-Thirring-Hardy inequalities and the stability of relativistic matter

by Rupert Frank (KTH Stockholm) in Skiles 268

We show that the Lieb-Thirring inequalities on moments of negative eigenvalues of Schrodinger-like operators remain true, with possible different constants, when the critical Hardy-weight is subtracted from the Laplace operator. Similar results are true for fractional powers of the Laplacian and, in particular, for relativistic Schr\"odinger operators. We also allow for the inclusion of magnetic vector potentials. As an application, we extend the proof of stability of relativistic matter with magnetic fields to the critical value of the nuclear charge. The talk is based on joint works with T. Ekholm and with E. H. Lieb and R. Seiringer.

2:00 pm Monday, November 6, 2006

Theory of Computation Colloquium: List Decoding with Optimal Rate: Folded Reed-Solomon Codes

by Venkat Guruswami (University of Washington) in MiRC 102A

Suppose you want to communicate a message of k packets on a noisy communication channel. So you judiciously encode it as a redundant collection of n = ck packets and transmit these. What is the fewest number of correct packets one needs to receive in order to have any hope of recovering the message? Well, clearly one needs at least k correct packets. In this talk, I will describe an encoding scheme that attains this information-theoretic limit: for any desired constant eps > 0, it enables recovery of the message as long as at least k(1+eps) packets are received intact. The location of the correct packets and the errors on the remaining packets can be picked adversarially by the channel. This achieves the optimal trade-off (called "capacity") between redundancy and error-resilience for a malicious noise model where the channel can corrupt the transmitted symbols arbitrarily subject to a bound on the total number of errors. These results are obtained in an error-recovery model called list decoding. The talk will introduce and motivate the problem of list decoding, and then give a peek into the algebraic ideas and constructions that lead to the above result. Based on joint work with Atri Rudra.

3:30 pm Monday, November 6, 2006

Geometry-Topology Seminar: The 4-vertex theorem and its converse

by Shea Vick (Univeristy of Pennsylvania, visiting Ga Tech) in Skiles 269

4:30 pm Monday, November 6, 2006

CDSNS Colloquium: Synchronization in complex networks of dynamical systems

by Igor Belykh [mail] (Georgia State University) in Skiles 255

A particularly interesting form of dynamical behavior occurs in networks of coupled oscillators when all of the subsystems behave in the same fashion; that is, they all do the same thing at the same time. A central dynamical question is: When is such synchronous behavior stable, especially with respect to coupling strengths and coupling configurations of the network? In this talk, I will address this question and present a new general method to prove global stability of synchronization in complex networks of linearly coupled oscillators. This rigorous method combines the Lyapunov function approach with graph theoretical reasoning. I will show how the method can be applied to undirected and directed networks. I will also present surprising results from the study of synchronization in pulse-coupled networks of bursting neurons.

3:00 pm Tuesday, November 7, 2006

PDE Seminar: Hyperbolic conservation laws and spacetimes with limited regularity

by Philippe G. LeFloch (Universit� Pierre et Marie Curie) in Skiles 269

I will present recent work on the existence and qualitative behavior of solutions to nonlinear hyperbolic systems of partial differential equations posed on manifolds, especially when both the solutions and the manifold have limited regularity. The discussion will include the following topics: -- Shock wave theory for hyperbolic conservation laws (joint work with M. Ben-Artzi, Jerusalem). -- Gowdy-type spacetimes with compressible matter in the bounded variation class (joint work with J. Stewart, Cambridge). -- Injectivity radius estimates for Lorentzian manifolds under curvature bounds (joint work with B.-L. Chen, Guang-Zhou)

4:30 pm Tuesday, November 7, 2006

ACO Colloquium: Towards a proof of Seymour's 1-flowing conjecture

by Bertrand Guenin (University of Waterloo) in Skiles 269

Refreshments at 4:00PM in Skiles 236.

The well known max-flow min-cut theorem states that the maximum amount of flow that can be sent from vertex s to a vertex t in a graph (with capacities) is equal to the capacity of smallest cut which separates s and t. While the notion of flows in graphs extends naturally to binary matroids, the max-flow min-cut relation does not hold for binary matroids in general. However, Seymour conjectured that the aforementioned minimax relation holds as long as the binary matroids do not contain any one of three special obstructions. This conjecture if true would generalize many classical results on multi-commodity flows and matchings. Our approach is to try to give a structural characterization of the binary matroids for which the minimax relation holds. I will review known cases of the conjecture and give a brief sketch of our strategy for solving the conjecture. A more technical description of the tools we are developing will be presented on November 10 during the Combinatorics seminar.

This is joint work with Irene Pivotto and Paul Wollan.

2:00 pm Wednesday, November 8, 2006

Research Horizons Seminar: Density Dependent Population Models: dynamical systems and real-life

by Howie Weiss (Georgia Institute of Technology) in Skiles 255

For non-microscopic size animals (including humans), almost every age-structured population forecasting model in current use is based on the linear Leslie model. I will begin by quickly discussing this simple model. I will then discuss our long-term program to understand the dynamics of general classes of density dependent, age structured population models. For some families we find a plethora of extremely complicated dynamical behaviors, several of which have not been previously observed in age structured population models, and which may give rise to new paradigms in population biology. We also attempt to provide a rigorous foundation for the population biologist's notion of "ergodicity" by constructing SRB or natural measures for some of these models. Using 20 years of population data from a local research stream, my Penn State fisheries colleague Robert Carline has discovered density (and seasonal stream flow) dependence of some survival probabilities and fertilities for brown trout. I will discuss our very preliminary progress in modeling this trout population using a density dependent, age structured model. I will not assume any background in dynamical systems during this talk.

3:00 pm Wednesday, November 8, 2006

Mathematical Biology & Ecology Seminar: The Morphogenesis of Patterned Silica Structures in Diatoms

by Nils Kr�ger [mail] (School of Chemistry and Biochemistry, Georgia Tech) in Skiles 255

The formation of inorganic materials by organisms (biomineralization) is a widespread biological phenomenon ranging from bone and teeth formation in humans to biogenesis of magnetic nanoparticles in certain bacteria. The structures of biominerals are species specific characteristics, implicating that the mineral morphogenesis process is controlled by a genetic program. Diatoms (a large group of single-celled algae) are probably the most remarkable biomineral forming organisms. They produce an incredible variety of SiO2 (silica) structures that exhibit different shapes and highly regular porous patterns (see figure http://www.math.gatech.edu/~heitsch/KrogerAbstract.pdf ). I will present an overview about the characteristics of diatom silica structures and current insights into the mechanism of diatom silica morphogenesis.

4:30 pm Wednesday, November 8, 2006

Analysis Seminar: Recent developments in vector-valued rational interpolation

by Avram Sidi (Technion and GT) in Skiles 255

In this talk, we discuss a recent approach to interpolation of vector-valued functions by vector-valued rational functions. Given the vector-valued function $F(z)$, $F:\mathbb{C}\rightarrow \mathbb{C}^N$, in this approach, we devise rational interpolants of the form $R(z)=U(z)/V(z)$, where $V(z)$ is a scalar polynomial, whereas $U(z)$ is a vector-valued polynomial, such that $R(z)$ interpolates $F(z)$ at a given set of points in the $z$-plane. We will explain how this can be achieved in different ways and discuss the algebraic properties of three interpolants. We will also present a de Montessus type convergence theory in case $F(z)$ is meromorphic in a domain of the $z$-plane.

4:30 pm Wednesday, November 8, 2006

ACM: Rare events in spatially extended media

by Eric Vanden-Eijnden (Courant Institute) in Skiles 269

The dynamical behavior of many systems arising in physics, chemistry, biology, etc. is dominated by rare but important transition events between long lived states. Important examples include nucleation events during phase transition, conformational changes of macromolecules, or chemical reactions. Understanding the mechanism and computing the rate of these transitions is a topic that has attracted a lot of attention for many years. In this talk, I will discuss recent computational techniques based on large deviations theory and extension thereof which allow one to determine the pathways and rate of these rare events. I will illustrate these techniques on the specific example of some reaction-diffusion equations driven by white-noise arising e.g. in the context of population dynamics and in the description of the kinetics of phase transitions. I will also discuss examples which involve stochastic resonance effects and/or large domains and which require going beyond large deviations theory.

10:00 am Thursday, November 9, 2006

QCF Seminar: An Introduction to Commercial Mortgage Backed Securities (CMBS)

by Kieran P. Quinn and David F. Smith (Credit Suisse) in Skiles 269

This talk will give an introduction to Commercial Mortgage Backed Securities (CMBS) from the perspective of practitioners in the field of real estate finance.� The talk will cover the CMBS process of securitization, the structure of CMBS tranches by credit rating, the market for CMBS, and the investors that are interested in this market.

3:00 pm Thursday, November 9, 2006

Graph theory seminar: On the structure of even-cycle and even-cut matroids

by Bertrand Guenin (University of Waterloo) in Skiles 255

The class of even-cycle matroids is the smallest minor-closed class of matroids containing co-extensions of graphic matroids. Similarly the class of even-cut matroids is the smallest minor-closed class of matroids containing co-extension of co-graphic matroids. Hence, these are fairly natural classes of matroids. We are interested in finding excluded minor characterization for the following classes of matroids:

(1) even-cycle matroids which are AG(3,2)-free,

(2) even-cut matroids,

(3) classes of matroids obtained by taking the union/intersection of (1), (2) or their dual.

Our main motivation is Seymour's long standing conjecture on 1-flowing matroids. The main tools I will present are Stabilizer theorems and Escape theorems for the classes (1) and (2).

This is joint work with Irene Pivotto and Paul Wollan.

4:30 pm Thursday, November 9, 2006

School of Mathematics Colloquium: Invariant-Based Face Recognition

by Nigel Boston [mail] (University of South Carolina and University of Wisconsin -- Madison) in Skiles 269

After a brief review of recent striking applications of algebra to engineering and computer science, the currently significant problem of face recognition is addressed. We introduce a new approach to obtaining invariants of Lie groups adapted to this problem and describe its success in implementations.

9:00 am Friday, November 10, 2006

Algorithms and Randomness Center and ThinkTank: Various Lectures

by Distinguished Lecturers in Klaus Building Auditorium, Georgia Tech

Instructors of graduate courses, especially those with any algorithmic or probabilistic content, are encouraged to allow their students to attend these lectures. This day-long affair will be enacted in the Klaus Auditorium (with probability rapidly approaching 1). Here are the details: http://www.cc.gatech.edu/arc

3:00 pm Friday, November 10, 2006

Mathematical Physics: On the overlap in the multiple spherical SK models

by Dmitry Panchenko [mail] (Department of Mathematics, M.I.T.) in Skiles 268

Joint work with Michel Talagrand In order to study certain questions concerning the distribution of the overlap in the Sherrington-Kirkpatrick type models, such as the chaos and ultrametricity problems, it seems natural to study the free energy of multiple systems with constrained overlaps. One can write analogues of Guerra's replica symmetry breaking bound for such systems but it is not at all obvious how to choose informative functional order parameters in these bounds. We were able to make some progress for spherical pure $p$-spin SK models where many computations can be made explicitly. For pure 2-spin model we prove ultrametricity (absence of frustration) and chaos in an external field. For the pure $p$-spin model for even $p>4$ without an external field we describe two possible values of the overlap of two systems at different temperatures. We also prove a somewhat unexpected result which shows that in the 2-spin model the support of the joint overlap distribution is not always witnessed at the level of the free energy and, for example, ultrametricity holds only in a weak sense.

4:00 pm Friday, November 10, 2006

Combinatorics: Information inequalities: interpretations & applications

by Prasad Tetali (Georgia Tech (and Microsoft Research)) in Skiles 255

We review and refine classical inequalities (of Shannon, Han, Shearer etc.) for the joint entropy of a collection of random variables in terms of an arbitrary collection of subset joint entropies. A duality between the upper and lower bounds for joint entropy is developed, as are connections to entropy power inequalities. Applications include a new upper bound on the number of independent sets in an arbitrary graph, and new determinantal inequalities. This is current work with Mokshay Madiman (Yale University).

2:00 pm Monday, November 13, 2006

Theory of Computation Colloquium: Approximation-Algorithms Turned Bandit

by Adam Kalai (Weizmann Institute) in MiRC 102A

In a multi-armed bandit problem, a decision maker makes a sequence of decisions, online, in the face of uncertainty. For example, consider a traveling salesman who repeatedly must choose a route to visit a set of cities. Each period, she decides on a route and then finds out the total time that it took. Her goal is to achieve low average time over many periods. More generally, the decision-maker has a known (possibly infinite) set of decisions D, and there is an unknown sequence of cost functions chosen from a known family of functions F. Unfortunately, many problems like the TSP example above, cannot be solved efficiently offline, even if everything was known in advance, and one must use approximation algorithms instead. We give a general reduction, showing how to solve *online* linear optimization problems (ones where costs scale linearly like the online TSP example), using an approximation algorithm designed for the *offline* (full-information) setting. Performance is guaranteed to be nearly as good as the average performance of the best single decision, e.g., best route, if one had to use the same decision each period, where the best is chosen with the benefit of hindsight. Our results generalize prior work which gave similar results for the case where the offline problem could be solved exactly, or for special types of approximation algorithms. The use of approximation algorithms presents interesting new difficulties for which we provide geometric solutions.

3:30 pm Monday, November 13, 2006

Geometry-Topology Seminar: Hyperbolic dynamics and Riemannian geometry

by Gerhatd Kneiper (U. Bochum) in Skiles 269

The purpose of this talk is to give a survey on geodesic flows with some weak conditions on hyperbolicity. In the first part we will talk about genericity of positive topological entropy for compact surfaces. In the second part we will focus on compact manifolds of nonpositive curvature. Their geodesic provide important examples of nonuniform- and patial hyperbolicity. In particular we will talk about the equidistribution of closed geodesics and the uniqueness of the measure of maximal entropy. Furthermore, we will provide some applications to rigidity questions.

4:30 pm Monday, November 13, 2006

CDSNS Colloquium: Carrying simplices in nonautonomous and random competitive Kolmogorov systems

by Wenxian Shen [mail] (Auburn University) in Skiles 255

This talk is concerned with the asymptotic behavior of positive solutions of nonautonomous and random competitive Kolmogorov systems. Via the skew-product flows approach, it is shown that associated with a nonautonomous (random) competitive Kolmogorov system, there exists an unordered carrying simplex which attracts all nontrivial positive solutions of the nonautonomous (random) competitive Kolmogorov system.

3:00 pm Tuesday, November 14, 2006

PDE Seminar: Existence and regularity of the free boundary in optimal design problems

by Eduardo Teixeira (Rutgers University) in Skiles 255

We will discuss a series of ideas and techniques for treating a class of free boundary problems that arise from the mathematical formulation of optimal shape problems in potential theory.

2:00 pm Wednesday, November 15, 2006

Research Horizons Seminar: KAM theory in Hamiltonian systems.

by Yingfei Yi (Georgia Institute of Technology) in Skiles 255

The lecture will review the classical KAM theory and its recent development in Hamiltonian systems. Related problems like the Poincar\'e mechanism of splitting resonant tori and the Melnikov persistence problem of lower dimensional tori will also be discussed.

3:00 pm Wednesday, November 15, 2006

Mathematical Biology & Ecology Seminar: Evolutionary Systems Biology: Prospects and Challenges for Formalism

by King Jordan (School of Biology, Georgia Tech) in Skiles 255

In this talk, I will: 1-argue that the implications of the systems perspective for the study of molecular and genome evolution have yet to be fully realized, 2-provide evidence for distinct modes of biological systems evolution and 3-outline some of the corollary prospects and challenges facing the systems level view of evolution. Systems biology entails a change in perspective from the function of individual players, e.g. genes and proteins, to the functional relationships among multiple interacting players. For the most part, the molecular evolution of function has been considered in the context of sequence and structural changes between homologous genes (proteins). However, evolutionary change in function must be due to combinatoric changes in the molecular interactions that underlie biological systems. Interaction space is likely to be far more dynamic and flexible than sequence/structure space and thus more amenable to evolutionary changes of functional and adaptive significance. I will present paradoxical results from two of our studies on the evolution of gene expression and regulation that are consistent with this world view. In the first case, we have shown that mammalian gene coexpression networks are essentially conserved at the global level yet highly divergent at the local level. Distinct sets of functionally related genes are coexpressed in the human versus the mouse. The second study revealed that eukaryotic genomes employ fundamentally disparate regulatory machineries to achieve conserved and specific expression patterns of core histone genes. These results raise a number of questions concerning how to measure evolutionary changes in biological systems. I will attempt a colloquial formulation of these questions in the hopes that this may stimulate some ideas on more formal treatments of evolutionary systems biology.

4:30 pm Wednesday, November 15, 2006

Analysis Seminar: Stabilization and Corona Theorems for the Real Disc Algebra and Real H^\infty

by Brett Wick (Vanderbilt) in Skiles 255

The real disc algebra and Real H^\infty are Banach algebras. As such a mix of algebraic and analytic questions can be asked. Certain algebraic quantities associated to these algebras are well understood, for example the maximal ideal structure. However, other invariants, such as the stable rank of the algebra, are a little harder to study. In this talk we will discuss connections between Corona Theorems for these algebras their Bass stable rank. These results are motivated by questions arising in complex function theory and Control Theory.

4:30 pm Wednesday, November 15, 2006

Applied & Computational Mathematics Seminar: An Interoperable Front Tracking Code and Applications to Various Scientific Problems

by Xiaolin Li (Department of Applied Mathematics and Statistics, SUNY) in Skiles 269

We introduce an enhanced front tracking method and its software implementation with an easy-to-use user interface. New algorithms include conservative coupling in ND and locally grid based topological bifurcation for 3D. Objective mathematical and computational assessment are given with comparison to other interface methods such as the level set method and volume of fluid method. We emphasize the interoperability of front tracking with other scientific software including the combined operation with AMR and combustion packages. Scientific applications include the study of turbulent mixing due to acceleration driven instabilities, fuel injection jet, shock flame interaction, extending to cell motion in biology. This work is in collaboration with the FronTier Group at Stony Brook University and Brookhaven National Laboratory.

10:00 am Thursday, November 16, 2006

Thesis Defense/Combinatorics Seminar: Maximum Codes with the Identifiable Parent Property

by Wen Jiang (Georgia Tech) in Skiles 255

We study codes that have identifiable parent property. Such codes are called IPP codes. The research on IPP codes is motivated by design of schemes that protect against piracy of digital products. Construction and decoding of maximum IPP codes have been studied in rich literatures. General bounds on $F(n,q)$, the maximum size of IPP codes of length $n$ over an alphabet with $q$ elements, have been obtained through techniques from graph theory and combinatorial design. Improved bounds on $F(3,q)$ and $F(4,q)$ are obtained recently. We prove a precise formula for $F(3,q)$, construct maximum IPP codes with size $F(3,q)$, and give an efficient decoding algorithm for such codes. The main techniques used in this thesis are from graph theory and nonlinear optimization. We begin by associating to each code an edge colored graph. Then a code has the IPP if and only if its associated graph has certain structural conditions. We study the underlying structure of graphs associated with IPP codes of maximum size. Using this approach, we present explicit construction of classes of IPP graphs associated with IPP codes of length 3, which gives a lower bound on $F(3,q)$. By further investigating the structure of graphs associated with IPP codes of length 3, we show that there exist maximum IPP codes of length 3 whose associated graphs have a very simple structure. Using that structure, we are able to convert the problem of deciding $F(3,q)$ to a nonlinear programming problem. Based on our nonlinear programming formulation, we give an algorithm which determines $F(3,q)$ numerically for each $q\ge 15$. We also describe how to construct maximum IPP codes from optimal solutions to the nonlinear programming problem, and show that such IPP codes allow for efficient tracing. Using techniques from nonlinear programming, we prove a precise formula for $F(3,q)$ when $q\ge 15$. Our approach may be used to improve bounds on $F(2k+1, q)$. For example, we characterize the structure of the associated graphs of maximum IPP codes of length 5, and obtain bounds on $F(5,q)$.

3:00 pm Thursday, November 16, 2006

Graph theory seminar: Circle graph obstructions under pivoting

by Sang-il Oum (Math, GT) in Skiles 255

Circle graphs are interesection graphs of chords of a circle; two vertices are adjacent if and only if the corresponding chords in a chord diagram cross. Circle graphs have many applications, in particular to knot theory. Bouchet [1994] characterized circle graphs by providing a list of 3 excluded vertex-minors (graphs obtainable by local complementation and vertex deletions). We showed that there is a list of 15 excluded pivot-minors (graphs obtainable by pivoting and vertex deletions) for circle graphs. As a corollary, our theorem implies the Kuratowski-Wagner theorem on excluded minors for planar graphs.

3:30 pm Friday, November 17, 2006

Algebra Seminar: The dendrology of PolyZeta values

by Stavros Garoufalidis (School of Mathematics, Georgia Tech) in Skiles 269

The zeta values are the vales of the Riemann zeta function at positive integers. The PolyZeta values is a natural generalization (due to Euler and Zagier), such that the vector space (over the rationals) generated by the PolyZeta values is an algebra A. It is conjectured that A is a polynomial algebra on a well-descibed set of generators, which among other things implies that zeta(3), zeta(5), ... are algebraically independent over the rationals (and in particular, transcendental). We will describe a dendrological form of the algebra A, and of a correspondent graded Lie (Ihara) algebra. Our motivation comes from motivic cohomology, knot theory, arithmetic, quantum field and combinatorics. Time permitting, we will also describe the relation between the Ihara algebra and a lesser known (but fundamental) cousin of it: the ARI algebra of Ecalle.

3:30 pm Monday, November 20, 2006

Geometry-Topology Seminar: Resurgence in quantum topology: wishes and facts

by Stavros Garoufalidis (GaTech) in Skiles 269

A resurgent function (due to Ecalle) is one that has an endless analytic continuation (without natural boundaries) in the complex plane. Examples of resurgent functions are meromorphic functions, algebraic functions, and more generally solutions to differential equations (linear or not). A little noticed fact is that the Taylor coefficients of a resurgent function have asymptotic expansions, with exponentially small terms included. The talk will explain some resurgence conjectures in quantum topology and how they imply the Volume Conjecture. We will also indicate a proof of our resurgence conjectures for the two simplest knots: the 3_1 and 4_1 knots.

4:30 pm Monday, November 20, 2006

CDSNS Colloquium: Independence in Dynamics

by Hanfeng Li [mail] (University at Buffalo, SUNY) in Skiles 255

I will show a systematic study of independence in topological and measure-theoretical dynamics. This includes a combinatorial approach to local and functional-analytic analysis of entropy and related mixing properties. This is a joint work with David Kerr.

3:00 pm Tuesday, November 21, 2006

PDE Seminar (Postponed): To Be Announced

by Nader Masmoudi (Courant Institute, NYU) in Skiles 255

4:30 pm Tuesday, November 21, 2006

Applied & Computational Mathematics Seminar: Extrapolation Methods with Applications to Solution of Large Systems of Equations and to PageRank Computations

by Avram Sidi (School of Mathematics, Georgia Tech and Technion, Israel Institute of Technology) in Skiles 269

An important problem that arises in different areas of science and engineering is that of computing limits of sequences of vectors x_n, where x_n\in\C^N with N very large. Such sequences arise, for example, in the solution of systems of linear or nonlinear equations by fixed-point iterative methods, and \lim_{n\to\infty}x_n are simply the required solutions. In most cases of interest, however, these sequences converge to their limits extremely slowly. One practical way to make the sequences x_n converge more quickly is to apply to them vector extrapolation methods. In this work, we review two polynomial-type vector extrapolation methods that have proved to be very efficient convergence accelerators; namely, the minimal polynomial extrapolation (MPE) and the reduced rank extrapolation (RRE). We discuss the derivation of these methods, describe the most accurate and stable algorithms for their implementation along with the effective modes of usage in solving systems of equations, nonlinear as well as linear, and present their convergence and stability theory. We also discuss their close connection with the method of Arnoldi and with GMRES, two well-known Krylov subspace methods for linear systems. We show that they can be used very effectively to obtain the dominant eigenvectors of large sparse matrices when the corresponding eigenvalues are known, and provide the relevant theory as well. One such problem is that of computing the PageRank of the Google matrix, which we discuss in detail. In addition, we show that a recent extrapolation method of Kamvar et al. that was proposed for computing the PageRank is very closely related to MPE. We present a generalization of the method of Kamvar et al. along with a very economical algorithm for this generalization. We also provide the missing convergence theory for it.

3:30 pm Monday, November 27, 2006

Geometry-Topology Seminar: Partially Anosov systems and bi-Engel structures

by Thomas Vogel (Stanford Univeristy) in Skiles 269

4:30 pm Monday, November 27, 2006

Mathematical Physics Seminar: Phase Transitions and Mesoscopic Structures in Systems with Long Range Interactions

by Joel Lebowitz (Rutgers University) in Skiles 256

I will describe the statistical mechanical derivation of mesoscopic free energy functionals for systems interacting via both short range and long range (Kac) potentials. These describe the spatial structures of the different phases coexisting at a phase transitions. They give, in particular, criteria for the formation of droplets, the wetting of a demixed binary fluid and the formation of periodic structures. The behavior of systems with power-law decay of the interactions (dipole, Coulomb) will also be discussed.

4:30 pm Monday, November 27, 2006

CDSNS Colloquium: Noise and Order: explaining patterns in stochastic population systems

by Shandelle Henson [mail] (Andrews University) in Skiles 255

Animal population systems, with their pervasive noise yet identifiable dynamic patterns, afford special opportunities to study the interplay between stochasticity and low-dimensional deterministic trends. A powerful paradigm for analyzing the mix of order and noise in population time-series is the "deterministic skeleton", i.e., what would remain of the system if one could tune the unexplained stochastic variability down to zero. The skeleton is amenable to the tools of dynamical systems theory. It fixes the geometry of state space providing a "stage" for the dance of stochasticity. Chance events allow the system to visit and re-visit the various deterministic entities on the stage, including unstable invariant sets which would have little or no influence on the system in the absence of noise.

3:00 pm Tuesday, November 28, 2006

PDE Seminar: Large deviations for stochastic Navier-Stokes equations: A PDE approach

by Andrzej Swiech (School of Mathematics, Georgia Tech) in Skiles 255

We will present a PDE approach to large deviations for solutions of 2-dimensional stochastic Navier-Stokes equations with small noise intensities. The approach is based on the notion of viscosity solution for infinite dimensional Hamilton-Jacobi-Bellman equations and the appropriately adapted method of half-relaxed limits.

11:00 am Wednesday, November 29, 2006

ECE & BME & Biology Seminar: On the Way to Intellectual Brain-Computer Interface (IBCI)

by Alexander Kaplan (Department of Human Physiology, Moscow State University) in IBB 1128

Evolution creates only one output of the brain realized via the neuronal motor system (motor cortex - spinal motor neurons) connected with muscles. The brain computer interface (BCI) summaries all components necessary to establish a radically new output from the brain to environment based on the brain-computer communication channel. A popular type BCI is based on conscious control of EEG parameters (Birbaumer et al. 1999; Pfurtscheller et al. 1993; Wolpaw et al. 1991). The core of the BCI is the algorithm of interpretation of EEG events as discrete signs of simple personal intentions (switch on/off, go ahead/back). Much research has been focused on the problem how to extract EEG features correlated with mental signs induced by standard cognitive loads.We reformulated this problem to the following one: how to use the spontaneous natural EEG patterns in the BCI learning to reach the goal of controlling the environment. We introduced structural analysis of EEG signal to find its quasi-stationary patterns. We combined this data analysis with algorithms using brain plasticity to construct the intellectual BCI (IBCI). This new IBCI uses interactive training to detect EEG patterns correlated with discrete personal intentions. At the moment we have first versions of IBCI: one for two-dimensional control of moving object (toy car), another one - 3-dimensional (RGB) brain control the color of computer screen (Kaplan et al, 2005).

3:00 pm Wednesday, November 29, 2006

Mathematical Biology & Ecology Seminar: Mechanisms of neuron and neuronal network robustness

by Astrid Prinz [mail] (Dept. of Biology, Emory University) in Skiles 255

Most neuronal networks function reliably throughout our lives, in spite of on-going molecular turn-over and developmental and environmental changes on much faster time scales - weeks, days, or hours. This is particularly evident in the central pattern generator circuits that underlie rhythmic behaviors such as breathing or walking. What are the cellular and network mechanisms that stabilize network function? My lab is examining this question with electrophysiological experiments on pattern-generating circuits and with brute-force computational parameter exploration of neuron and network models. I will present experimental and modeling evidence to show that a variety of mechanisms support robustness at the cellular and neuronal network levels. Stable neuronal systems output is supported by the fact that similar neuron or network behavior can arise from different underlying cellular and network properties, meaning that neural systems do not have to be fine-tuned to a unique "solution", but rather have entire "solution spaces" at their disposal. Furthermore, activity-dependent and activity-independent homeostatic regulation mechanisms exist that allow a neuron or network to compensate for perturbations by adjusting its properties to return to a target activity.

4:30 pm Wednesday, November 29, 2006

Analysis Seminar: The norm of the maximal function on L^p(R^n), a Bellman function approach

by Leonid Slavin (U Conn) in Skiles 255

The problem of computing the norm of the Hardy-Littlewood maximal operator on L^p gives rise to a transparent Bellman function setup on a convex domain. An explicit expression for the Bellman function gives much more information about the operator in question than just its norm; however, only a handful of such computations exist. We outline the basics of the method and discuss the development of the Bellman PDE and its solution for two model cases: the dyadic n-dimensional and continuous one-dimensional. Both have been solved (the first one, by Melas; the second one, by Grafakos and Montgomery-Smith) but neither solution used the full power of the method. At the end, we discuss the challenges and perspectives for the higher-dimensional continuous case. This is joint work with Alex Stokolos.

4:30 pm Wednesday, November 29, 2006

Applied & Computational Mathematics Seminar: Fully discrete finite element approximations of parabolic optimal control problems

by Steve Hou (Iowa State) in Skiles 269

In this talk we discuss fully discrete finite element approximations of optimal control problems for semilinear parabolic PDEs, using the forced Fisher equation as a prototype example. We define a nonstandard one-step time- discretization scheme with standard spatial finite elements, and derive fully discrete error estimates. We also present some related results concerning fractional-ordered fully discrete approximations of linear parabolic PDEs and continuous embeddings of time-dependent Sobolev spaces.

10:00 am Thursday, November 30, 2006

QCF Seminar: Econometric Evaluation of Asset Pricing Models with No-Arbitrage Constraint

by Haitao Li (University of Michigan) in Skiles 269

We develop econometric methods for evaluating asset pricing models that explicitly require that a correct asset pricing model has to be arbitrage free. In particular, we develop the asymptotic distribution of the second Hansen-Jagannathan distance which measures the least-square distance between a given asset pricing model and a set of positive stochastic discount factors that correctly price all assets. Simulation evidence shows that our test has good finite sample performance for typical sample sizes considered in the literature. The no-arbitrage constraint makes significant differences in empirical studies of asset pricing models using the Fama-French size and book-to-market portfolios or hedge fund portfolios that exhibit option-like returns. Without the no-arbitrage constraint, we fail to reject certain models using existing methods. However, our test overwhelmingly rejects these models because their stochastic discount factors take negative values with high probabilities and therefore are not arbitrage free.

3:05 pm Thursday, November 30, 2006

Graph theory seminar: On Forbidden Subdivision Characterization of Graph Classes

by Zdenek Dvorak (Charles University, Prague) in Skiles 255

We provide a characterization of several graph parameters (acyclic chromatic number, arrangeability, and sequence of parameters related to the expansion of graph) in terms of forbidden subdivisions. We also relate these results to several open problems regarding game chromatic number.

3:30 pm Friday, December 1, 2006

Geometry-Topology Reading Seminar: The Kauffman bound on the Bennequin number (after Rutherford)

by Elena Bogdan (Univeristy of Pennsylvania, visiting Ga Tech) in Skiles 269

4:00 pm Friday, December 1, 2006

Combinatorics: Random Linear Extensions of Grids

by Joshua Cooper (University of South Carolina) in Skiles 255

A grid poset -- or "grid" for short -- is a product of chains. We ask, what does a random linear extension of a grid look like? This problem generalizes now-classical work on random plane partitions, and has surprising connections with the theory of random Ferrer diagrams, poset order dimension, representability theory in qualitative probability, and conjoint analysis (a subfield of marketing research). We show that the average "jump number," i.e., the number of times that two consecutive elements in a linear extension are incomparable in the poset, is close to its maximum possible value. The techniques employed rely on entropy arguments. We mention several interesting questions about this wide-open area.

3:30 pm Monday, December 4, 2006

Geometry-Topology Seminar: Trace fields and commensurability of link complements

by Eric Chesebro (Rice University) in Skiles 269

I will present a family of 2-component links in S^3 who have hyperbolic complements for which many algebraic and geometric invariants can be easily computed. Among other things, this allows us to to show that these links are all pairwise incommensurable yet they all have the same trace field. Our work is inspired by a question of Alan Reid's about the existence of infinitely many knot complements all with the same trace field. (This project is joint with Jason Deblois.)

3:00 pm Tuesday, December 5, 2006

PDE Seminar: Some Remarks on W^{2,2} Mappings

by Bob Hardt (Rice University) in Skiles 255

Consider a mapping u from a smooth domain to a compact Riemannian manifold N whose Hessian energy \int | D2 u|^2 dx is finite. In joint work with T. Riviere, relations between the Hessian energy, the topological singularity of the map, and approximability by smooth maps is studied for W^{2,2}(B5,S3) and other topologically related cases. We also describe constructions needed for this work and for joint work with CY Wang on proving the optimal partial regularity of Hessian energy-minimizers with various target manifolds N.

4:30 pm Tuesday, December 5, 2006

ACO Colloquium: Perfect Implementation of Normal-Form Mechanisms

by Sergei Izmalkov (MIT) in Skiles 255

Privacy and trust affect our strategic thinking, yet they have not been precisely modeled in mechanism design. In settings of incomplete information, traditional implementations of a normal-form mechanism ---by disregarding the players' privacy, or assuming trust in a mediator--- may not be realistic and fail to reach the mechanism's objectives. We thus investigate implementations of a new type.

We put forward the notion of a perfect implementation of a normal-form mechanism M: in essence, an extensive-form mechanism exactly preserving all strategic properties of M, without relying on a trusted party or violating the privacy of the players.

We prove that any normal-form mechanism can be perfectly implemented via envelopes and an envelope-randomizing device (i.e., the same tools used for running fair lotteries or tallying secret votes).

3:30 pm Friday, December 8, 2006

Geometry-Topology Reading Seminar: The Kauffman bound on the Bennequin number (after Rutherford)

by Elena Bogdan (Univeristy of Pennsylvania visiting Ga Tech) in Skiles 269

4:00 pm Friday, December 8, 2006

Combinatorics: Dispersion of Mass and Lower Bounds for Randomized Algorithms

by Luis Rademacher (M.I.T.) in Skiles 255

How much can randomness help computation? Motivated by this general question and by volume computation, one of the few instances where randomness provably helps, we analyze a notion of dispersion and connect it to asymptotic convex geometry. We obtain a nearly quadratic lower bound on the complexity of randomized volume algorithms for convex bodies in R^n (the current best algorithm has complexity roughly n^4, conjectured to be n^3). Our main tools, dispersion of random determinants and dispersion of the length of a random point from a convex body, are of independent interest and applicable more generally; in particular, the latter is closely related to the variance hypothesis from convex geometry. This geometric dispersion also leads to lower bounds for matrix problems and property testing.

3:30 pm Monday, December 11, 2006

Geometry-Topology Seminar: sl(N) link homology from trivalent TQFT

by Blanchet Christian in Skiles 269

For each integer N, we construct a TQFT for planar trivalent graphs and trivalent surfaces. Following Khovanov construction for sl(2) and sl(3), we use this TQFT to define an homology of links. This homology is defined over integers; over rational numbers, it should be equivalent to Khovanov-Rozansky sl(N) link homology.

2:00 pm Monday, January 8, 2007

Georgia State Mathematics Colloquium: Stability in Extremal Set Theory

by Dhruv Mubayi (University of Illinois, Chicago) in COE 796, 30 Pryor St, Atlanta

I will begin by defining the notion of stability for monotone properties of set systems. This formulation encompasses the classical definition in extremal graph theory initiated by Erdos and Simonovits in the 60's. I will then give stability theorems for old problems of Erdos, Frankl, Furedi, Katona, Sos, and some new problems as well. If time permits, a brief description of the proof techniques will be provided.

3:00 pm Tuesday, January 9, 2007

PDE Seminar: Recent progress in Conformal Geometry

by Abbas Bahri (Rutgers University) in Skiles 255

We will describe in this talk some recent work on Sign-changing Yamabe-type problems leading to the completion of a Morse Lemma at infinity in this framework.Accordingly, the search for infinitely many solutions for this family of problems on Riemannian compact manifolds becomes possible. Several meaningful objects, some belonging to Geometry, others to Topology or Analysis are described. A discrete inequality involving the Green's function of the conformal Laplacian is conjectured to hold.

11:00 am Wednesday, January 10, 2007

Combinatorics Seminar: Efficient algorithms for computing all low s-t edge connectivities and related problems

by Debmalya Panigrahi (Bell Labs Research, Bangalore, India) in Skiles 269

Given an undirected unweighted graph on n nodes and m edges, we consider the problem of finding all the k-edge-connected components of the graph for any given k, and the edge connectivity between these components. In essence, the problem asks for identifying (and thereafter finding the pairwise edge connectivity of) all pairs of vertices whose connectivity is less than k. We present an algorithm with expected time complexity \tilde{O}(m + nk^3), which represents the first algorithm with sub-quadratic (in n) time complexity for this problem and improves upon the previous best bound of \tilde{O}(n^2 k^2) achieved by Nagamochi and Watanabe. In this algorithm, we introduce a new data structure called a "partial Gomory-Hu tree" and also present a new paradigm for designing edge connectivity algorithms by replacing the classical approach of iteratively running maxflow computations by a set of overlapping minimum steiner cut computations. We also show that this new approach can be used to efficiently solve other problems like the minimum T-cut problem. Finally, we report that using this new technique in conjunction with an improved minimum steiner cut computation algorithm, we can improve the time complexity of the Gomory-Hu tree construction problem to \tilde{O}(nm) from \tilde{O}(nm^{1/2} min(n^3/2, m)).

3:00 pm Wednesday, January 10, 2007

Special Geometry Seminar: Nondegeneracy and Moduli Space Theory for CMC Surfaces

by Rob Kusner (University of Massachusetts, Amherst) in Skiles 255

Properly embedded constant mean curvature (CMC) surfaces -- the model for complete noncompact soap bubbles or equilibrium fluid droplets -- have a particularly nice asymptotic behavior. This leads to a pair of natural questions: 1) How well do these asymptotic data determine the surface? 2) Can one describe the moduli space of all CMC surfaces with a given (finite) topology, using these asymptotic data as natural parameters? The key to answering these questions is the nondegeneracy of the linearized mean curvature, namely, the Jacobi (or second variation of area) operator. We will report on recent joint work with K. Grosse-Brauckmann, N. Korevaar, J. Ratzkin and J. Sullivan, in which we have found good estimates for the (tempered) nullity of the Jacobi operator. For the special class of coplanar CMC surfaces, we have also proven that the natural classifying homeomorhism from CMC moduli space to k-pointed spherical metrics is, in fact, a diffeomorphism. It follows that all 3-ended CMC surfaces of genus 0 are nondegenerate, and we hope to use this to show the same is true for an number of ends k > 3 (at least in the coplanar case). Some applications to CMC gluing constructions and implications for the structure of CMC moduli space will also be discussed.

3:05 pm Thursday, January 11, 2007

Stochastic Seminar: Structure adaptive approach to dimension reduction

by Anatoli Juditsky (Grenoble, France) in Skiles 269

The problem of estimating the effective subspace in the multi-index regression model with deterministic design and additive noise is considered. A new procedure for recovering the directions of the subspace is proposed. Under mild assumptions, sqrt(n)-consistency of the proposed procedure is proved (up to a logarithmic factor) in the case when the effective dimension does not exceed 4. The empirical behavior of the algorithm is studied through numerical simulations.

4:05 pm Thursday, January 11, 2007

Mathematics Colloquium (Job Candidate): Flows, bumps, and flexibility: fish fins, whale flippers, and more

by Silas Alben (Division of Engineering and Applied Sciences, Harvard University) in Skiles 269

I will discuss a few recent studies on how organisms propel themselves through water, focusing on the appendages that allow them to do so efficiently. I will begin with fish fins, which have evolved over millions of years in a convergent fashion, leading to a highly-intricate fin-ray structure that is found in half of all fish species. This fin ray structure gives the fin flexibility plus one degree of freedom for shape control. I will present a linear elasticity model of the fin ray, based on experiments performed in the Lauder Lab in Harvard's Biology department. In conjunction with this work, I will present numerical simulations of a fully-coupled fin-fluid model, based on a new method for computing the dynamics of a flexible bodies and vortex sheets in 2D flows. The simulations are applied to the most common mode of fish swimming, based on tail fin oscillations. In the passive case, an optimal flexibility for thrust is identified, and we consider also the optimal distribution of flexibility, with reference to recent measurements of tapering of insect wings and fish fins. We also briefly present work on fundamental instabilities of a flexible body aligned with a flow (the "flapping flag" problem). I will then discuss work on the role of bumps on the leading edge of humpback whale flippers, in collaboration with Ernst van Nierop and Michael Brenner at Harvard. Bumps have been shown in wind tunnels to increase the angle of attack at which flippers lose lift dramatically, or "stall." This stall-delay is thought to enable greater agility. In this study we propose an aerodynamic mechanism which explains why the lift curve flattens out as the amplitude of the bumps is increased, leading to potentially desirable control properties. Finally, I will briefly describe results on a recent problem in self-assembly: the formation of 3D structures from flat elastic sheets with embedded magnets. The ultimate utility of this method for assembly depends on whether it leads to incorrect, metastable structures. We examine how the number of metastable states depends on the sheet shape and thickness. Using simulations and the theory of dislocations in elastic media we identify out-of-plane buckling as the key event leading to metastability. The number of metastable states increases rapidly with increasing variability in the boundary curvature and decreasing sheet thickness.

4:45 pm Tuesday, January 16, 2007

PDE Seminar: Amazing Results in Optimal Transport

by Allen Tannenbaum (Electrical and Computer Science and Bio Medical Engineering, Georgia Tech) in Skiles 269 (Note different room and time)

In the talk, we will give some amazing new results in optimal tranpsort. Applications to medical imaging will be included.

11:00 am Wednesday, January 17, 2007

Mathematical Biology & Ecology Seminar: Fundamental Questions in Genome Evolution

by Soojin Yi [mail] (School of Biology, Georgia Tech) in Skiles 255

Adaptive diversity on earth originates from genomic mutations at the molecular level. There are several classes of genomic mutations, including point mutations, insertions and deletions, and larger scale rearrangements and duplications. An extreme case of a larger scale rearrangement is when the whole genome duplicates, which happened numerous times during evolutionary history. The frequency of different types of mutations, how they are initially retained, and how they lead to evolution of different genomic traits are fundamental problems in genome evolution, yet mostly remain unknown at the current state. In this presentation, I will review some of recent work in my group, investigating the relationship between different types of mutations and genome evolution. These include (i) our ongoing investigation into the effect of mutational origins on molecular clocks and regional heterogeneity in mammalian genomes, and (ii) relationship between sequence and functional evolution of proteins following whole genome duplication, in yeast.

2:00 pm Wednesday, January 17, 2007

Theory of Computation Colloquium: Quadratic Forms on Graphs

by Konstantin Makarychev (Princeton University) in Klaus 1116

We introduce a new graph parameter, called the Grothendieck constant of a graph, which measures the integrality gap of certain integer programs, and is motivated by various algorithmic applications. Its study leads to several extensions of the classical inequality of Grothendieck, to an improvement of a recent result of Kashin and Szarek, and to a solution of a problem of Megretski and of Charikar and Wirth. Based on a joint work with Noga Alon, Yury Makarychev and Assaf Naor.

2:00 pm Wednesday, January 17, 2007

Analysis/PDE: Applications of the Cahn-Hilliard Equation

by Amy Novick-Cohen (Technion) in Skiles 269

Historically the Cahn-Hilliard equation was developed to model phase separation in binary alloys, and among its successes was its ability to qualitatively describe both the early stages (spinodal decomposition) and the later stages (coarsening) of phase separation. Recently, it has been found to be a good model in numerous contexts in which pattern formation is accompanied by a mass constraint. These include: population dynamics, biofilm structure, thin films with thermal-capillary and gravitational effects, and stratifications which have been sited in the rings of Saturn. The plan of the lecture is to outline the assumptions behind the modeling which leads to the Cahn-Hilliard equation in these different contexts, with particular emphasis on the importance of including a possibly degenerate mobility in the modeling.

3:00 pm Wednesday, January 17, 2007

Analysis : The Schrodinger Equation with a Large Magnetic Potential

by Michael Goldberg (Johns Hopkins) in Skiles 269

We study the Schrodinger operator H = (i\nabla + A)^2 + V, where both the magnetic and electric potentials are time-independent and have polynomial pointwise decay. The potentials need not be small with respect to any norm, nor do we assume any additional regularity or sign conditions. Many properties of the free Schrodinger operator H_0 = -\Delta are still present in this case, including: *There is no singular spectrum of H on the positive half-line *The resolvent of H is a bounded map between weighted L^spaces, with similar asymptotics to H_0 in the high-energy limit. *The linear propagator e^ itH satisfies Strichartz inequalities, once all bound states are projected away. These results are obtained in collaboration with Burak Erdogan and Wilhelm Schlag.

4:30 pm Wednesday, January 17, 2007

Joint Applied & Computational Mathematics/Mathematical Biology Seminar: Prediction of Protein Structure, Function and Druggability on a Proteomic Scale

by Jeffrey Skolnick (School of Biology, Georgia Tech) in Skiles 255

A novel method for the prediction of protein structure, function and druggability based on the sequence-to-structure-to-function paradigm has been developed. We first show recent results that suggest that for compact single domain proteins, the PDB is most likely complete and that the completeness can be explained by the packing of compact, hydrogen bonded, secondary structural elements. We next present results from the application of our structure prediction algorithm, TASSER to all GPCRs in the human genome. Based on confidence criteria, 90% should have approximately correct structures, and clustering shows that structurally similar GPCRs have similar function even when their sequences are diverse. We then describe our multimeric structure prediction algorithm, m-TASSER, and its application to the prediction of protein-protein interactions. Next, a newly developed method for the accurate inference of protein biochemical function is presented and results of the comprehensive analysis of all sequenced genomes and the automated assignment of proteins to metabolic pathways are shown. Finally, we combine these approaches into a pathway-based method for the prediction of druggable protein targets and apply the resulting methodology to the human genome.

11:00 am Thursday, January 18, 2007

Theory of Computation Colloquium: Approximation Algorithms for Unique Games

by Yury Makarychev (Princeton University) in Klaus 1116

Unique games are constraint satisfaction problems that can be viewed as a generalization of MAX CUT to a larger domain size: We are given a graph G = (V,E) on n vertices and a permutation P_{uv} for every edge (u,v) on the set of labels {1,...,k}. Our goal is to assign a label X_u in {1,...,k} to each vertex u, so as to maximize the number of satisfied constraints P_{uv} (X_u) = X_v. Given an instance where all constraints are satisfiable, it is easy to find such a satisfying assignment. However, given an instance where a (1-epsilon) fraction of all constraints is satisfiable, the Unique Games Conjecture of Khot says that it is NP-hard to satisfy even a delta fraction of the constraints (for k large enough). This conjecture has attracted a lot of recent attention since it has been shown to imply hardness of approximation results for several important problems that were difficult to approach by other means. We present three approximation algorithms for Unique Games that satisfy roughly k^{-epsilon/2}, 1 - O(sqrt{epsilon log k}) and 1 - epsilon * O(sqrt{log k log n}) fraction of all constraints if a (1-epsilon) fraction of all constraints is satisfiable. This talk is based on joint papers with Moses Charikar, Eden Chlamtac, and Konstantin Makarychev.

1:30 pm Thursday, January 18, 2007

Graph theory seminar: The Tau Constant, An Invariant of Metrized Graphs

by Zubeyir Cinkir (University of Georgia) in Skiles 255

On a weighted graph G, thought of as a metrized graph, the diagonal value of the Arakelov-Green's function g_{m}(x,y) is a constant for a certain "canonical measure" m on G. This constant is called "the tau constant", denoted as tau(G), introduced by M.Baker and R.Rumely, studied in Summer 2003 REU at UGA and related to S. Zhang's work on the reduction of curves.

There are a number of ways to describe tau(G). In terms of spectral theory, it is the trace of the inverse of the continuous Laplacian on G. Also, it is closely related to the resistance and the voltage functions on G when G is considered as an electrical network. Our main focus is to show the existence of a universal positive lower bound to tau(G) for any G with length(G)=1.

We will show how tau(G) changes under various graph operations e.g., doubling edges, deleting and contracting edges, union of graphs along one or two points. We will establish some identities which we call "the deletion and the contraction identities". We will show that tau(G) >= length(G)/12*(1-4/N)2 , where the edge connectivity N > 4. We will show how tau(G) is related to the discrete Laplacian and present an effective Matlab program computing tau(G). We will present several families of graphs with equal edge lengths and having tau constants asymptotically approaches to length(G)/108. We will also present an application by giving the relation between tau(G) and an another constant that was used in one of S. Zhang's papers.

3:05 pm Thursday, January 18, 2007

Stochastic Seminar: Generalized Multiparameter Likelihood Models

by Ming-Yen Cheng (Department of Mathematics, National Taiwan University) in Skiles 269

Traditional parametric modelling assumes the samples come from a specific family of distributions, and estimates the unknown parameters based on the samples. Maximum likelihood estimation is most powerful in that case. In reality, we don't know which family of distributions the samples come from, and model misspecification can result in larger biased estimators. Nonparametric modelling makes no assumptions on the distribution; however, it would pay a price on variance side since there are more unknowns to be estimated. Even worse, when the dimension of the covariate is large, nonparametric modelling has the ``curse of dimensionality" problem. A promising idea is to relax the restrictions imposed by parametric models to make model specification more flexible. In this paper, we develop a hybrid of parametric and nonparametric models which has a wide range of applications. It copes with multiple covariates and adapts to dynamic structural changes well. The associated estimation problem is solved by a simple and effective method. The proposed estimator of the parametric part has root-n convergence rate, and the estimator of the nonparametric part enjoys an oracle property. Model selection is discussed. Simulation studies are conducted to demonstrate the performance of the proposed methods. Finally, the proposed model is used to analyse the infant mortality data of China.

4:30 pm Thursday, January 18, 2007

Mathematics Colloquium (Job Candidate): Design and Application of Optimal Algorithms for Systems of Partial Differential Equations

by Jinchao Xu (Mathematics, Penn State) in Skiles 269

A number of recent results, including special discretization schemes, adaptive methods and multilevel iterative methods for the resulting algebraic systems, will be presented in this talk for various partial differential equations (PDE). With a careful and combined use of qualitative properties of PDEs, the underlying functional spaces and their discretizations, many different kinds of equations will be treated with the same or similar techniques. After an introduction to some practically efficient methods such as the algebraic multigrid method for the Poisson equations, it will be shown how more complicated systems such as linear elasticity equations, electro-magnetic equations, porous media, Stokes equations and more general newtoninan-nonnewtonian models can be reduced to the solution of a sequence of Poisson equation and its simple variants. The efficiency of these algorithms will be illustrated by theoretical analysis, numerical examples and engineering applications.

8:30 am Monday, January 22, 2007

Workshop on Complex Networks and Their Applications: Joint Dimacs-Georgia Tech Special Year on Discrete Random Systems

in Georgia Tech TSRB Auditorium (Tech Square)

Complex networks and more generally complex systems are pervasive in today's science and technology. They include examples like the Internet, the WWW, peer-to-peer, sensor and ad-hoc networks, as well as biological networks such as gene and protein interactions and others. The study of such networks spans across mathematics, computer science, engineering, biology and the social sciences. In this three day workshop we want to bring together researchers from mathematics, theoretical computer science, networking, systems and biology. With the common theme of complex networks and systems, we want to share the state of the art in these fields, the work done so far, the vision and major challenges of the future. See http://www-static.cc.gatech.edu/~mihail/indexcomplex.html for details.

3:00 pm Monday, January 22, 2007

Geometry-Topology Seminar: Realizing 4-manifolds as achiral Lefschetz fibrations

by Terry Fuller (Cal State, Northridge) in Skiles 269

Work of Donaldson and Gompf in the late 90s characterized symplectic 4-dimensional manifolds as those admitting a particular sort of fibration by surfaces known as a Lefschetz fibration. Following this, it was natural to ask to what extent arbitrary smooth 4-manifolds admit similar structures. In this talk, I will discuss joint work with John Etnyre which demonstrates that any smooth 4- manifold contains an embedded curve such that the manifold obtained by surgery along that curve admits an achiral Lefschetz fibration.

4:05 pm Monday, January 22, 2007

Geometry-Topology Seminar: A topological proof of Mess's theorem

by Dan Margalit (University of Utah) in Skiles 269

The Torelli group is the subgroup of the mapping class group consisting of elements which act trivially on the homology of the surface. In his 1989 thesis Geoff Mess proved the celebrated theorem that the Torelli group in genus 2 is an infinitely generated free group. While his proof is algebro-geometric, we give a new, topological, approach, with a view towards the higher genus cases.

4:30 pm Monday, January 22, 2007

CDSNS Colloquium: Estimating density dependence, process noise, and observation error in populations models

by Brian Dennis [mail] (University of Idaho (Department of Fish and Wildlife Resources and Department of Statistics)) in Skiles 255

In this presentation I describe a discrete time, stochastic population model with density dependence, environmental-type process noise and lognormal observation or sampling error. The model, a stochastic version of the Gompertz model, can be transformed into a linear Gaussian state space model (Kalman filter) for convenient fitting to time series data. The model has a multivariate normal likelihood function and is simple enough for a variety of uses ranging from theoretical study of parameter estimation issues to routine data analyses in population monitoring. A special case of the model is the discrete time, stochastic exponential growth model (density independence) with environmental-type process error and lognormal observation error. Two methods for estimating parameters in the Gompertz state space model, maximum likelihood based on time series observations and restricted maximum likelihood based on first differences, are compared with computer simulations. Both offer adequate statistical properties. Because the likelihood function is identical to a repeated measures analysis of variance model with a random time effect, parameter estimates can be calculated using PROC MIXED of SAS. Data sets from the Breeding Bird Survey provide illustrative analyses. For one data set, the fitted model suggests that over 70% of the noise in the population's growth rate is due to observation error. The model describes the autocovariance properties of the data especially well. While observation error and process noise variance parameters can both be estimated from one time series, multimodal likelihood functions can and do occur. For data arising from the model, the statistically consistent parameter estimates do not necessarily correspond to the global maximum in the likelihood function. Maximization, simulation, and bootstrapping programs must accomodate the phenomenon of multimodal likelihood functions to produce statistically valid results.

5:10 pm Monday, January 22, 2007

Geometry-Topology Seminar: Crossing changes and slice knots

by Brendan Owens (LSU) in Skiles 269

The slicing number of a knot is the minimum number of crossing changes required to convert to a slice knot. I will show that for many knots, previous bounds on unknotting number obtained by Ozsvath and Szabo and the speaker are in fact bounds on slicing number. I will also discuss a refined slicing invariant due to Livingston and exhibit an infinite family of knots for which Livingston's invariant is strictly greater than the slice genus.

3:00 pm Tuesday, January 23, 2007

PDE Seminar: Structures of Local Minimizers of a One-dimensional Higher Order Variational Problem

by Aaron Yip (Purdue University) in Skiles 255

We present results on the structures of local minimizers of a one-dimensional variational problem. The investigation of the energy functional is motivated by the study of coherent solid-solid phase transformations and the competition between the effects from regularization and formation of small scale structures. With a special choice of a bilinear double well potential function, we can prove the periodicity of all local minimizers with low enough energy. This is due to some intricate interactions between the boundaries of different phases. Some connections with the dynamics and other types of phase transition phenomena will also be discussed.

11:00 am Wednesday, January 24, 2007

Mathematical Biology & Ecology Seminar: Mathematical models of drug resistance emergence

by Andreas Handel [mail] (Department of Biology, Emory University) in Skiles 255

Drug resistant pathogens pose a significant public health problem. While numerous studies have taught us a lot about specific aspects of drug resistance emergence, a 'big-picture' framework is still missing. In this talk, I present some work on within-host and between-host modeling of drug resistance and show how data combined with models might bring us closer towards a comprehensive and quantitative understanding of the processes that lead to drug resistance emergence.

2:00 pm Wednesday, January 24, 2007

Geometry: Nonpositively Curved Surfaces in Euclidean 3-space

by Hsungrow Chan (National Pingtung University of Education, Taiwan) in Skiles 269

In this talk we discuss generalizations of some minimal surface theorems in Euclidean three space to surfaces with nonpositive Gauss curvature and square integrable second fundamental form. For example we show that if a complete simply connected nonpositively curved surface that is twice continuously differentiably embedded in Euclidean three space has square integrable second fundamental form, then it must be a plane. Further, if a complete nonpositively curved surface with one end is embedded near infinity and has square integrable second fundamental form, then it must lie between two parallel planes. These results may be considered generalizations of Bernstein's minimal surface theorems.

4:30 pm Wednesday, January 24, 2007

Analysis : Discrete multivariable orthogonal polynomials satisfying second-order difference equations

by Plamen Iliev (GT) in Skiles 255

Orthogonal polynomials $p_n(x)$ of a single variable $x$, which are eigenfunctions of a second-order difference or differential operator $D$ in $x$, have numerous applications in mathematics and physics. They provide special solutions of the so called ``bispectral problem'', and within this framework, the existence of $D$ amounts to a hidden symmetry between the variables $n$ and $x$. In this talk, I will characterize discrete multivariable orthogonal polynomials, which are eigenfunctions of a (partial) second-order difference operator. The results are based on joint work with Yuan Xu.

10:00 am Thursday, January 25, 2007

QCF Seminar: Impact of PD and LGD correlation on Credit Portfolio

by Sungjoo Lee (SunTrust Banks) in Skiles 269

Risk Adjusted Return on Capital (RAROC) is one of the crucial tools in credit portfolio management. The use of risk based capital strengthens the risk management discipline within business as the methodologies employed quantify the level of risk within each line of business and attribute capital accordingly. We investigate how the correlation of Probability Default and Loss given Default changes the probability distribution of portfolio. Two risk measures expected loss (EL) and economic capital (EC) which are essential elements of RAROC and regulatory capital are studied.

3:05 pm Thursday, January 25, 2007

Job Candidate Seminar: Tridiagonal Random Matrix Models

by Ionel Popescu (Department of Mathematics, Northwestern University) in Skiles 269

We will discuss the generalization of the beta ensembles introduced by Dumitriu and Edelman and in particular the convergence of the empirical distributions and the fluctuations from. It turns out that the limit distribution is not in general the Wigner semicircle law as it happens in the case of the Wigner ensembles, but the fluctuations under general conditions converge to a Gaussian. We will also discuss the case of the convergence of multiple random matrices and the analogy with free probability theory.

4:30 pm Thursday, January 25, 2007

Mathematics Colloquium (Job Candidate): Drinfeld modular varieties and the Weil-Deligne bound

by Mihran Papikian (Department of Mathematics, Stanford University) in Skiles 269

To obtain good codes from Goppa's algebro-geometric construction, one needs curves over finite fields which have many rational points compared to their genera. It was shown in the 1980's that modular curves, which play a very important role in number theory, are examples of such curves. In this talk I will explain how this result can be generalized to the case of higher dimensional modular varieties.

3:05 pm Friday, January 26, 2007

Combinatorics Seminar: An algorithm for source location in directed graphs

by Mihaly Barasz (Department of Operations Research, Eotvos University) in Skiles 255

In my talk I will briefly introduce the class of combinatorial optimization problems known as "Source Location" and give an overview of the known results. Then I'll present a (strongly) polynomial algorithm for the following problem. Given a directed graph D = (V,E) and nonnegative integers k and l, find a minimum-cardinality subset S of nodes such that for any node v not in S, there exist k pairwise edge-disjoint directed paths from S to v, and l pairwise edge-disjoint directed paths from v to S. Based on a joint work with Johanna Becker and Andras Frank.

3:30 pm Friday, January 26, 2007

Geometry-Topology Seminar: Knot concordance and Heegaard Floer homology invariants of branched covers

by Eli Grigsby (Columbia University) in Skiles 269

Let $K \subset (S^3=\partial B^4)$ be a classical knot. The {\it smooth concordance order} of $K$ is defined as the smallest $n \in \mathbb{Z}_+$ for which the connected sum of $n$ copies of $K$ bounds a smoothly-imbedded disk in $B^4$. I will describe two new invariants which yield an obstruction to a knot having finite smooth concordance order. These invariants are defined by examining analogues of ``classical'' Heegaard Floer homology invariants in the double-branched cover of $K$. Of the fifty-nine $2$-bridge knots of 12 or fewer crossings for which the smooth concordance order was previously unknown, we use these invariants to show that all fifty-nine have infinite concordance order. This is joint work with Daniel Ruberman and Sa\v{s}o Strle.

3:30 pm Monday, January 29, 2007

Geometry-Topology Seminar: Open Books and Smooth 4-manifolds

by Jack Calcut (Univeristy of Texas) in Skiles 269

Open book decompositions are useful structures on 3-manifolds. Planar open books also lead naturally to the study of smooth structures on simply connected 4-manifolds.

4:30 pm Monday, January 29, 2007

CDSNS Colloquium: Stabilizing the Benjamin-Feir instability

by Diane Henderson [mail] (Penn State) in Skiles 255

Here we report experiments on permanent form gravity waves on deep water propagating in both one and two horizontal dimensions. We find that moderate amplitude, bi-periodic patterns are ``stable'' within the length of our wave basin. This result is surprising in light of classic instability results (the Benjamin-Feir instability) for deep-water waves. And large amplitude experiments do show evidence of what appears to be the Benjamin-Feir instability. However, recent numerical results by Fuhrman & Madsen (2006) provide a different explanation. Our further experiments show that their explanation is correct and the patterns are indeed ``stable''. To explain the unexpected persistence of these patterns mathematically, we reconsider the stability of a uniform wavetrain using the nonlinear Schroedinger (NLS) equation modified to include linear damping. We prove that the presence of damping, no matter how small, stabilizes (with linear and nonlinear stability) the uniform wavetrain solution. The predicted evolutions are in excellent agreement with our experiments. These stability results are then extended to the case of a permanent form solution of coupled NLS equations that model wave patterns.

2:00 pm Tuesday, January 30, 2007

Job Candidate Seminar: The Geometry of Grassmannians and Flag varieties

by Izzet Coskun (Mathematics, MIT) in Skiles 269

A Littlewood-Richardson rule is a positive rule for computing the structure constants of the cohomology ring of flag varieties with respect to their Schubert basis. In recent years new geometric Littlewood-Richardson rules have led to the solution of many important problems, including Klyachko, Knutson and Tao's solution of Horn's conjecture and Vakil's solution of the reality of Schubert calculus. In this talk I will survey some of the basic geometric ideas that underlie geometric Littlewood-Richardson rules.

11:00 am Wednesday, January 31, 2007

Mathematical Biology & Ecology Seminar: Rabies Emergence Within Host Adaptive Landscapes

by Charles Rupprecht (CDC/CCID/NCZVED) in Skiles 255

Rabies is an acute, progressive viral encephalitis. This ancient zoonosis continues to offer a wealth of both paradoxical and enigmatic challenges from the realms of molecular virology to population biology. The etiological agents are relatively simple, small, rod-shaped negative-stranded, non-segmented RNA viruses in the Family Rhabdoviridae, Genus Lyssavirus, yet have one of the longest and most variable and unexplained incubation periods on record, which can extend for years after initial exposure. These viruses lack proof-reading mechanisms with resultant creation of evolutionary bottlenecks, or replication errors may also yield abundant raw material for natural selection. Lyssaviruses are quintessential neurotropic agents, but are entirely dependent naturally upon glandular tissue invasion and salivary excretion for efficient host transmission. Various viral subversive strategies may offer a means of escape from immune surveillance, or exploit particular microenvironments in the face of host adaptive landscapes. Even though it is one of the oldest infectious diseases, rabies continues to emerge in the 21st century. Warm-blooded vertebrates are susceptible to experimental infection, but only mammalian meso-carnivores and bats serve as substantial reservoirs. Viral variants may be maintained in some regions for decades or more, or may not exist at all in other suitable portions of host range. The viruses persist in populations, but are not maintained in individuals. While rabies has the highest case to fatality ratio of any infectious disease, not all exposures lead to productive infections. The dog is responsible for the burden of human cases in the developing world, but wildlife is important for most human exposures in developed nations. Within certain mammalian populations herd immunity may be immeasurable, or approach seroprevalence levels in excess of 50%. In some cases rabies may engage only a single predominant taxon, or envelop a multi-species community. Outbreaks may spread rapidly in wave-like fashion or encounter geographical barriers that impede further spread. Oral vaccination programs targeting selected free-ranging Carnivora offer an important adjunct control measure that can result in local disease elimination, but are not anticipated to affect regional dynamics of affected Chiroptera. Thus, considering its worldwide distribution, public health and veterinary ramifications, and substantial economic consequences, rabies remains the most significant viral zoonosis from a global perspective, and predicated upon its curious microbiology, ecology, and epizootiology, an ideal candidate for modeling considerations at a variety of levels.

3:30 pm Wednesday, January 31, 2007

Mathematics Colloquium (Job Candidate): Moments of Dirichlet L-functions and ranks of elliptic curves

by Matt Young (AIM and Stanford University) in Skiles 255

An L-function is a special type of holomorphic function that encodes arithmetical information. For example, properties of the Riemann zeta function can give information about the distribution of the prime numbers. Similarly, Dirichlet L-functions are used to count primes in arithmetic progressions, and elliptic curve L-functions are conjectured to detect if cubic diophantine equations have infinitely many rational solutions or not. In this talk I will tell the story of how L-functions relate to arithmetic, and describe some recent advances in our understanding of L-functions. In particular, I will discuss some results on mean values of Dirichlet L-functions, and on average ranks of elliptic curves.

3:00 pm Thursday, February 1, 2007

Job Candidate Seminar: Reliable and unreliable dynamics in driven oscillator networks

by Kevin Lin (Courant Institute) in Skiles 269

This talk concerns the reliability of coupled oscillator networks in response to fluctuating inputs. Reliability means that repeated presentations of an input elicit essentially identical responses regardless of the system's state at the onset of the input. This work is motivated by basic questions from neuroscience.

I will begin by showing how the concept of reliability can be formulated precisely in the framework of random dynamical systems theory, and recall the well-known fact that single phase oscillators are reliable. Our first new result is that unreliability can occur even in a 2-oscillator system; we propose a geometric mechanism for the observed phenomena. The rest of the talk concerns larger networks, including a natural condition which precludes unreliable behavior.

This is joint work with Eric Shea-Brown and Lai-Sang Young.

3:00 pm Friday, February 2, 2007

Combinatorics Seminar: Connectivity Augmentation Algorithms

by Laszlo Vegh (Eotvos University, Budapest) in Skiles 255

In my talk, I will give an overview of combinatorial algorithms for connectivity augmentation. The connectivity augmentation problem is the following: for a given k, add a minimum number of edges to a graph to make it k-connected. As the graph can be both directed or undirected, and we may talk about either edge or vertex connectivity, this general scheme contains four basic problems. Edge connectivity augmentation can be done as a nice application of Mader's splitting-off theorems. For directed vertex connectivity augmentation, I will present the general min-max formula of A. Frank and T. Jordan for covering set-pair functions. Then I will show the first combinatorial polytime algorithm for directed vertex connectivity augmentation, which we have developed jointly with A. Benczur.

3:30 pm Friday, February 2, 2007

Geometry-Topology Reading Seminar: Small Exotic 4-manifold

by Anar Ahmadov (Georgia Tech) in Skiles 269

3:30 pm Monday, February 5, 2007

Algebra Seminar: Abel-Jacobi maps for higher Chow groups.

by Matt Kerr [mail] (University of Chicago) in Skiles 269

4:30 pm Monday, February 5, 2007

CDSNS Colloquium: Invariant manifolds for quasilinear parabolic equations with fully nonlinear boundary conditions

by Yuri Latushkin [mail] (University of Missouri) in Skiles 255

We investigate quasilinear systems of parabolic partial differential equations with fully nonlinear boundary conditions in the setting of Sobolev--Slobodetskii spaces. We establish local wellposedness and study the time and space regularity of the solutions. Our main results concern the asymptotic behavior of the solutions in the vicinity of an equilibrium. In particular, the center, stable and unstable manifolds are constructed.

3:00 pm Tuesday, February 6, 2007

PDE Seminar: Critical Threshold in Eulerian Dynamics

by Dongming Wei (University of Maryland at College Park) in Skiles 255

We study critical regularity phenomena in Eulerian dynamics, \mathbf{u}_t + \mathbf{u}\cdot \nabla \mathbf{u} = F(\mathbf{u},D\mathbf{u},\cdots), where F represents a general force on the flow. We begin with the one-dimensional Euler-Poisson system driven by the Poisson forcing together with the usual \gamma-law pressure, showing that global regularity depends on whether or not the initial configuration of the Riemann invariants and density crosses an intrinsic critical threshold. Next we discuss the vanishing viscosity limit of the two-dimensional pressureless flows. Finally we show the surprising characterizations of critical thresholds in 3D and 4D restricted Euler dynamics.

11:00 am Wednesday, February 7, 2007

Mathematical Biology & Ecology Seminar: Classifying two-locus disease models using regular polyhedral subdivisions

by Debbie Yuster [mail] (Columbia University) in Skiles 255

Genes play a complicated role in how likely one is to get a certain disease. Biologists would like to model how one's genotype affects their likelihood of illness. I will discuss some classical disease models, as well as our new classification of two-locus disease models, where each model corresponds to an induced subdivision of a point configuration (basically a picture of connected dots). Our models reflect epistasis, or gene interaction. This work is joint with Ingileif Hallgrimsdottir. For more information, see our preprint at arXiv:q-bio.QM/0612044

12:00 pm Wednesday, February 7, 2007

Research Horizons Seminar / Gradstudent Pizza Seminar: What is a hyperlogarithm?

by Stavros Garoufalidis (Georgia Institute of Technology) in Skiles 269

We all know what is a logarithm, some may know what is a dilogarithm (or polylogarithm). But what is a hyperlogarithm? And how can one use hyperlogarithms to solve "simple" ODEs of the form: y'=y-1/x(A + B y^2)? I mean really solve! Tired of existence theorems with no concrete formulas and no effective constants? Tired of applications in this and that? Is this number theory? Or numerical analysis? Or Topology? Come and listen.

4:30 pm Wednesday, February 7, 2007

Applied & Computational Mathematics Seminar: Efficient Computation of Strained Heteroepitaxy Using Kinetic Monte Carlo

by Peter Smereka (University of Michigan) in Skiles 255

The growth of strained heteroepitaxial films using a Solid-on-Solid model KMC is discussed. Elastic effects are included by using a ball and spring model. Discrete models of this form naturally include nanoscale effects, such as nucleation, which are difficult to incorporate in continuum models. On the other hand, it is more computationally intensive to use these discrete models to simulate film growth on experimentally relevant length scales. This talk will discuss some of the computational challenges and approaches we have developed for simulation of heteroepitaxy. In particular, I will discuss a multigrid-Fourier method for rapid solution of the displacement field, an expanding box method which allows very quick local corrections to thedisplacement field, and a rejection-reduced KMC method which is combined with fast estimates of the hopping rates. Preliminary results of film growth will be presented which shows that when the elastic effects are small the film grows in a layer-by-layer fashion. However, when the elastic effects become strong we observe mound formation. This is joint work with Giovanni Russo and Tim Schulze.

10:00 am Thursday, February 8, 2007

QCF Seminar: Developing Financial Applications with MATLAB: Value at Risk Calculators

by Jamie Winter and Oren Rosen (The MathWorks, Inc.) in Skiles 269

In this talk we will demonstrate how MATLAB can be used to create some simple graphical user interface based "VaR Calculators". These will import historical stock price data from Excel and compute estimates of the maximum likely loss using historical simulation and parametric methods. The goal of the demonstration is to educate the audience on the wide variety of capabilities MATLAB offers for data management, analysis and visualization.

3:05 pm Thursday, February 8, 2007

Stochastic Seminar: An Integrated Approach for Protein Function Prediction

by Fengzhu Sun (Molecular and Computational Biology Program, Department of Biological Sciences, University of Southern California) in Skiles 269

Assigning functions to novel proteins is one of the most important problems in the post-genomic era. Many different sources of genomic data, such as protein-protein physical interactions, genetic interactions, gene expression, and domain structure, contain information about protein function. We developed a novel approach that employs the theory of kernel based Markov random fields to infer a protein's functions using an integrated approach combining various sources of biological data. The model is flexible in that other protein pairwise relationship information and features of individual proteins can be easily incorporated. We apply our integrated approach to predict functions of yeast proteins and in mouse.

4:15 pm Thursday, February 8, 2007

Math Dept. Tea:

in Skiles 236

All math faculty, staff, and students are invited for food and beverages.

10:30 am Friday, February 9, 2007

ACO/ARC Theory Day:

in Klaus 1116

A day-long event of talks covering subjects such as unbalanced expanders and randomness extractors, refuting dense random 3CNF formulas, Chernoff-style direct product theorem, and minimum bounded degree spanning trees. For details, see http://www.arc.gatech.edu/event_theoryday.php

3:30 pm Friday, February 9, 2007

Geometry-Topology Reading Seminar: Small Exotic 4-manifold

by Anar Ahmadov (Georgia Tech) in Skiles 269

3:00 pm Monday, February 12, 2007

Theory of Computation Colloquium: Efficient Algorithms using the Multiplicative Weights Update method

by Satyen Kale (Princeton University) in Klaus 1116 West

***Refreshments at 2:45pm in Klaus 2222***

Algorithms based on convex optimization, especially linear and semidefinite programming, are ubiquitous in Computer Science. While there are polynomial time algorithms known to solve such problems, quite often the running time of these algorithms is very high. Designing simpler and more efficient algorithms is important for practical impact. In my talk, I will describe applications of a Lagrangian relaxation technique, the Multiplicative Weights Update method in the design of efficient algorithms for various optimization problems. We generalize the method to the setting of symmetric matrices rather than real numbers. The new algorithm yields the first truly general, combinatorial, primal-dual method for designing efficient algorithms for semidefinite programming. Using these techniques, we obtain significantly faster algorithms for approximating the Sparsest Cut and Balanced Separator in both. Directed and undirected weighted graphs, and the Min UnCut problem. In addition, we also obtain an efficient derandomization of the Alon-Roichman theorem, a deterministic O(log n) approximation to the Quantum Hypergraph Covering problem, and an alternative proof of Aaronson's result on the learnability of quantum states.

3:30 pm Monday, February 12, 2007

Geometry-Topology Seminar: Near-symplectic broken Lefschetz fibrations

by Inanc Baykur (Michigan State University) in Skiles 269

The focus of this talk is on extending several techniques and ideas originated from the study of symplectic 4-manifolds to the much broader class of near-symplectic 4-manifolds. Auroux, Donaldson, and Katzarkov defined a generalization of Lefschetz fibrations, which we call 'broken Lefschetz fibrations', and showed that they are to near-symplectic 4-manifolds what Lefschetz fibrations are to symplectic 4-manifolds. Relying on this fact, Perutz defined an invariant, called 'Lagrangian matching invariant', which generalizes the Donaldson-Smith standard surface count to near-symplectic broken Lefschetz fibrations, and conjectured it to be equivalent to the Seiberg-Witten invariant. In this talk, I will present simplified representations of near-symplectic broken Lefschetz fibrations via Kirby diagrams and monodromies, and construct some examples. I will then describe some new operations in the near-symplectic setting, and discuss their effects on Seiberg-Witten and Heegaard-Floer invariants, as well as on Lagrangian matching invariants. Using these, we will derive some results regarding near-symplectic 4-manifolds with nontrivial invariants.

4:30 pm Monday, February 12, 2007

CDSNS Colloquium: Homoclinic chaos on routes into bursting in slow-fast models of neurons

by Andrey Shilnikov [mail] (GSU) in Skiles 255

Bursting is a manifestation of the complex, multiple time scale dynamics observed in diverse neuronal models. A description list of the nonlocal bifurcations leading to its onset is far from being complete and presents a dare need for cross-disciplinary neuroscience and the dynamical systems theory.There has been a recent breakthrough in this direction that explains a few novel mechanisms of transitions between tonic spiking and bursting activity, as well as their co-existence in models of leech interneurons through homoclinic saddle-node bifurcations of periodic orbits including a blue sky catastrophe. We will discuss the bifurcation theory that underlies theses transitions, as well as one on a spike adding route: as a parameter shifting the membrane potential of half-inactivation slow potassium current is monotonically changed, a sequence of bifurcations occurs causing incremental change of the number of spikes in a burst. Of our special interest is the origin of the sequence, where each transition is accompanied by chaos. To figure out the transition dynamics we construct a one-parameter family of the onto Poincare return mappings on the central manifolds of slow motions. We show that the transitions in question are due to the bifurcations of homoclinics of a repelling point of the map setting a threshold between tonic spiking and hyperpolarized states of the neuron model.

4:30 pm Tuesday, February 13, 2007

ACO Colloquium: Controllable random permutations

by Yuri Nesterov (Universit� catholique de Louvain) in Skiles 255

***Refreshments at 4PM in Skiles 236***

In this talk, we propose a new procedure for generating random permutations. We discuss applications of this technique to some scheduling problems. In particular, we show that by applying this procedure to a single machine, it is possible to ensure a desired average completion time for every job, or to prove that this is not feasible. The parameters of the procedure can be found in polynomial time by solving a simple nonlinear convex optimization problem.

11:00 am Wednesday, February 14, 2007

Mathematical Biology & Ecology Seminar: Mechanistic modeling to design or evaluate and possibly improve public health interventions.

by John Glasser (CDC/CCID/NCIRD) in Skiles 255

To illustrate the breadth of mathematical epidemiology at the CDC, I will contrast two ongoing projects. To deduce the long-term impact of vaccination on varicella and zoster caused by wild-type and vaccine-strain virus and to consider a second childhood dose, we modeled the transmission of varicella-zoster virus. We based parameters on the literature or independent analyses insofar as possible and adjusted others by fitting typical and modified varicella time-series from active surveillance since before vaccination began in Antelope Valley, CA. Our zoster results differ from those of earlier modelers, none of whom included internal boosting via controlled reactivations of latent virus through middle age, after which they lead increasingly to zoster. While such biological details affect projected long-term consequences of varicella vaccination, authorities must respond to outbreaks of new human diseases with limited information that may be inaccurate. Motivated by how little modelers contributed to the control of severe acute respiratory syndrome (SARS) and by the looming influenza pandemic, we modeled a generic emerging infectious disease. Our model�s stages are clinical, so sojourns can be guesstimated from initial case-series; infected people may become infectious at any time; social phenomena may occur at time-varying specific rates; when cases are diagnosed affects how long contacts remain at large before being quarantined; isolation may be inadequate until suspects are reclassified and hospitalized; and medications may shorten clinical course if begun early enough, and reduce infectiousness whenever begun. We derived an expression for the reproductive number, which must be less than one for control, and took its partial derivatives with respect to available interventions. We also analyzed responses to the outbreaks in Singapore and Taiwan, where quarantine substantially underperformed its potential. Insofar as movement restriction � versus social distancing and anti-viral drugs � contributed, recently-published projections of the containment of avian influenza in Southeast Asia, where person-to-person transmission likely would begin, may be overly optimistic. If policymakers involved modelers in decision-making, we could assist in designing or evaluating and improving public health interventions, including those to control future outbreaks of emerging infectious diseases.

12:00 pm Wednesday, February 14, 2007

Research Horizons Seminar / Gradstudent Pizza Seminar: Partial differential equations in infinite dimensions

by Andrzej Swiech in Skiles 269

I will give a brief introduction to second order PDE in infinite dimensional Hilbert spaces. I will discuss the main challenges, and difficulties of the theory and applications of such equations.

4:00 pm Wednesday, February 14, 2007

Analysis Seminar: Long arithmetic progressions in sumsets of random sets

by Izabella Laba (University of British Columbia) in Skiles 269

Let $A$ be a random subset of $\{1,...,N\}$, where the events $x\in A$ are independent and have probability $N^{-\epsilon}$ for a fixed $\epsilon>0$. Fix $k>3$. We prove that if $N$ is large enough, depending on $k$ and $\epsilon$, then the sumset $A+A$ contains a $k$-term arithmetic progression. Unlike in a recent result of Croot, Ruzsa and Schoen for deterministic sets, no dependence between $k$ and $\epsilon$ is assumed. The proof is based on Fourier-analytic arguments. (Joint work with Mariah Hamel.)

4:30 pm Wednesday, February 14, 2007

Applied & Computational Mathematics Seminar: An Efficient Real-Space Method for Orbital-Free Density Functional Theory

by Carlos Garcia-Cervera (UCSB) in Skiles 255

I will describe an efficient implementation of the Truncated-Newton method for energy minimization in the context of orbital-free density functional theory. I will illustrate the efficiency and accuracy of the method with numerical simulations in an Aluminium FCC lattice.

1:30 pm Thursday, February 15, 2007

Combinatorics Seminar: Phase Transitions in a random 3-SAT problem and the random NK landscape model

by Jeong Han Kim (Yonsei University (Seoul, Korea) and Microsoft Research) in Skiles 255

We consider and analyze the satisfiability problem of a certain random 3-SAT problem in which the appearances of 3-clauses are not independent. The random 3-SAT model is motivated by the random NK landscape, which is used as a test bed of many genetic algorithms. Proposed by Kauffman, the NK model is one of the most notable mathematical models to study the evolution on a fitness landscape, where a fitness landscape is a function that maps each genetic composition (genotype) to the fitness of the expression (phenotype) of the genetic composition in an environment. Though the idea of fitness came from actual biological environment, we may consider the fitness of an assigment to a given (SAT) problem, and try to design an algorithm that keeps increasing the fitness by changing a part of the assignment. Gao and Culberson introduced a random NK model and observed that the solubility problem of the random NK model is equivalent to the satisfiability problem of a certain random 3-SAT problem in which the appearances of 3-clauses are not independent. In this talk, we discuss a phase transition phenomenon for the random 3-SAT problem. In the course of the analysis, we also introduce a generalized random 2-SAT formula and discuss its phase transition phenomenon. No knowledge about genetic algorithms or NK landscape will be assumed. (Joint work with Sung-Soon Choi, Kyomin Jung)

3:00 pm Thursday, February 15, 2007

Stochastic Seminar: A Multivariate Statistical Approach to Performance Analysis of Wireless Communication Systems

by Siamak Sorooshyari (Lucent Technologies - Bell Laboratories) in Skiles 269

The explosive growth of wireless communication technologies has placed paramount importance on accurate performance analysis of the fidelity of a service offered by a system to a user. Unlike the channels of wireline systems, a wireless medium subjects a user to time-varying detriments such as multipath fading, cochannel interference, and thermal receiver noise. As a countermeasure, structured redundancy in the form of diversity has been instrumental in ensuring reliable wireless communication characterized by a low bit error probability (BEP). In the performance analysis of diversity systems the common assumption of uncorrelated fading among distinct branches of system diversity tends to exaggerate diversity gain resulting in an overly optimistic view of performance. A limited number of works take into account the problem of statistical dependence. This is primarily due to the mathematical complication brought on by relaxing the unrealistic assumption of independent fading among degrees of system diversity. We present a multivariate statistical approach to the performance analysis of wireless communication systems employing diversity. We show how such a framework allows for the statistical modeling of the correlated fading among the diversity branches of the system users. Analytical results are derived for the performance of maximal-ratio combining (MRC) over correlated Gaussian vector channels. Generality is maintained by assuming arbitrary power users and no specific form for the covariance matrices of the received faded signals. The analysis and results are applicable to binary signaling over a multiuser single-input multiple-output (SIMO) channel. In the second half of the presentation, attention is given to the performance analysis of a frequency diversity system known as multicarrier code-division multiple-access (MC-CDMA). With the promising prospects of MC-CDMA as a predominant wireless technology, analytical results are presented for the performance of MC-CDMA in the presence of correlated Rayleigh fading. In general, the empirical results presented in our work show the effects of correlated fading to be non-negligible, and most pronounced for lightly-loaded communication systems.

3:30 pm Friday, February 16, 2007

Geometry-Topology Seminar: Non-nullhomologous knots in lens spaces with \chi at least -1.

by Kenneth Baker (Georgia Tech) in Skiles 269

4:00 pm Friday, February 16, 2007

CANCELLED Analysis Seminar: Spectral phase transition in a class of Jacobi matrices

by Sergey Naboko (St. Petersburg State University (Russia) / UAB) in Skiles 255

We consider so-called first order spectral phase transition phenomena when the spectrum of a family of Hermitian Jacobi Matrices changes from discrete to pure absolutely continuous provided the matrix parameters cross some critical surfaces.Connection between this phenomenon and asymptotic behavior of the generalized eigenvectors of Jacobi matrices( = orthogonal polynomials) to be analyzed.The transition point corresponds to the case where the limit transfer matrix is a Jordan box or equivalently we have a "double root" case for the corresponding recurrent relation. Some examples illustration the topic including one coming from Markov birth and death processes to be presented.

11:00 am Monday, February 19, 2007

Mathematics Colloquium (Job Candidate): Repeating patterns for one-dimensional random walk in random environment. A law of the iterated logarithm.

by Dimitris Cheliotis (University of Toronto) in Skiles 269

We take a random walk (or diffusion) in a random one-dimensional environment, and we look at its graph at different, increasing scales natural for it. What are the patterns that we will see appear repeatedly? This is a classical problem in the theory of stochastic processes. Surprisingly, despite the complexity of our process, there is a neat characterization of the repeating patterns. The analogous result for random walk in a flat, deterministic environment is the well known functional law of the iterated logarithm of Strassen. The first half of the talk will be introductory. We will describe the model and state some fundamental results for it.

2:00 pm Monday, February 19, 2007

Job Candidate Seminar: Connectivity bounds in combinatorics

by Patricia Hersh (Department of Mathematics, Indiana University) in Skiles 269

Over the past few decades, several important problems (from algebra, discrete geometry, theoretical computer science, etc.) have been solved by associating to the problem a suitable simplicial complex and proving a lower bound on its connectivity. Lovasz pioneered this strategy, proving a much-sought chromatic number lower bound. I will discuss a discrete Morse theoretic approach to connectivity bounds, with applications to commutative algebra and representation theory. Then I will turn to very recent work proving a connectivity bound for "Coxeter-like complexes" and in the process giving a homological obstruction to greedy sorting of data by inversion elimination for many types of networks.

3:00 pm Monday, February 19, 2007

Theory of Computation Colloquium: Towards Universal Semantic Communication

by Madhu Sudan (MIT) in TBA

Consider the following fantastic scenario: Earth has just started receiving some signals from outer space. These signals don't seem like usual cosmic noise. Potentially an intelligent alien civilization is trying to make contact. How should Earth respond? How can we (earthlings) tell if the aliens are receiving our response and reacting to it? Are they really intelligent, or are we talking to sunspots? If they are intelligent, will we ever be able to achieve meaningful interaction in this setting? The classical theory of communication, typically ignores the issue of semantics of communication, and has focussed principally on quantitative measures in syntactic settings. Increasingly, however, it is becoming clear that practical challenges to communication arise due to semantic gaps between senders and receivers. The fictional problem above, merely, carries this gap to the extreme. In this talk, I will describe what complexity theory has to say about such interactions. Most of the talk will focus on how some of the nebulous notions, such as intelligence and understanding, should be defined in this setting. We'll also show how interactive proofs, randomness and one-way functions, can play an enabling role in establishing some meaningful communication. One of the consequences of our examination is a proposal for a mathematical test of intelligence, which is quite different from the "human-oriented" test proposed by Turing. Joint work with Brendan Juba (MIT).

3:30 pm Monday, February 19, 2007

Geometry-Topology Seminar: Open books and cut-and-paste constructions of symplectic 4-manifolds

by Jeremy van Horn (Univeristy of Texas) in Skiles 269

The most straight forward cut-and-paste operation on a Symplectic manifold involves cutting and pasting along hypersurfaces of contact type. We'll take a look at how open book decompositions can inform this procedure. As a nice example, we'll use open book decompositions to show rational blowdown exists in the symplectic setting.

4:30 pm Monday, February 19, 2007

Geometry-Topology Seminar: The fundamental class of a cusped hyperbolic 3-manifold

by Christian Zickert (Columbia University) in Skiles 269

A compact hyperbolic 3-manifold gives rise to a fundamental class in H_3(PSL(2,C)) by means of the discrete faithful representation. If the manifold has cusps we get a class in H_3(PSL(2,C),P), where P is the parabolic subgroup. We shall discuss a direct method for producing this class for a cusped hyperbolic manifold with an ideal triangulation, and discuss how this can be used to compute the Chern-Simons invariant.

4:30 pm Monday, February 19, 2007

CDSNS Colloquium: Physical measures and chaotic attractors for some multidimensional maps with applications to population models

by Ilie Ugarcovici [mail] (DePaul University) in Skiles 255

Given a dynamical systems, a physical measure describes the asymptotic distribution of all orbits starting from a positive Lebesgue measure set of initial conditions. Based on recent work of Q. Wang and L.-S. Young we show the existence of chaotic attractors and physical measures for some multidimensional nonlinear maps which have found applications to population dynamics. This is joint work with Howie Weiss.

11:00 am Tuesday, February 20, 2007

Guest Seminar: CFD by First-Order PDE's

by Bram van Leer (Department of Aerospace Engineering, University of Michigan) in Room 317, Weber Space Science & Technology Bldg

Over the past dozen years we have developed in our CFD Lab the view that flow physics would be best expressed in PDEs of the first order. Such PDEs only contain flux derivatives and local, possibly stiff source terms; these are called hyperbolic-relaxation (HR) equations. The flux terms represent advection and wave propagation, the source terms are responsible for damping, traditionally the role of second- or higher-order dissipation terms. A hierarchy of equation systems can be derived by taking successively higher moments of the collisional Boltzmann equation and using a BGK approximation of the collision terms. Best known is the 10-moment system, which describes viscous, nonconducting flow and is valid up to intermediate Knudsen numbers. A validation study based on a MEMS flow will be presented.Discretizations of HR systems have a host of advantages, among which: * stencils size is minimal; * stiffness is only local; * convergence is faster than for NS; * accuracy problems on adaptive grids are minimized; * suited for decomposition by equations; * suited for flow at intermediate Knudsen numbers, reacting flow, radiative hydrodynamics. A grand numerical challenge is to develop algorithms for HR systems that remain accurate whether or not the relaxation time-scale is resolved; this property is called "Asymptotic Preserving" (AP). It requires strong coupling between flux and source terms. We have chosen to pursue the AP property with Discontinuous Galerkin methods, since these achieve maximum compactness of the computational stencil for any the order of accuracy, and therefore are a good match to the PDE's chosen. I will describe the development of DG methods for HR systems and illustrate it with numerical results. A second challenge lies in the detailed physical modeling of shock waves, in particular, steady shocks in supersonic flow. Viscous shock structures obtained with HR systems for most Mach numbers show an incorrect embedded inviscid-like discontinuity. I shall explain the nature of such discontinuities, and present an example of how to prevent their appearance.

3:00 pm Tuesday, February 20, 2007

PDE Seminar: Sharp integral inequalities for harmonic function

by Xiaodong Yan (Michigan State University) in Skiles 255

Motivated by Carleman's proof of isoperimetric inequality in the plane, we study some sharp integral inequalities for harmonic functions on the upper halfspace. We also derive the regularity for nonnegative solutions of the associated integral system and some Liouville type theorems.

11:00 am Wednesday, February 21, 2007

Mathematical Biology & Ecology Seminar: Population genetics and vector biology of Amblyomma americanum

by Tonya Mixton (CDC/CCID/NCZVED) in Skiles 255

Amblyomma americanum has emerged as an important vector of human disease over the last decade, and the geographic range of this vector has expanded due to unknown factors. Genetic studies of arthropod vectors seek to describe the genetic variability between populations in order to predict the introduction of disease agents into new areas, the spread of resistant alleles, and potential epidemics. I describe the population structure of A. americanum and �Rickettsia amblyommii� from 32 populations throughout the range of this tick using denaturing high performance liquid chromatography to analyze single nucleotide polymorphisms and insertion deletion events (indels) at two loci. The prevalences of four bacteria transmitted by this tick that are potential or known pathogens of humans were also analyzed.

12:00 pm Wednesday, February 21, 2007

Research Horizons Seminar / Gradstudent Pizza Seminar: Combinatorics of RNA Secondary Structures

by Christine Heitsch (Georgia Institute of Technology) in Skiles 269

Complex biological problems can reveal surprising mathematical structure. In this talk, I will highlight some of the mathematical challenges that arise in the design, analysis, and prediction of RNA secondary structures. I'll also illustrate how the interaction between discrete mathematics and molecular biology motivates new combinatorial theorems as well as advancing biomedical applications.

4:30 pm Wednesday, February 21, 2007

Emory University Colloquium: From Condorcet to Fourier; An analytical approach to voting systems

by Ehud Friedgut (Department of Mathematics, Hebrew University, Jerusalem) in MSC W301 (Emory University)

***Refreshments will be served at 4:00 pm in the Department Lounge***

The full details of this seminar may be found at: http://www.mathcs.emory.edu/Seminar/pdf_files/seminar_134.pdf Emory University Department of Mathematics and Computer Science

4:30 pm Wednesday, February 21, 2007

Applied & Computational Mathematics Seminar: Discontinuous Galerkin for Diffusion

by Bram van Leer (Department of Aerospace Engineering, University of Michigan) in Skiles 255

Discontinuous Galerkin methods are the Finite Element analyst's answer to Finite Voluime methods. Originally inspired by upwind (Godunov-type) methods for the advection equation and hyperbolic systems, the DG community soon turned to the diffusion equation, with much less success. It seems that the DG approach is fundamentally unsuited for second-order operators. Not surprisingly, recent developments such as the Local Discontinuous Galerkin method of Shu and Cockburn require that the diffusion equation be rewritten as a system of first-order equations. While working with first-order systems is computationally advantageous, and a general trend in CFD, it evades the question how to directly discretize a second-order operator. In this lecture I will first show there are essential differences between discretizing the advection and diffusion equations: what works for one does not work for the other, and vice versa. This means that, when formulating a DG method for diffusion, one can not blindly copy what's done for the advection equation. I will show, however, there is absolutely no conflict between the DG approach and the diffusion equation. In order to make it work two insights are needed: (1) the realization that there are multiple representations of the numerical solution which all are equivalent in the weak sense, and that one may have to switch between these for the sake of getting useful schemes; (2) for a second-order PDE integration by parts should be done TWICE in order to obtain the most accurate DG equations - which is not standard DG practice. Next, I will present the Recovery method, developed from the above starting points. Specifically, a smooth locally recovered solution is used that in the weak sense is indistinguishable from the discontinuous discrete solution. The recovery principle creates schemes that are not included in the family of traditional DG diffusion schemes, and are potentially more accurate. A procedure is presented to extend the family so that recovery-based schemes are included. The latest development is the so-called "recovery basis", resulting from applying the recovery principle to the discontinuous basis functions from which the solution is built. To make the computation of diffusive fluxes possible one simply replaces the original basis by the corresponding unique recovery basis. This practically blurs the difference between discontinous and continuous Galerkin methods.

4:30 pm Wednesday, February 21, 2007

Analysis Seminar: To Be Announced

by Eric Sawyer (McMaster University) in Skiles 269

10:00 am Thursday, February 22, 2007

QCF Seminar: Efficient Simulation for Risk Measurement in Portfolios of CDOs

by Michael B. Gordy (Federal Reserve Board) in Skiles 269

We consider a portfolio containing CDO tranches as well as ordinary bonds. Our interest is in large loss probabilities and risk measures such as value-at-risk. When loss is measured on a mark-to-market basis, estimation via simulation requires a nested procedure: In the outer step one draws realizations of all risk factors up to the horizon, and in the inner step one re-prices each instrument in the portfolio at the horizon conditional on the drawn risk factors. Practitioners perceive the computational burden of such nested schemes to be unacceptable, and adopt a variety of somewhat ad hoc measures to avoid the inner simulation. In this paper, we question whether such short cuts are necessary. We show that a relatively small number of trials in the inner step can yield accurate estimates, and analyze how a fixed computational budget may be allocated to the inner and the outer step to minimize the mean square error of the resultant estimator.

1:30 pm Thursday, February 22, 2007

Combinatorics seminar: Enumeration and uniform sampling of planar structures

by Mihyun Kang (Humboldt Universitat zu Berlin) in Skiles 255

Planar structures, particularly planar graphs, have been extensively studied during the last few decades, including the proofs of the famous four colour theorem. /Random/ planar structures, however, have been investigated only during the last few years. In this talk we study the following aspects of (random) planar structures:

* How many of them are there (exactly or asymptotically)?

* How can we efficiently sample a random instance uniformly at random?

* What properties does a random planar structure have? E.g., what is the probability of connectedness? How many edges are there in average? What is the chromatic number?

To this end, we decompose planar structures along the connectivity, which yields a decomposition tree. Along the decomposition tree we derive recursive counting formulas, from which we compute the exact numbers and design a uniform sampling algorithm to generate a random planar structure as a reversed procedure of the decomposition. Furthermore, we interpret the decomposition in terms of generating functions. Using singularity analysis we determine the dominant singularities of generating functions and their singularity types, from which we estimate the asymptotic numbers. Using the asymptotic numbers and the probabilistic method we investigate typical asymptotic properties of random planar structures. We also discuss Gaussian matrix integral method to enumerate graphs embedded/embeddable on a 2-dimensional surface.

4:30 pm Thursday, February 22, 2007

School of Mathematics Colloquium: Solution of Polyhedra

by Idzhad Sabitov (Moscow State University) in Skiles 269

By an analogy with the term "solution of triangles" we use the term "solution of polyhedra" for problems concerning the calculation of geometric characteristics of a polyhedron on the base of its metric and combinatorial structure. We show that the volume and diagonals of an orientable polyhedron $P$ (with triangle faces) are roots of some polynomial equations with coefficients completely determined by the combinatorial structure and lengths of edges of $P$. Moreover, polynomials for the volume are monic that is their leader coefficient is equal to 1 and therefore volumes of all polyhedra with the same combinatorial structure and the same lengths of edges can have only a finite number of values. This gives us a positive solution of so-called "Bellows conjecture" affirming the invariance of volume of a flexible polyhedron during its flexion. The existence of such polynomial equations has many other consequences in the questions of isometric immersions and rigidity of polyhedra too.

11:00 am Friday, February 23, 2007

Stochastic Seminar: The Abyss Between Financial Engineering and Value Investing

by Donald Richards (Statistics, Penn State University) in Skiles 269

In financial engineering, numerous mathematical assumptions are made in order to render the mathematical models tractable. I shall argue that many of these assumptions are not based in reality and perhaps can never be made to produce models which even vaguely resemble the real world. Moreover, the nature of the assumptions have attracted incredulity from eminent practitioners of value investing, a style of money management whose principles were elucidated most prominently by Benjamin Graham. We shall discuss the vast abyss which exists between financial engineering and value investing.

2:00 pm Friday, February 23, 2007

Stochastic Seminar: Minimax estimation for the deconvolution problem on the space of positive definite matrices

by Donald Richards (Statistics, Penn State University) in Skiles 269

We consider the problem of statistical deconvolution on the space of positive definite symmetric matrices. This problem is motivated by research in human tissue imaging, notably in diffusion tensor image (DTI) analysis, a novel method of magnetic resonance imaging which enables non-invasive investigation of the molecular structure of human tissue, such as white matter in the brain. The statistical deconvolution problem also arises in mathematical models for the flow of signals over very long transmission lines with random inhomogeneities. We construct a nonparametric estimator of the density function of the underlying random variable in the deconvolution problem. Using the Helgason-Fourier transform in harmonic analysis on the space of positive definite matrices, we derive estimates of the rate of convergence of the nonparametric estimator to the population density.

2:05 pm Friday, February 23, 2007

Algebra Seminar: Iterated monodromy groups, Julia sets, Schreier graphs, and applications

by Zoran Sunic (Texas A&M) in Skiles 255

In the first, slow and informal, part of the talk we introduce the notion of iterated monodromy group (due to Nekrashevych) and we consider several concrete examples that are associated to post-critically finite rational maps (Basilica group, Hanoi Towers group, airplane and rabbit group, etc.). Through these examples we will see and explain the explicit connections relating the Julia set of a post-critically finite rational map f of degree d and the Schreier graphs of the action of the iterated monodromy group IMG(f) on a regular rooted d-ary tree. In the second (and shorter) part of the talk, as an application, we show how iterated monodromy groups associated to the double self-cover of the unit interval by the tent map lead to a class of groups of intermediate growth parametrized by the invertible polynomials over GF(2). This class of groups will also be used to discuss briefly the topic of Hausdorff dimension of finitely constrained groups (analogs of shifts of finite type in symbolic dynamics) and, in particular, Hausdorff dimension of groups of p-adic tree automorphisms.

3:05 pm Friday, February 23, 2007

Combinatorics Seminar: Minimal triangulations of cubes and simplotopes

by Francis Edward Su (Harvey Mudd College) in Skiles 255

A simplotope is a polytope that is a product of simplices; such objects arise naturally as strategy spaces for n-person games. Thus a cube, which is a product of line segments, is a special kind of simplotope. What is the minimal number of simplices needed for a triangulation of a cube or simplotope? We present a new approach to obtain lower bounds for minimal triangulations of cubes and simplotopes. In our bounds, we allow triangulations to admit interior vertices, and our work improves earlier known bounds for dimensions 4 and higher. These results are joint work with Adam Bliss and Tyler Seacrest.

3:30 pm Friday, February 23, 2007

Algebra Seminar: Multisums, singularities and arithmetic

by Stavros Garoufalidis [mail] (Georgia Tech) in Skiles 269

A special term is a product of binomials of linear forms in many variables. It is known that the generating series of a special term is convergent near the origin and satisfies a linear ODE with polynomial coefficients. There are three proofs of this property, although they are all computationally costly, for multisums. In the talk we will give a simple ansatz for the singularities of the generating series. We will also give a proof of our ansatz in case the special term is positive, and an illustration our ansatz with the Apery series, responsible for the irrationality of zeta(3). Our ansatz is reminiscent of the Bethe ansatz, and has a motivic interpretation using additive K-theory, and the entropy function. If you don't know what the Bethe ansatz, entropy, or motivic cohomology are, you won't loose much! The talk is inspired by the Stirling formula for n! and by a two hour conversation with M. Kontsevich in Miami.

1:00 pm Monday, February 26, 2007

Theory of Computation Colloquium: Learning, Regret minimization, and Equilibria

by Avrim Blum (Carnegie Mellon) in Klaus Advanced Computing Bldg 2447

3:00 pm Monday, February 26, 2007

Theory of Computation Colloquium: Some Simple Randomized Algorithms for Auction and Pricing Problems

by Avrim Blum (Carnegie Mellon) in Klaus Advanced Computing Bldg 1116 West

***Refreshments at 2:35pm in Klaus 2222***

Consider the problem of a retailer with various goods for sale, attempting to set prices to maximize revenue. If customers have separate valuations over the different goods, and these are known to the retailer, then the goods can be priced separately and the problem is not so difficult. However, when customers have valuations over *sets* of items, this becomes a combinatorial auction problem, and the problem becomes computationally hard even when valuations are fully known in advance. In this talk I will discuss some simple randomized algorithms for a number of interesting cases of this problem, along with a few annoyingly simple-looking open questions. I will also talk about the setting where valuations are not known in advance and so the goal is to produce an incentive-compatible mechanism, as well as the dynamic case where customers arrive over time and one would like to adapt prices in a way that performs comparably to the best fixed setting of prices in hindsight. Various portions of this talk are based on joint work with Nina Balcan, Jason Hartline, and Yishay Mansour.

3:00 pm Monday, February 26, 2007

Job Candidate Seminar: Hardy-Sobolev and Lieb-Thirring inequalities

by Rupert Frank (KTH Stockholm) in Skiles 255

We consider the relation between inequalities on eigenvalue moments of Schrodinger-like operators and Sobolev-like inequalities. We present some new Hardy-Sobolev inequalities for the Laplacian in a convex domain and for fractional powers of the Laplacian. As a consequence we deduce that the Lieb-Thirring inequalities on moments of negative eigenvalues of Schrodinger-like operators remain true, with possibly different constants, when the critical Hardy-weight is subtracted from the Laplace operator. The talk is based on joint works with T. Ekholm, with E.H. Lieb and R. Seiringer, and with R. Benguria and M. Loss.

3:30 pm Monday, February 26, 2007

Geometry-Topology Seminar: Infinitesimal and continuous deformations of surfaces

by I. Kh. Sabitov (Moscow State University, Russia) in Skiles 269

Infinitesimal deformation of a surface S is a smooth one-parameter deformation under which the lengths of curves on S are stationary at the initial value of parameter. A continuous isometric deformation of S is a continuous family S_t of surfaces with S = S_0 such that all curves on S keep their length unchanged. If we suppose additionally that the family S_t is smooth relatively to t then the velocity of deformation S_t in any moment t = t_0 gives the infinitesimal deformation of the surface S_{t_0}. Starting from this and other relations between infinitesimal and derivable isometric deformations of a surface one can obtain the rigidity of many surfaces using their infinitesimal rigidity which is much easier to prove. We intend to discuss other questions of theory of infinitesimal deformations too including the problem of a correct definition of a high order infinitesimal deformation.

3:00 pm Tuesday, February 27, 2007

PDE Seminar: From Navier Stokes to Thin Film Equation

by S. Ulusoy (School of Mathematics, Georgia Tech) in Skiles 255

I am going to derive the general form of the thin film equation as a reduction of leading order Navier Stokes Equation. This is known as the "lubrication approximation." The applications arising in physical processes will also be discussed briefly to motivate the study. Mathematically interesting problems with methods in solving these problems, and interesting open problems will be briefly discussed. In the second part of the talk I will introduce our accomplishments in this field. Finally, if time permits, we will go over some open problems. Part of the results is a joint work with my advisor, Prof. Eric A. Carlen

11:00 am Wednesday, February 28, 2007

Mathematical Biology & Ecology Seminar: CANCELLED

by Pejman Rohani [mail] (U. GA) in Skiles 255

You say pertussis, I say petussis: the epidemiology of whooping cough in the USA and the UK In this talk, I will use whooping cough as a case study in order to demonstrate (i) how models and ecological perspectives can inform epidemiology and (ii) how case notification data can provide invaluable ecological insights. Whooping cough is caused by a bacterium and remains a significant source of childhood mortality responsible for an estimated 300-500,000 annual infant deaths. While pediatric whooping cough immunizations have substantially reduced incidence in the developed world, there is widespread belief that vaccines prevent disease and not transmission, with important implications for potential eradication. The issue is further complicated by documented loss of infection- and vaccine-derived immunity. Using mathematical models and time-series data, I will first examine the epidemiological consequences of vaccination and will attempt to obtain bounds for the period of immunity. I will then briefly explore the spatial pattern of transmission cycles in the US.

4:30 pm Wednesday, February 28, 2007

Applied & Computational Mathematics Seminar: Efficient Methods for Elliptic Moving Interface Problems

by John Strain (UC Berkeley) in Skiles 255

Many moving interface problems evolve complex material interfaces through topological changes, under velocities determined by elliptic systems of partial differential equations. A combination of semi-Lagrangian contouring, fast distance finding and Ewald summation yields robust efficient methods for such problems. High-resolution computations with geometric, Stokes and viscoelastic flows exhibit merging, anisotropy, faceting, curvature, dynamic topology and nonlocal interactions. The interface motion is converted to a contouring problem with an explicit second-order semi-Lagrangian advection formula. Grid-free adaptive refinement resolves complex interface geometry. A fast new Voronoi-based algorithm computes distances to high-order interface patches. Elliptic systems are solved by a fast new locally-corrected boundary integral formulation derived by Ewald summation and accelerated by new geometric nonequidistant fast Fourier transforms.

4:30 pm Wednesday, February 28, 2007

Analysis Seminar: Radon transforms, orthogonal polynomials and CT

by Yuan Xu (University of Oregon) in Skiles 269

The central problem for computered tomography (CT) is to reconstruct a function (an image) from a finite set of its Radon projections. We propose a reconstruction algorithm, called OPED, based on Orthogonal Polynomial Expansion on the Disk. The algorithm works naturally with the fan data and can be implemented efficiently. We proved that the algorithm converges uniformly under a mild condition on the function. Numerical experiments have shown that the method is fast, stable, and has a small global error.

9:00 am Thursday, March 1, 2007

Computational Homology & Fluid Dynamics Workshop: To Be Announced

in Wardlaw Center

This workshop will bring together a select group of physicists, applied mathematicians and engineers with an interest in characterizing quantitatively structures and dynamics of spatio-temporally chaotic and turbulent fluid flows. In addition to providing a forum for discussing current and future applications of topological techniques, the workshop includes tutorial lectures on computational homology and the publicly available software package CHomP. We want to give the participants ample opportunity to interact, share ideas, and explore the potential for future collaborations. For this reason, we are limiting the number and length of the lectures in order to provide adequate time for informal discussions. Details are posted at http://www.physics.gatech.edu/workshops/chomp/

1:30 pm Thursday, March 1, 2007

Graph Theory Seminar: A characterization of affine Steiner triple systems and Hall triple systems and their application to edge-colorings of cubic graphs

by Daniel Kral (Charles University, Prague, Czech Republic) in Skiles 255

A Steiner triple system is a combinatorial design formed by triples of points such that any two distinct point lie in a single common triple. There are two important classes of Stiener triple systems that arise from projective and affine geometries, called projective and affine Steiner triple systems. It is known that a Steiner triple system is projective if and only if it does not contain the four- triple configuration C_14. We find three configurations such that a Steiner triple system is affine if and only if it does not contain any of these configurations. Similarly, we characterize Hall triple systems, a superclass of affine Steiner triple systems, using two forbidden configurations.

We then apply our results to edge-colorings of cubic graphs. A cubic graph G is S-edge-colorable for a Steiner triple system S if its edges can be colored with points of S in such a way that the points assigned to three edges sharing a vertex form a triple in S. We show that a cubic graph is S-edge-colorable for every non-trivial affine Steiner triple system S unless it contains a well-defined obstacle called a bipartite end. In addition, we show that all cubic graphs are S-edge-colorable for every non-projective non-affine point-transitive Steiner triple system S.

3:00 pm Thursday, March 1, 2007

Stochastic Seminar: The Dantzig Selector and Sparsity Oracle Inequalities

by Vladimir Koltchinskii (School of Mathematics, Georgia Tech) in Skiles 269

The Dantzig Selector is a new method of estimation of regression function based on its representation as (or approximation by) a linear combination of given functions from a large dictionary. The method was recently introduced by Emmanuel Candes and Terry Tao and it consists of minimizing the \ell_1-norm of the vector of coefficients subject to constraints that the gradient of the L_2 empirical loss belongs to the \ell_{\infty}-ball around 0 of certain radius, which can be solved using linear programming. We will discuss several inequalities that go further than initial results of Candes and Tao in showing (in the case of models with random design) that this method performs in a nearly optimal way when the regression function has a reasonably good sparse approximation in the dictionary.

4:15 pm Thursday, March 1, 2007

Math Dept. Tea:

in Skiles 236

All math faculty, staff, and students are invited to attend. We will be honoring those of us recently promoted. Food and beverages will be served.

3:00 pm Friday, March 2, 2007

Combinatorics Seminar: Combinatorial offsprings of Schnyder's work

by Stefan Felsner (TU-Berlin) in Skiles 255

W. Schnyder used certain decompositions of the edges of a plane triangulation into three trees to prove two by now classical theorems. These Schnyder woods continue to find new applications in graph drawing, enumeration and in dimension theory of orders. Orthogonal surfaces allow a very natural approach to Schnyder woods and help explaining their usefulness. We give an overview on the basic theory and explain some more recent results and problems in the area. This includes a slick proof of the Brightwell-Trotter Theorem about the order dimension 3-polytopes and some surprising lattice structures on planar objects.

12:00 pm Monday, March 5, 2007

Combinatorics Seminar: Equitable Coloring

by Hal Kierstead (Arizona State University) in Skiles 269

An equitable r-coloring of a graph is a proper coloring with r colors such that the sizes of any two of the r color classes differ by at most one. Hajnal and Semeredi answered a question of Erdos by proving that any graph with maximum degree r has an equitable (r + 1)-coloring. We have found a simpler proof of this theorem that leads to further results and the formulation of new conjectures. (Joint work with Alexandr Kostochka, UICU)

3:15 pm Monday, March 5, 2007

Geometry-Topology Seminar (joint seminar held in Athens): Cubic surfaces and SL(2)-character varieties of surfaces

by Bill Goldman (University of Maryland) in Boyd 304 (in UGA Math Department)

This talk will survey the relationship between elementary invariant theory of SL(2) and moduli spaces of geometric structures over surfaces. In particular I will describe a relationship (first noted by Fricke and Klein) between affine cubic surfaces, the one-holed torus and the four-holed sphere.

4:20 pm Monday, March 5, 2007

Geometry-Topology Seminar (joint seminar held in Athens): Construction of New Symplectic Cohomology S^2 x S^2 and Small Exotic Manifolds

by Anar Akhmedov (Georgia Tech) in Boyd 304 (in UGA Math Department)

In this talk we present new examples of symplectic manifolds with same integral cohomology as S^2 x S^2. We also discuss the generalization of these examples as well as its application in the construction of simply connected exotic 4-manifolds with small Euler characteristic.

4:30 pm Monday, March 5, 2007

CDSNS Colloquium: Looking for order within chaos: Locating and understanding coherent structures within Kuramoto-Sivashinsky equation

by Ruslan L. Davidchack [mail] (University of Leicester) in Skiles 255

In this talk I will discuss our efforts to understand the properties of chaotic solutions of the Kuramoto-Sivashinsky equation by locating and analyzing coherent structures (equilibria, traveling waves, periodic and relative periodic orbits) embedded within the chaotic dynamics. Our approach for locating the coherent structures is based on transforming the original equation in such a way that the unstable structures are stabilized by the transformation. The approach has proved very successful in low dimensional chaotic systems and has now been extended to high dimensional flows such as discretized PDEs. Using this approach we have found many coherent structures within a moderately chaotic regime of the Kuramoto-Sivashinsky dynamics and are now attempting to understand the chaotic dynamics in terms of these structures.

11:00 am Tuesday, March 6, 2007

Job Candidate Seminar: Sobolev mappings with geometric type constraints

by Mohammad Reza Pakzad (Max Planck Institute for Mathematics in the Sciences) in Skiles 269

Classes of weakly differentiable mappings with geometric type constraints, such as "Sobolev isometric immersions", arise for example in continuum mechanics of solids as the set of admissible deformations. The main questions regarding these mappings concern singularities, density of smooth mappings (in the given class), regularity and rigidity. We will discuss these questions alongside some motivatation from the theory of elasticity and geometric analysis.

4:30 pm Tuesday, March 6, 2007

DixieLand Analysis Seminar: Finite Energy Solutions to the Isentropic Euler Equations

by Michael Westdickenberg (School of Mathematics, Georgia Tech) in Mathematics and Sciences Center (MSC), W301, Emory Univ

***REFRESHMENTS will be served at 4:00PM in W427, Department Lounge on the 4th floor.***

We consider the isentropic Euler equations of compressible fluid dynamics and establish the existence of globally defined entropy solutions with arbitrary large amplitude, whose mass and energy are finite. This is the natural functional space suggested by the physics. We propose a versatile framework based on three main steps. First, we establish a higher-integrability estimate for the mass density variable. Second, we establish an equi-integrability property for the total energy of the fluid. Third, we rely on the theory of compensated compactness and Young measures, suitably extended to sequences with finite mass and energy and with vacuum states, and we characterize the class of entropy admissible Young measures. This analysis is based on a new study of singular products involving measures and principal values.

4:30 pm Tuesday, March 6, 2007

ACO Colloquium: Correlation Inequalities

by Jeffrey Kahn (Rutgers University) in Skiles 255

Correlation inequalities --- e.g. the Harris, FKG and BK inequalities --- are basic tools in probability and have also had some beautiful combinatorial consequences. Here we will mention a few relatively recent inequalities, but mainly want to give some indication of how little we actually know about such things. *** Refreshments at 4pm in Skiles 236 before the lecture ***

4:30 pm Tuesday, March 6, 2007

Special Analysis Seminar: Extremal problems for vector potentials and their applications to the asymptotics of Hermite-Pade approximants

by Sasha Aptekarev (Mathematics Department, Vanderbilt & Moscow State Universty) in Skiles 269

There is a vector of measures supported on the fixed system of curves and arcs. Extremal problem of minimization of logarithmic potential energy for the system of measures with a given interaction matrix is considered. Then doing variation of the system of curves and arcs we find locations for the curves where the local maxima of the energy functional is achived. As a result an extremal system of curves and arcs appears. These extremal contours and the extremal measures on them play a crucial role for investigation of the asymptotics of Hermite-Pade approximants. The application of the Matrix Riemann-Hilbert problem method for the asymptotic analysis uses the extremal system of contours for finding an approprate deformation of the boundaries for the solution of BVP for matrix analytic functions.

11:00 am Wednesday, March 7, 2007

Mathematical Biology & Ecology Seminar: What do we optimize when select activity of individual muscles to perform a skilled motor task?

by Boris I. Prilutsky (School of Applied Physiology, Georgia Tech) in Skiles 255

Patterns of muscle activity in many skilled motor tasks are rather stereotyped and similar among different subjects despite muscle redundancy -- an excessive number of muscles compared to the number of kinematic degrees of freedom at the joints. It is not clear why these specific activation patterns are selected by the nervous system over other possible patterns to perform the task. In this talk I address the functional significance of the stereotyped muscle activation during walking, cycling and other skilled motor tasks by comparing recorded muscle activation patterns with the activation patterns computed by optimizing different physiological cost functions. Muscle electromyographic activity (EMG) or muscle forces, movement kinematics, and external forces were recorded during these motor tasks. Different cost functions (e.g., the sum of muscle forces, the sum of muscle stresses, muscle fatigue, metabolic cost, etc.) and experimentally determined joint moments were used to calculate muscle forces or muscle activation in studied motor tasks and to compare them with the recorded muscle forces and EMG activity. The results suggest that the functional significance of the observed activation patterns in the studied tasks is minimization of muscle fatigue.

12:00 pm Wednesday, March 7, 2007

Research Horizons Seminar / Gradstudents Pizza Seminar: Dynamics of Surface group representations

by William Goldman (University of Maryland) in Skiles 269

The space of representations of the fundamental group of a surface in a Lie group is a rich geometric object, with an algebraic structure enjoying much symmetry. The simplest examples include symplectic vector spaces, Jacobi varieties, and moduli spaces of holomorphic vector bundles. Fricke-Teichmueller spaces also arise as representation spaces. They are a special case of deformation spaces of locally homogeneous geometric structures in the sense of Ehresmann and Thurston. The underlying algebraic structure of deformation spaces closely relates to the geometric structures they parametrize. Understanding the geometric structures is often a key for understanding the topology and dynamics of these spaces. The mapping class group of the surface acts on this space preserving a natural Poisson geometry. Natural Hamiltonian flows on the deformation space generalize the classical Fenchel-Nielsen twist flows on Teichmueller space. For compact Lie groups, the mapping class group action is chaotic. The proof of ergodicity can be regarded as an analog of the Fenchel-Nielsencoordinates for Teichmuller space. For representations corresponding to uniformizations by geometric structures, the action is proper. In general the dynamics falls between these two extremes. In the case of a one-holed torus, the dynamics reduces to an action of the modular group on cubic surfaces related to the Markoff equation, where both chaotic and proper dynamics coexist.

2:30 pm Wednesday, March 7, 2007

Applied Computational and Mathematics Seminar: Translational Biomedical Informatics and Systems Biology

by May D Wang (BME, Georgia Tech) in Skiles 255

Dr. Wang has established a translational biomedical informatics and systems biology program, which is integrated with bionanotechnology and lipidomics to solve problems in personalized and predictive oncology. The program consists of (1) identification of diagnostic and prognostic biomarkers from high throughput genomics, proteomics, and lipidomics data, (2) quantification of molecular imaging and profiling data for diagnosis and drug efficacy study, (3) visualization of complex molecular pathways for therapeutics study, as well as (4) interfacing with the National Cancer Institute's Cancer Biomedical Informatics Grid (caBIG). Dr. Wang initiated the integration of biomedical informatics with bionanotechnology and the establishment of several large interdisciplinary programs at Georgia Tech and Emory University BRP (R01CA108468), P20 (P20GM072069), and Emory-Georgia Tech Center of Cancer Nanotechnology Excellence (U54CA119338). This talk will discuss several challenging issues existing in the personalized and predictive medicine field, and opportunities for applying informatics and systems biology approaches.

4:00 pm Wednesday, March 7, 2007

Analysis: Universality Limits for General Measures

by Doron Lubinsky (Georgia Tech) in Skiles 269

In the theory of random matrices, universality limits play a key role. We show that for measures with compact support, continuity at just one point, is sufficient for universality limits at that point. Previously, strong global assumptions, typically analyticity, have been required.

10:00 am Thursday, March 8, 2007

QCF Seminar : CANCELED Nonparametric Regression Models for Nonstationary Variables with Applications in Economics and Finance

by Zongwu Cai (Department of Mathematics & Statistics and Department of Economics, University of North Carolina at Charlotte) in Skiles 269

In this talk, I will talk about how to use a nonparametric regression model to do forecasting for nonstationary economic and financial data, for example, to forecast the inflation rate using the velocity variable in economics and to test predictability efficiency in stock returns using the log dividend-price ratio and/or the log earnings-price ratio and/or the three-month T-bill and/or the long-short yield spread. A local linear approach is developed to estimate the unknown functionals. The consistency and asymptotic normality of the proposed estimators are obtained. Our asymptotic results show that the asymptotic bias is same for all estimators of coefficient functions but the convergence rates are totally different for stationary and nonstationary covariates. The convergence rate for the estimators of the coefficient functions for nonstationary covariates is faster than that for stationary covariates with a factor of n^&ob;-1/2&cb;. This finding seems new and it leads to a two-stage approach to improve the estimation efficiency. When the coefficient function is a function of nonstationary variable, our new findings are that the asymptotic bias term is the same as that for stationary case but the convergence rate is different and further, the asymptotic distribution is not a normal but a mixed normal associated with the local time of a standard Brownian motion. Moreover, the asymptotic behaviors at boundaries are investigated. The proposed methodology is illustrated with an economic time series, which exhibits nonlinear and nonstationary behavior. This is a joint work with Qi Li, Department of Economics, Texas A&M University and Peter M. Robinson, Department of Economics, London School of Economics.

1:30 pm Thursday, March 8, 2007

Graph Theory Seminar: Hamilton cycles in tough chordal graphs

by Daniel Kral (Charles University, Prague, Czech Republic) in Skiles 255

The notion of toughness is a well-established notion closely related to hamiltonian graphs. A graph G is B-tough if for every set A of its vertices, G\A is connected or the number of the components of G\A does not exceed |A|/B. Clearly, if G contains a Hamilton cycle, then G is 1-tough. A famous conjecture of Chvatal asserts that there exists a constant B such that every B-tough graph is hamiltonian. It is known that the conjecture is not true with B<9/4.

The conjecture was verified for several special classes of graphs, among those interval graphs, split graphs and chordal graphs. Interval graphs coincide with the intersection graphs of subpaths of a path, split graphs with the intersection graphs of subtrees of a star and chordal graphs with the intersection graphs of subtrees of a tree. It is known that 1-tough interval graphs and 3/2-tough split graphs are hamiltonian and the bounds are tight. On the other hand, it is known that 18-tough chordal graphs are hamiltonian but there exist (7/4-epsilon)-tough chordal graphs that are not hamiltonian. Motivated by these results, we study the existence of Hamilton cycles in spider graphs, intersection graphs of subtrees of a subdivision of a tree. We show using the matroid intersection theorem that 3/2-tough spider graphs are hamiltonian. Since the class of spider graphs includes both interval and split graphs, our result is tight.

3:00 pm Thursday, March 8, 2007

Stochastic Seminar: Fast rates for plug-in estimators of density level sets

by Philippe Rigollet (School of Mathematics, Georgia Tech) in Skiles 269

In the context of density level set estimation, we recall the notion of \gamma-exponent of a density at a certain level. This notion is similar to Tsybakov's margin assumption and allows us to prove fast rates of convergence for general plug-in methods, up to order $n^{-1}$ when the density is supposed to be smooth in a neighborhood of the level under consideration. We consider two measures of performance: one is a measure of the symmetric difference between the sets and another is related to the excess mass, a quantity that naturally arises while considering density level sets. Lower bounds proving optimality of the rates in a minimax sense are also provided.

4:30 pm Thursday, March 8, 2007

School of Mathematics Colloquium: Renormalization and rigidity in circle dynamics

by Konstantin Khanin (University of Toronto) in Skiles 269

We shall present several recent results related to a rigidity problem. The first result (joint with A. Teplinsky) proves the phenomenon of robust rigidity for critical circle map. We show that two critical circle maps with the same irrational rotation number are C1-smoothly conjugated to each other. This result demonstrates that in the presence of critical points rigidity is much stronger. Indeed, similar statement in the diffeomorphisms case requires Diophantine-type conditions on the rotation number. The second result (joint with B. Fayad) is related to Moser's problem. We prove that commuting circle diffeomorphisms can be linearized provided that their rotation numbers satisfy the joint Diophantine condition.

3:00 pm Friday, March 9, 2007

Combinatorics Seminar: The spectrum of symmetric boolean functions

by Mihalis Kolountzakis (School of Mathematics, Georgia Tech and Crete, Greece) in Skiles 255

We study a problem coming from theoretical computer science. When one tries to "learn" a boolean function on k variables (that is a function f:{0,1}^k --> {0,1}) it turns out that, for a certain learning algorithm, the crucial parameter of the function which determines the running time is the order of the first Fourier non-zero of the function. We show that if a symmetric boolean function (this means that the function does not depend on the order of its inputs) is not constant or a parity function then that crucial quantity is bounded above by Ck/log k, improving on previous upper bounds which were of the form Ck. The problem translates into a number-theoretic problem about binomial coefficients. This is joint work with E. Markakis and A. Mehta, done in 2005 here at GA Tech.

1:00 pm Monday, March 12, 2007

Combinatorics Seminar: Packing cubes into a torus

by Tom Bohman (Carnegie Mellon University) in Skiles 255

We consider the following packing problem. How many d-dimensional cubes of side length 2 can we pack into a d-dimensional torus of odd side length? In this talk we present some of the best known constructions, detail the connection between this packing problem and the problem of determining the Shannon capacities of graphs, and discuss some recent techniques for establishing upper bounds.

3:30 pm Monday, March 12, 2007

Geometry-Topology Seminar: Bounding the Betti numbers of semi-algebraic sets defined by quadratic polynomials

by Michael Kettner (Georgia Tech) in Skiles 269

A semi-algebraic set $S$ in $\mathbb{R}^k$ is one defined by finitely many polynomial inequalities; it is a fundamental object in Real Algebraic Geometry. The Betti numbers $b_i(S )$ are an important measurement of the topological complexity of a semi-algebraic set $S$ . For example, the 0-th Betti number $b_0(S )$ is equal to the number of connected components of the semi-algebraic set, whereas one can think of the other Betti numbers $b_i(S )$ as the number of i-dimensional holes of $S$. We consider the problem of bounding the Betti numbers of a semi-algebraic set $S\subset\mathbb{R}^k$. In this talk we present a new bound on the Betti numbers $b_i(S )$ for a semi-algebraic set defined by quadratic polynomials. We recall all necessary results (and their geometric meaning) in order to prove our new bound.

4:30 pm Monday, March 12, 2007

CDSNS Colloquium: Geometry of state space of a turbulent plane Couette flow

by Predrag Cvitanovic >[mail] (School of Physics, GT) in Skiles 255

We propose to use a hierarchy of exact unstable invariant solutions of the Navier-Stokes equations -- corresponding to the recurrent coherent structures observed in experiments -- to construct a description of the spatio-temporally chaotic dynamics of turbulent fluid flows as a walk through the space of such structures. This description should allow us to obtain quantitative predictions of transport properties of fluid flows such as bulk flow rate and mean wall drag. (Joint work with J. F. Gibson, J. Halcrow, and F. Waleffe)

3:00 pm Tuesday, March 13, 2007

PDE Seminar: Infinite Time Aggregation for the Critical Patlak-Keller-Segel model in R^2

by N. Masmoudi (Courant Institute, NYU) in Skiles 255

The Patlak-Keller-Segel (PKS) model describes the collective motion of cells which are attracted by a self-emitted chemical substance. The long time behavior depends on the total mass which is assumed here to be conserved. It was conjectured by S. Childress and K. Percus that in two space dimensions there is a threshold number above which there is a chemotactic collapse. One can prove that if the initial mass is below a critical mass 8 pi then the solution is global and spreads when t goes to infinity. If the initial mass is above the critical mass 8 pi then there is blow up in finite time. For the critical mass 8 pi, there is infinite time aggregation. One of the main tools in proving these results is the use of the free-energy of the system combined with a Logarithmic Hardy-Littlewood-Sobolev inequality with a sharp constant. We will also discuss the derivation of the model from kinetic models and show some numerics.

11:00 am Wednesday, March 14, 2007

Mathematical Biology & Ecology Seminar: IBSI Conference

by Integrative BioSystems Institute in Klaus Atrium & Conference Room 1116E and 1116W

An all day poster session to be held between 8:00AM-5:30PM.

12:00 pm Wednesday, March 14, 2007

Research Horizons Seminar / Gradstudent Pizza Seminar: Subadditivity of entropy and related inequalities of Brascamp-Lieb type

by Eric Carlen (Georgia Institute of Technology) in Skiles 269

This talk will explain some recent joint work with Lieb, Loss and Cordero on the proof of Brascamp-Lieb inequalities via entropy inequalities and heat flow.

4:30 pm Wednesday, March 14, 2007

Applied Computational and Mathematics Seminar: Interface Tracking Using Face Offsetting and Anisotropic Mesh Adaptation

by Xiangmin Jiao (College of Computing, Georgia Tech) in Skiles 255

Dynamic moving surfaces are central to many scientific, engineering, and graphics applications. A surface triangulation is often used to represent the dynamic surface or moving interface, posing significant challenges in interface tracking and mesh adaptation. We present a new method for moving interfaces, called the face offsetting, and an anisotropic mesh adaptation technique to meet these challenges, based on a unified asymptotic and singularity analysis. We report applications of our methods in multi-physics simulations of solid- rocket combustion.

4:30 pm Wednesday, March 14, 2007

Analysis: Norms of products of polynomials and the farthest distance function

by Igor Pritsker (Oklahoma State University) in Skiles 269

We study inequalities connecting the product of uniform norms of polynomials with the norm of their product. This circle of problems include the Gelfond-Mahler inequality for the unit disk and the Kneser-Borwein inequality for the segment $[-1,1]$. We find the asymptotically sharp constant in such inequalities over an arbitrary compact set in the complex plane. The constant is expressed in terms of potential theoretic notions and the distance function to the farthest point in the set. This leads to interesting interplay between extremal problems in potential theory and convex geometry. Furthermore, we show that the best constant is smallest (namely: 2) for a disk. We also conjecture that it takes its largest value for a segment, among all compact connected sets in the plane, and prove this fact for some classes of sets.

10:00 am Thursday, March 15, 2007

QCF Seminar: Option Pricing by Fast Convolution Transform

by Shijie Deng (ISyE, Georgia Tech) in Skiles 269

Financial option pricing problems often boils down to the computation of the probability density function (PDF) convoluting with given distributions using N discretized points which requires O(N^2) work by a direct method. We propose a scheme which approximates the probability density function utilizing combinations of the Fourier series expansion and sums of the double exponential distributions. A recursive relation is then derived for each term which reduces the total amount of computational work to the asymptotically optimal O(N). Our numerical technique is a generalization of the alternative fast Gauss transform first introduced by Greengard and Sun (1998) for the efficient evaluation of Gaussian sums. Numerical results show that our method is accurate, efficient, stable, and can be easily applied to option pricing problems involving probability density functions of general form. This is a joint work with Yuanqing Li (Chinese Academy of Sciences) and Jingfang Huang (Univ. of North Carolina, Chapel Hill).

1:30 pm Thursday, March 15, 2007

Stochastic Seminar: Reduced Bias Tail Index and Quantile Estimation

by M. Ivette Gomes (Faculty of Science DEIO and CEAUL, Portugal, University of Lisbon) in Skiles 269

We are here essentially interested in reduced-bias estimators of high quantiles. We deal only with heavy tails, trying to improve the performance of the classical high quantile estimators. High quantiles depend on the tail index gamma. Recently, new classes of reduced-bias estimators for gamma have been introduced, where the second order parameters in the bias are estimated at a level k_1 of a larger order than that of the level k at which we compute the tail index estimators. Doing this, we keep the asymptotic variance of the new estimators equal to the one of the Hill estimator, the ML tail index estimator under a strict Pareto model. The use of one of those classes of gamma-estimators in quantile estimation enables us to introduce new classes of high quantile estimators. The asymptotic distributional properties of the proposed classes of estimators are derived and the estimators are compared with alternative ones, not only asymptotically, but also for finite samples through Monte Carlo techniques. An illustration of the behavior of these estimators for sets of real data in the field of finance is also provided.

3:05 pm Thursday, March 15, 2007

Stochastic Seminar: Two new results on the logarithmic Sobolev inequality

by Maria Reznikoff (School of Mathematics, Georgia Tech) in Skiles 269

The logarithmic Sobolev inequality (LSI) is a powerful tool for studying convergence to equilibrium in spin systems. The Bakry-Emery criterion implies the LSI in the case of a convex Hamiltonian. What can be said in the nonconvex case? We present two new sufficient conditions for LSI. The first is in some sense a generalization of the Bakry-Emery criterion. The second is a two-scale condition: an LSI on the microscopic scale (conditional measures) and an LSI on the macroscopic scale (marginal measure) are combined to prove a global LSI. We discuss a couple of applications. The talk includes joint work with Natalie Grunewald, Felix Otto, and Cédric Villani.

4:30 pm Thursday, March 15, 2007

Mathematics Colloquium: Counting Connected Graphs via Erdös Magic

by Joel Spencer (Courant Institute (New York University)) in Skiles 269

Let C(k,l) be the number of labelled connected graphs with k vertices, k-1+l edges. We employ random graphs and breadth first search techniques to find the asymptotics of C(k,l) whenever k and l tend to infinity. (This was first done by Rod Canfield et al.) The probabilistic analysis has, at its heart, a biased bridge. We place k-1 balls into k bins, ball j going into bin i with probability p(1-p)^{i-1}/(1-p^k), a truncated exponential with "tilt" p. With Z_t balls in bin t the "queue walk" has Y_0=1, Y_t = Y_{t-1}+Z_t-1 and thus Y_k = 0. When p\gg k^{-3/2} we analyze the probability that the walk has Y_t > 0 for all 0 \leq t < k. In combinatorial terms, the placement is a parking function, allowing movement to the right the k-1 "cars" can fill the first k-1 "parking places." The techniques allow analysis of the "giant component" and "dominant component" of Erdös and Rényi. They further yield a randomized algorithm that (in many cases) efficiently generates a uniformly distributed connected graph with k vertices and k-1+l edges.

9:30 am Monday, March 19, 2007

GA Tech-DIMACS Workshop: Phase Transitions in Random Structures and Algorithms

by Various Speakers in Klaus Advanced Computing Building, Klaus Main Auditorium, Room 1443

The GaTech-DIMACS Workshop on Phase Transitions in Random Structures and Algorithms will be happening at Georgia Tech during spring break week (March 19-23). There are a lot of great speakers discussing topics at the intersection of randomized algorithms, probabilistic combinatorics, and statistical physics. For complete details, see http://www-static.cc.gatech.edu/~vigoda/workshop/

3:00 pm Monday, March 19, 2007

Analysis Seminar: A look at some of Euler's work and how it is still relevant

by Dick Askey (Mathematics Department, University of Wisconsin) in Skiles 269

What do the sum of (-1)^n * n2/(1+n3), cyclic quadrilaterals and Fibonacci numbers have to do with each other. All were either worked on by Euler or use results developed by Euler, and have come up in my work on mathematics education. Try to find the sum of this series, find the diagonals of a cyclic quadrilateral in terms of the sides and find out not only what F(n+3)F(n-3)-F(n)2 equals, how it can be generalized, and how the generalization can be explained using ideas of Euler. F(n) is the n'th Fibonacci number.

3:00 pm Monday, March 26, 2007

Colloquium: The geometrical basis of numerical stability

by Doug Arnold (Institute for Mathematics and its Applications, University of Minnesota) in Skiles 255

The accuracy of a numerical solution to a partial differential equation depends on the consistency and stability of the discretization method. While consistency is usually elementary to establish, stability of numerical methods can be subtle, and for some key PDE problems the development of stable methods is extremely challenging. After illustrating the situation through simple but surprising examples, we will describe a powerful new approach--the finite element exterior calculus---to the design and understanding of discretizations for a variety of elliptic PDE problems. This approach achieves stability by developing discretizations which are compatible with the geometrical and topological structures, such as de Rham cohomology and Hodge decompositions, which underlie well-posedness of the PDE problem being solved.

4:30 pm Monday, March 26, 2007

CDSNS Colloquium: Stability of Vortex Solutions of the Two and Three Dimensional Navier-Stokes Equations

by Eugene Wayne [mail] (Boston University) in Skiles 255

Vortex solutions play an important role in organizing the behavior of viscous fluid flows. Indeed they have been referred to as the "sinews" of turbulence. In this talk I will explain how using ideas from dynamical systems theory and kinetic theory one can understand the stability of certain vortices in both the two and three dimensional Navier-Stokes equations.

3:00 pm Tuesday, March 27, 2007

PDE seminar: Convergence of phase-field approximations to the Gibbs-Thomson law

by Yoshihiro Tonegawa (Hokkaido University) in Skiles 255

The Modica-Mortola (or Allen-Cahn) functional of phase transitions is well-known to Gamma converge to the hypersurface area functional as the small parameter tends to zero. Even without energy minimality, one can show that the control of the mean curvature-like quantity is sufficient for the good convergence, and more subtle questions can be asked about its property. I will give a brief summary of the results related to the Modica-Mortola energy in the framework of geometric measure theory and also a recent result (joint work with Matthias Roeger) on the rigorous proof of the Gibbs-Thomson law for the limit interface.

11:00 am Wednesday, March 28, 2007

Mathematical Biology & Ecology Seminar: Efects of direct vs indirect and cascading interactions in structuring biotic systems

by Mark Hay [mail] (GA Tech)

12:00 pm Wednesday, March 28, 2007

Research Horizons Seminar / Gradstudents Pizza Seminar: Basic Fourier Analysis in Additive Combinatorics

by William McClain (Georgia Institute of Technology) in Skiles 269

I'll show some problem solving techniques in Additive Combinatorics that are derived from Fourier Analysis. I'll offer different perspectives on interpreting the problem solving techniques and present them in the most elementary way possible. First we'll see a basic proof of Roth's theorem addressing 3-term Arithmetic Progressions in subsets of the integers. Then we'll see the complications and adaptations of the problem solving techniques used to tackle problems concerning 4-term arithmetic progressions in subsets of the integers.

3:00 pm Wednesday, March 28, 2007

Annual Joseph Ford Commemorative Lecture: The Content of Shape

by Edward A. Spiegel (Columbia University) in Howey/Physics Lecture Room 5

A reception will be held at 2:30 p.m. in room N201.

First we shall look at some pictures of the kinds of patterns that we see around us in Nature and that experimentalists are busy producing in a controlled way. Such patterns are often built up from recurrent basic structures that arise in biology, fluid dynamics, chemistry and other interesting fields. These structures are thought to be subject to general physical laws though details may vary from case to case. Pattern theory attempts to rationalize the apparent universality of the patterns in terms of elementary notions such as instability and nonlinearity. The aim of the lecture is to discuss how relatively simple theoretical descriptions of pattern formation can emerge from the study of basically complex situations.

4:30 pm Wednesday, March 28, 2007

Applied Computational and Mathematics Seminar: Global dynamics of three dimensional totally competitive Lotka - Volterra system

by Dongmei Xiao (Shanghai Jiaotong University, visiting Harvard University) in Skiles 255

In this talk, we consider three Lotka-Volterra systems modeling a community of three mutually competing species, each of which , would exhibit logistic growth in the absence of other species. Based on the theorem of M.W.Hirsch, every trajectory in the interior of the first octant is asymptotic to one in a carrying simplex which is a two dimensional Lipschitz submanifold. The compact limit sets of the system are either equilibria or periodic orbits. We introduce some results on global stability of a positive equilibrium, discuss the finiteness of number of limit cycles, present a method to study the number of limit cycles, and fine classifications of M. L. Zeeman.

4:30 pm Wednesday, March 28, 2007

Analysis Seminar: Bounds on the Star Discrepancy in the Unit Square

by Matthew Darnall ((UW-Madison)) in Skiles 269

(t,m,s) nets are sets of points in the unit hypercube that are "well distributed" in a precise sense, i.e they achieve the best known star discrepancy. In this talk, I will show that the discrepancy at any point of a random (0,m,2) net in base 2 behaves no worse than a simple symmetric random walk with O(m) steps. I will also show that an analogous result can't be extended to higher dimensions.

3:05 pm Thursday, March 29, 2007

Stochastic Seminar: Optimization principles in statistical learning and the ranking problem

by Nicolas Vayatis (School of Mathematics, Georgia Tech) in Skiles 269

The classification problem has been at the center of recent advances in statistical learning. However in many applications such as information retrieval, or credit risk screening, ranking the instances can be much more significant than simply classifying them. In a first approach, ranking can be considered as being equivalent to classification with pairs of observations. Then, the empirical risk takes the form of a U-statistic and classification theory can be developed in order to fit with this framework. In the talk, I will also discuss the possibilities of generalizing the ranking error in order to include priors on the ranking output, for instance, when one wants to focus only on the "best" instances. In such extensions, statistical functionals of a different nature arise and novel stochastic processes have to be studied. [The talk is based on joint work with Stephan Clemencon and G'abor Lugosi]

4:30 pm Thursday, March 29, 2007

School of Mathematics Colloquium: Searching for the Origin of Life at Georgia Tech

by Nick Hud [mail] (School of Chemistry and Biochemistry, Georgia Tech) in Skiles 269

It is widely accepted that an early form of life used RNA for both information storage and chemical catalysis, before the advent of proteins and DNA. However, major gaps exist in our knowledge between this "RNA World" and the process that gave rise to the first RNA polymers. For the past several years we have been conducting experimental investigations of our hypothesis regarding how RNA-like polymers might have self-assembled on the prebiotic Earth. I will discuss our progress to date, and suggest some problems associated with the origin of life that could benefit from the participation of mathematicians.

3:30 am Friday, March 30, 2007

Geometry-Topology Seminar: An introduction to quantum invariant link invariants

by Nathan Geer (GaTech) in Skiles 269

This will be the first of two talk on quantum invariants of link. The goal of these talks is to define the colored Jones polynomials using the representation theory of quantum sl(2). The first talk will be a review of Lie algebras and their quantum analogs. In particular, it will contain the classification of irreducible sl(2)-modules and discuss how this classification can be extended to the quantum algebra associated to sl(2). The second talk will contain the construction of the Reshetikhin-Turaev quantum group invariant and then the definition of the Colored Jones polynomial.

3:00 pm Friday, March 30, 2007

Combinatorics Seminar: Percolation on the 2D Hamming graph: the supercritical phase

by Malwina Luczak (London School of Economics) in Skiles 255

Abstract: We study random subgraphs of the 2-dimensional Hamming graph $H(2,n)$, which is the Cartesian product of two complete graphs on $n$ vertices. Suppose that every edge is independently present with probability $p$, where $p=\frac&ob;1+\vep&cb;&ob;2(n-1)&cb;$ for some $\vep\in \R$. In a sequence of three papers, Borgs, Chayes, van der Hofstad, Slade and Spencer studied phase transitions in a large class of random graphs including $H(2,n)$. They obtained precise estimates for the maximum size of a connected component in the subcritical and critical regimes, as well as upper bounds in the supercritical phase. No general lower bounds on the largest supercritical component exist; indeed, it is thought that the geometry of a given graph will play a crucial role in upper bounding its giant component. Here, we complement the results of Borgs et al. for the supercritical phase in the Hamming graph $H(2,n)$. We prove that, when $\epsilon \gg (\log&ob;n&cb;)^&ob;1/3&cb; n^&ob;-2/3&cb;$, then the largest connected component has size close to $2\epsilon n^2$ with high probability. We thus obtain a law of large numbers for the largest connected component size, and show that the corresponding values of $p$ really are supercritical. Except for the factor $(\log&ob;n&cb;)^&ob;1/3&cb;$, this identifies the size of the largest connected component all the way down to the critical $p$ window. (This is joint work with Remco van der Hofstad.)

3:00 pm Monday, April 2, 2007

Theory of Computation Colloquium: A Hybrid Approach to Coping With Hard Problems

by Ryan Williams (Carnegie Mellon University) in Klaus Advanced Computing Bldg 1116 West

Refreshments at 2:30pm in Klaus 2222

A hybrid algorithm is a finite collection of heuristics, paired with a polynomial time selector that runs on the input to decide which heuristic should be executed to solve the problem. We study the interesting case where the selector must decide between heuristics that are "good" with respect to different complexity measures. For example, one heuristic may be an approximation algorithm, and another heuristic may be an exact algorithm. Ideally, we only want the approximation algorithm to be run if a good approximation is possible, and the exact algorithm should only run if it does not take too long. In this talk, we present "hardness-defying" hybrid algorithms for some well-studied NP-hard problems, such as Longest Path, Maximum Clique, and Bandwidth. These hybrids choose between an approximation and an exact algorithm. When the hybrid algorithm decides to run its approximation algorithm on an instance, the approximation guarantee is better than the worst-case, assuming P != NP. When the hybrid decides to run its exact algorithm, the runtime is substantially better than the current state-of-the-art (e.g. subexponential). Most of the results are joint with Virginia Vassilevska and Maverick Woo.

4:30 pm Monday, April 2, 2007

CDSNS Colloquium: Computing the Lyapunov exponent for Random Matrix Products

by Mark Pollicott [mail] (University Warwick) in Skiles 255

Given two matrices A(1) and A(2), we can consider the random products A(i_1) A(i_2) ... A(i_n), for i_1, i_2, ... in {1,2}. The Lyapunov exponent L is the rate of growth of the norm of a typical such product, as n tends to infinity. In the case of positive matrices we propose a way to efficiently estimate L. For example, with the matrices A(1) = ( 2 1 //1 1) and A(2) = (3 1 // 2 1) we estimate that L = 1.1433110351029492458432518536555882994025 ...

11:00 am Wednesday, April 4, 2007

Mathematical Biology & Ecology Seminar: Control of substrate properties to discover principles of locomotion

by Dan Goldman [mail] (School of Physics, Georgia Tech) in Skiles 255

Please see http://www.math.gatech.edu/~heitsch/GoldmanAbstract.pdf

12:00 pm Wednesday, April 4, 2007

Research Horizons Seminar / Gradstudent Pizza Seminar: The Discrepancy Theory: Classical and New Results

by Dmitriy Bilyk (Georgia Institute of Technology) in Skiles 269

Given an arbitrary set P_N of N points in the d-dimensional unit cube, one can measure how well-distributed this set is by means of the discrepancy function: D(x)=#{P_N \cap [0,x)} - N x_1 x_2 ... x_d , where [0,x) denotes a rectangle with sides parallel to the axis, and corners at the origin and at x. This function is just the difference between the actual and expected number of points in [0,x). The classical theory of Irregularities of Distributions says that this function cannot be `too small' in various senses; tight bounds are known in many situations. In this talk, I will describe some of important ideas, methods, examples in the theory. I'll also mention some new results and connections to other areas.

2:00 pm Wednesday, April 4, 2007

PDE/Analysis: On some geometric problems coupling motion by surface diffusion and motion by mean curvature

by Amy Novick-Cohen (Technion) in 255 Skiles

Many modern materials are in fact polycrystalline and quite naturally also have an "external surface." Roughly speaking, the boundaries of the polycrystalline grains move by motion by mean curvature, and the exterior surface evolves according to motion by surface diffusion. These motions can be formulated in terms of various specific geometries in order to isolate various features of interest. We discuss a variety of these formulations, some of their domiant features and some stability issues. Joint work with J. Kanel and A. Vilenkin.

3:00 pm Wednesday, April 4, 2007

Analysis Seminar: Logarithmic Estimates for singular oscillatory Integrasl

by Ioannis R. Parissis (University of Crete, Iraklion) in Skiles 269

We prove a new estimate for the logarithmic measure of the sublevel set of a +polynomial. As a consequence, we obtain a sharp bound for the Stein-Wainger oscillatory integral: $$ \sup_ {P} \bigg|p.v.\int_\re {e^{iP(t)}\frac{dt}{t}}\bigg|\sim \log d. $$ Here, the supremum is over all polynomials $P$ of degree d. We can also extend the $\log d-$ bound on $R^n$: \begin{eqnarray*} \sup_ {P} \bigg| p.v.\int_{\re^n} {e^{iP(x)}\frac{\Omega(x/|x|)}{|x|^n}dx} \bigg| \lesssim \log+d\,\|\Omega\|_{L\log L(S^{n-1})}, \end{eqnarray*} where the supremum is over polynomials $P$ of degree at most $d$ on $R^n$ and the function $\Omega$ has zero mean value on the unit sphere. The constants in the symbols $\sim$ and $\lesssim$ are absolute.

4:30 pm Wednesday, April 4, 2007

Analysis Seminar: Calderon commutators and the Cauchy integral on Lipschitz

by Camil Muscalu (Cornell) in Skiles 269

The plan of the talk is to revisit some of the most important operators of harmonic analysis: Calderon commutators and the Cauchy integral on Lipschitz curves. We shall speak about a proof of their L^p boundedness properties, which is conceptually simpler and avoids the usual theory of BMO functions and Carleson measures.

4:30 pm Wednesday, April 4, 2007

Applied & Computational Mathematics Seminar: Shape Optimization of Swimming Sheets

by Jon Wilkening (Berkeley) in Skiles 255

Motivated by the propulsion mechanisms adopted by gastropods, we consider shape optimization of a flexible sheet which propels itself over a thin layer of viscous fluid by propagating deformation waves along its body. We use a lubrication approximation to model the dynamics and derive the relevant Euler-Lagrange equations to optimize swimming speed and efficiency. We present a fast, highly accurate method for solving the optimization equations and explore the solution in various singular limits. We also monitor the validity of the model using a new rigorous error estimate for Reynolds' approximation.

10:00 am Thursday, April 5, 2007

QCF Seminar: Does Noise Create the Size and Value Effects?

by Jun Liu (University of California, San Diego) in Skiles 269

Black (1986) and Summers (1986) suggest that there is noise in stock prices in a sense that the price of a stock can be randomly different from its intrinsic value. Such noise can arise from economic models (e.g., Grossman and Stiglitz (1980) and De Long, Shleifer, Summers, and Robert J. Waldmann (1990)), market microstructure (e.g., Stambaugh (1983) and Roll (1983)), among other sources. In this talk, we show that when there is noise in the price of a stock, its expected return conditional on the price or the price-dividend ratio decreases with the price or the price dividend-ratio. These higher expected returns associated with lower price or price-dividend ratios are not compensation for risk, but are generated because a stock with a low price or a price-ratio is more likely to have a negative price noise thus to be undervalued. Fama and French (1992) use the matrix of expected returns conditional on size-value deciles as a demonstration of size and value effects. This matrix can be computed in closed form using our model and, for plausible parameters, is similar to its empirical counterpart (Table V of Fama and French). In our model, small and value stocks have slightly higher betas and positive alphas. Our study suggests that noise creates the size and value effect.

3:05 pm Thursday, April 5, 2007

Stochastic Seminar: CANCELLED Complete and Incomplete financial markets, martingales, and bubbles

by Philip Protter (ORIE, Cornell University) in Skiles 269

We will present an abstract framework for mathematical finance theory, focusing on the two fundamental theorems of finance, and their implications. We will then show how a typically unasked question, when asked, results in a study of financial bubbles through the analysis of the nuanced differences of martingales and local martingales. There are surprisingly no bubbles possible in complete markets, but they live, die, and are reborn naturally in incomplete markets, through a process roughly analogous to phase change in Ising models.

4:30 pm Thursday, April 5, 2007

School of Mathematics Colloquium: CANCELLED Discrete Minimal Energy Problems

by Edward Saff [mail] (Vanderbilt University )

For a compact set A in Euclidean space we shall investigate the asymptotic behavior of optimal (and near optimal) N-point configurations that minimize the Riesz s-energy (corresponding to the potential 1/r^s for s>0 and log(1/r) for s=0) over all N-point subsets of A, where r denotes Euclidean distance. If A has finite and positive d-dimensional Hausdorff measure and sd or s=d ? In such cases, the classical theory does not apply since A has s-capacity zero and so new techniques are needed to analyze the behavior of minimal energy configurations. We shall describe these techniques, which also yield information about "best-packing points" on A; that is, N points of A for which the minimal pairwise distance is as large as possible. The talk represents joint work with Doug Hardin and Sergiy Borodachov, and is meant for a broad audience from undergraduates on up.

3:00 pm Friday, April 6, 2007

Combinatorics Seminar: Calculus and Other Things Graphs Shouldn't Be Doing

by Joel Friedman (University of British Columbia, Vancouver, Canada) in Skiles 255

We give some surprising connections between analysis (e.g., wave equations) and graph theory. There are plenty of connections that are not very surprising: although graph theory has its origins in discrete problems, the attempt to study "connectivity" or "expansion" of a graph has yielded many nice but fairly straightforward analogies between graph theory and Laplacians in analysis or Riemannian geometry. Our "calculus on graphs" makes these analogies more direct. We shall discuss "calculus on graphs," that (1) gives new theorems in both graph theory and analysis from theorems in the other, (2) shows how to create wave equations on graphs, (3) relates Laplacians from graphs and analysis fairly directly, (4) simplifies the proofs of many theorems on graphs involving Laplacians or adjacency matrices, and (5) allows the nonlinear p-Laplacian theory in analysis to carry over immediately to graphs. We shall also discuss fundamental differences between analysis and graph theory that our "calculus on graphs" cannot resolve.

3:00 pm Friday, April 6, 2007

Geometry-Topology Seminar: Totally Skew Embeddings and Their Generalizations

by Gordana Stojanovic (Penn State) in Skiles 269

A submanifold M of an affine space is called totally skew if any two lines, tangent to M at two distinct points, are neither parallel nor intersecting. The following is a natural question: for a given smooth manifold M what is the smallest dimension of an affine space that can contain M as a totally skew submanifold? This simple, but nontrivial question is related to some well studied problems in algebraic topology. Totally skew embeddings are introduced by Ghomi and Tabachnikov as a projective variant of the notion of skew embeddings which goes back to Steinhaus and Segre. I will define a generalization, the notion of embeddings wih multiple regularity, which also generalize a certain other class of maps defined in the context of approximation theory. I will survey some of the available results on the subject and if time permits, will also discuss some other close relatives, such as skew maps and T-embeddings.

3:30 pm Monday, April 9, 2007

Geometry-Topology Seminar: Cohomology of the Adjoint Map

by Scott Carter (University of South Alabama) in Skiles 269

This is based on Joint work with Alissa Crans, Mohamed Elhamdadi, and Masahico Saito There are two fundamental identities that are satisfied by the adjoint map in a Hopf algebra. Because of these one can construct a solution to the Yang-Baxter equation from the adjoint map. This equation is intimately related to conjugation in a group. On the other hand, we can used these maps to write a deformation of the adjoint map and define an associated 2-cocycle. In the case of some specific Hopf algebras such as the bozonization of the superline, we can compute the two dimensional cohomology. It is our hope to construct knot invariants from such cocycles.

4:30 pm Monday, April 9, 2007

CDSNS Colloquium: Mean curvature in the sub-Riemannian setting

by LUCA CAPOGNA [mail] (University of Arkansas ) in Skiles 255

I will discuss the notion of (sub-Riemannian) horizontal mean curvature, the analogue of the Riemannian mean curvature. Most of the talk will be focused on analytical aspects of the mean curvature operator in connection with the mean curvature flow and regularity of minimal surfaces. I will also discuss two of the main motivations behind the study of the mean curvature PDE, namely the recently proposed models for the geometry of the first layer V1 of the visual cortex and the Pansu conjecture on Isoperimetric profile of the Heisenberg group.

11:00 am Tuesday, April 10, 2007

ISyE Seminar: Optimal Control of Parallel Server Systems with Many Servers

by Jim Dai (School of Industrial and Systems Engineering, Georgia Tech) in Room 228, the Executive classroom, the Main Building of ISyE

We consider parallel server systems that consist of several customer classes and server pools with exponential service times. We propose a control policy that dynamically routes customers. The policy uses minimal state information; in particular, it does not require arrival rate information. In the Hafin-Whitt many-server heavy traffic regime, we show that this policy is asymptotically optimal to minimize the total linear holding and reneging costs. A key to the optimality proof is a state space collapse (SSC) result. We illustrate how a framework of Bramson (1998) can be adapted, from conventional heavy traffic to many-server heavy traffic, to prove the SSC result. This talk is based on a joint work with Tolga Tezcan from University of Illinois at Urbana-Champaign.

3:00 pm Tuesday, April 10, 2007

Special Seminar: On the spectral gap and isoperimetric constants for convex bodies and log-concave probability measures

by Sergey Bobkov (School of Mathematics, University of Minnesota) in Skiles 255

We will be discussing lower bounds on the spectral gap and isoperimetric constants for log-concave probability measures on the Euclidean spaces in terms of the distribution of the Euclidean norm. In particular, the localization approach is used to refine a result of Kannan, Lovasz and Simonovits.

4:30 pm Tuesday, April 10, 2007

School of Mathematics Colloquium: When the Oriental Ornaments, Quasicrystals and Non-Arnold Diffusion Meet Together

by George Zaslavsky (Courant Institute and Physics Dept. NYU) in Skiles 269

We consider a weak chaos dynamics in low dimensional case, when the stochastic web emerges, and discuss the symmetry properties of the web. A brief review will be given for the problem of dynamical generation of different symmetric tiling of plane using the two-dimensional web map. This problem will be generalized to the four-dimensional web map. The latter one permits to construct a generator of quasicrystal symmetry potential in three-dimensional space. We demonstrate the dynamics along the three-dimensional stochastic webs and compare the phase space tiling to the oriental ornaments.

11:00 am Wednesday, April 11, 2007

Mathematical Biology & Ecology Seminar: Mathematical models of disease spread

by Jonathan Dushoff (Princeton University) in Skiles 255

Simple models of how diseases spread through populations are one of the great success stories of population biology. I will discuss the history and implications of simple disease models, and a variety of extensions -- including seasonality, age structure and discrete individuals -- with applications to questions of strain competition and persistence, and spatiotemporal spread.

2:00 pm Wednesday, April 11, 2007

Stochastic Seminar: Concentration inequalities for Lévy processes and related topics

by P. Marchal (Ecole Normale Superieure, Paris, France) in Skiles 269

We are interested in concentration inequalities, that is, inequalities controlling how a random variable deviates from its typical value. We study this question for a natural class of probability distributions, namely infinitely divisible (ID) distributions, and their continuous-time counterparts, namely Lévy processes. We give results at a fixed time for various subclasses of ID distributions, including those with heavy tails, as well as estimations when time varies. We also investigate the question of independence of the dimension for ID distributions with iid components. This is partly joint work with Christian Houdré and Patricia Reynaud.

4:00 pm Wednesday, April 11, 2007

Analysis Seminar: Cauchy-type Singular integrals in Several Complex Variables

by Loredana Lanzani (U Ark) in Skiles 269

in this talk I will give an overview of known and new results concerning existence and regularity of the Cauchy-type integrals for bounded domains in several complex variables; the focus is on integral kernels that are holomorphic in the parameter (as these have several applications to the classical problems in higer dimensional complex function theory). Whereas all known results are for domains with C^infty- smooth boundary, in this talk I will present new results (separately joint with D. Barrett and E. M. Stein) that holds under rather weak assumptions on the domain's boundary regularity.

4:30 pm Wednesday, April 11, 2007

Applied Computational and Mathematics Seminar: Measures related to complexity functions

by Valentin Afraimovich (San Luis Potosi University, Mexico) in Skiles 255

We consider measures related to (epsilon,n)-complexity functions. They reflect asymptotic behavior of the functions as epsilon goes to 0 or/and n goes to infinity. The main result is that these measures are invariant for the systems with zero entropy.

10:00 am Thursday, April 12, 2007

QCF Seminar: A brief history of arbitrage

by Anda Gadidov (Department of Mathematics, Kennesaw State University) in Skiles 269

The notion of arbitrage is central in the mathematics of financial markets. In addition, the Fundamental Theorem of Asset Pricing, relates the notion of no arbitrage with the existence of an equivalent probability measure, called the risk-neutral measure, under which the stochastic process modeling the asset price is a martingale. In this talk I will present the evolution of this result, from its original form given by Harrison, Kreps and Pliska to the more recent results of Schachermayer and Delbaen.

11:00 am Thursday, April 12, 2007

Honeywell Nobel Laureate Lecture Series : From Micro- to Nanoelectronics

by Klaus von Klitzing (Max Planck Institut f�r Festk�rperforschung) in Howey Physics Building, Lecture Room 1

The National Nanotechnology Initiative in the United States triggered tremendous research activity in the field of nanoscience, the science of systems more than 1000 times smaller than the diameter of a single hair. At these dimensions, the arrangements of atoms, the smallest building blocks of matter, become increasingly important and the boundaries between physics, chemistry and biology disappear. More than one hundred faculty members at the Georgia Institute of Technology are involved in nanoscience and nanotechnology research projects. In his talk, Dr. von Klitzing will provide an overview of physics, technology and the application of semiconductor quantum structures and discuss some of the recent research activities undertaken by his group in this field.

This lecture is a campus-wide event intended for a broad audience.

11:00 am Thursday, April 12, 2007

Combinatorics seminar: From the Mahler conjecture to Gauss linking integrals

by Greg Kuperberg (UC Davis) in Skiles 255 (Please note change in time)

The Mahler volume of a centrally symmetric convex body K in n dimensions is defined as the product of the volume of K and the polar body K^o. It is an affinely invariant number associated to a centrally symmetric convex body, or equivalently a basis-independent number associated to a finite-dimensional Banach space. Mahler conjectured that the Mahler volume is maximized by ellipsoids and minimized by cubes. The upper bound was proven long ago by Santalo'. Bourgain and Milman showed that the lower bound, known as the Mahler conjecture, is true up to an exponential factor. Their theorem is closely related to other modern results in high-dimensional convex geometry.

I will describe a new proof of the Bourgain-Milman theorem that establishes the Mahler conjecture up to an exponential factor of (pi/4)^n. The proof minimizes a different volume at the opposite end of the space of convex bodies, i.e., at ellipsoids. The minimization argument is based on indefinite inner products and Gauss-type integrals for linking numbers.

3:05 pm Thursday, April 12, 2007

Stochastic Seminar: Rare events, large deviations and exact asymptotics

by Bob Foley (ISyE, Georgia Tech) in Skiles 269

Often, the stationary distribution \pi of a Markov process is known to exist, but cannot be computed exactly. The stationary probability of states that are visited frequently can usually be estimated through simulating the Markov process. However, for states that are visited rarely, even simulations have difficulty estimating the stationary probability. Through the use of a particular queueing network, we illustrate an approach to obtaining exact asymptotic estimates of the stationary distribution \pi(x,y). That is, for appropriate sequences of states (x_\ell, y_\ell) with \pi(x_\ell, y_\ell) tending to zero as \ell goes to infinity, we derive estimates a_\ell such that the ratio of a_\ell to \pi(x_\ell, y_\ell) converges to 1. We are able to derive estimates even when the fluid limit of excursions from the origin to the rare event is non-linear. For example, we are able to derive exact asymptotics for \pi(\ell, y) as \ell goes to infinity when the fluid limit of an excursion from the origin to the distant state (\ell, y) collapses to an initial segment climbing the y-axis to (0,c) before turning southeast to reach (1,0).

4:30 pm Thursday, April 12, 2007

SoM Colloquium: Numerical cubature from geometry and coding theory

by Greg Kuperberg (UC Davis) in Skiles 269

The numerical cubature problem is the generalization to higher dimensions of integration methods such as Simpson's rule. Given a measure \mu on R^n, a t-cubature formula is a finite set C such that the integral of any polynomial P of degree t with respect to \mu equals a weighted sum over values on C. The main interest is in cubature formulas with few points, with positive weights, and without points outside of the domain of \mu. Gaussian quadrature satisfies all three conditions in one dimension, but the problem is already open-ended in two dimensions and increasingly non-trivial in higher dimensions.

I will discuss new methods for the cubature problem coming from error-correcting codes, algebraic geometry, moment maps, combinatorial lattice packings, and a moment problem in univariate probability. of discretized convex bodies. The methods yield many new explicit, efficient, positive, interior, cubature formulas for the most standard choices of \mu. In one context, they also lead to an interesting local lower bound on the number of points needed for cubature.

2:00 pm Friday, April 13, 2007

Physics Colloquium: Quantum Hall Effect: Physics and Applications

by Klaus von Klitzing (Max-Planck-Institut f�r Festk�rperforschung) in Physics Lecture Room 5

Basic research on the most important device in microelectronics, a silicon field effect transistor, led in 1980 to the discovery of the Quantum Hall Effect (QHE). Electrical measurements on such a device demonstrated, that a new type of electrical resistor can be realized, a resistor with a well defined value which depends exclusively on fundamental constants. Today, the word QHE is a synonym for the more general topic of electrons in strong magnetic fields which has connections not only to solid state physics but also to other research areas like astrophysics (edge states in gravity and black hole physics), high energy physics (quantum Hall quarks) and metrology (fundamental constants). This broad interest in QHE physics explains the high publication rate of about one publication per day. The talk will focus on two topics, the application of the QHE in connection with our international system of units (SI units) and some new developments in quantum Hall physics. For applications in metrology, the QHE is primarily used as a resistance standard with a value of 25812.807 Ohm. This resistor is not only the base for all resistance calibrations but also important for the realization of the capacitance unit Farad, electrical current unit Ampere, and mass unit Kilogram.

Dr. Klaus von Klitzing is a Nobel Prize Laureate in Physics.

3:00 pm Friday, April 13, 2007

Combinatorics Seminar: The Finite Plane Kakeya Problem and Collinear Points in Permutations

by Josh Cooper (University of South Carolina) in Skiles 255

Abstract: We show that the problem of counting collinear points in a permutation (previously considered by the author and J. Solymosi) and the well-known finite plane Kakeya problem are intimately connected. Via counting arguments and by studying the hypergraph of collinear triples we show a new lower bound 5q/14 + O(1) for the number of collinear triples of a permutation of GF(q) and a new lower bound q(q + 1)/2 + 5q/14 + O(1) on the size of the smallest Besicovitch set in GF(q)^2. Several intriguing questions about the structure of the collinear triple hypergraph are discussed.

3:00 pm Friday, April 13, 2007

Geometry-Topology Seminar: An introduction to quantum invariant link invariants, Part II

by Nathan Geer (GaTech) in Skiles 269

4:00 pm Friday, April 13, 2007

General Colloquium: Soap Bubbles and Mathematics

by Frank Morgan [mail] (Williams College) in DM Smith 105

Soap bubbles continue to fascinate and confound mathematicians. The show will include demonstrations, explanations, and a little guessing contest with prizes. No prerequisites; friends and families welcome.

1:30 pm Monday, April 16, 2007

Combinatorics Seminar: Where do power laws come from?

by Josh Cooper (Mathematics, U. South Carolina) in Skiles 246 (Note: Not Usual Room/Time)

We discuss the question, "What does `scale-free' really mean?" in the context of complex networks. In particular, we show that a certain degree sequence arises from the unique asymptotically invariant distribution under taking random subgraphs. Since the distribution is of power-law type, this provides some explanation for the ubiquity of power-laws in "real world" massive graphs. We present several open questions that are natural extensions of the main results. Joint work with Lincoln Lu of USC.

3:35 pm Monday, April 16, 2007

Geometry-Topology Seminar: Mapping class groups of Heegaard splittings

by Jesse Johnson (Yale) in Skiles 269

The mapping class group of a Heegaard splitting is the group of automorphisms of the ambient manifold that take the Heegaard surface onto itself. There is a canonical homomorphism from this group into the mapping class group of the 3-manifold. I will outline a proof that for high distance Heegaard splittings this homomorphism is an isomorphism, then describe examples of low distance, irreducible Heegaard for which the kernel is infinite.

4:30 pm Monday, April 16, 2007

CDSNS Colloquium: Orbits homoclinic to periodic solutions and their application in hydrodynamics

by Tom Bridges [mail] (University of Surrey, UK) in Skiles 255

Degenerate periodic orbits in Hamiltonian systems provide a mechanism for the creation of orbits homoclinic to periodic solutions. Periodic orbits are degenerate when the energy considered as a function of the frequency has a critical point. The normal form for this bifurcation has an interesting geometric characterisation in terms of the curvature of the energy-frequency map. When the periodic orbits are spatial, say steady solutions of a Hamiltonian PDE, then the bifurcation leads to a class of dark solitary waves, with a geometric phase. It turns out that these observations are a special case of the class of homoclinic orbits generated by degenerate relative equilibria. A new theory for this homoclinic bifurcation is presented. Examples in dynamical systems include a new formulation of the saddle-center bifurcation of invariant tori. There are a range of applications of this bifurcation in hydrodynamics. For illustration, the generation of internal solitary waves by this mechanism will be discussed, and application to the classical water wave problem shows that there is a new pervasive class of steady dark solitary waves in shallow water.

3:00 pm Tuesday, April 17, 2007

PDE Seminar: Gamma-limit of the Folding Energy Problem

by Bo Su (Iowa State University) in Skiles 255

Abstract

11:00 am Wednesday, April 18, 2007

Mathematical Biology & Ecology Seminar: Evolutionary and Population Dynamics of Bacterial Viruses

by Joshua Weitz [mail] (School of Biology, Georgia Tech) in Skiles 255

Bacterial viruses, aka bacteriophage or phage, are ubiquitous in nature, yet many central aspects of host-phage biology have not been integrated into mathematical models. In this talk I present a series of theoretical efforts to understand the diversity, population dynamics and life history of phage. First, I discuss an evolutionary ecology model of host-phage diversification using the framework of adaptive dynamics and show how the principle of competition exclusion is modified in the context of coevolutionary arms races. Second, current models of host-phage population dynamics neglect important aspects of phage biology, including reduced lysis of hosts as hosts approach stationary phase. Incorporating reduced lysis into dynamics leads to a prediction of alternative stable states, which are discussed in the context of preliminary experiments. Finally, I explore ongoing experimental and theoretical efforts to understand how phage may optimally exploit their hosts by utilizing a variety of life history strategies.

12:00 pm Wednesday, April 18, 2007

Research Horizons Seminar / Gradstudent Pizza Seminar: Two variable orthogonal polynomial on the bicircle and and the geometry of polynomial factorization

by Jeff Geronimo (Georgia Institute of Technology) in Skiles 269

Orthogonal polynomials have a surprising number of applications in Mathematics and Physics. This talk will consist of a brief description of the theory of orthogonal polynomials on the bicircle and their application to the problem of spectral factorization of bivariate positive trignometric polynomials. In particular a geometrical condition is require in order for stable factorization to occur. Time permitting other applications of this theory will be discussed.

2:00 pm Wednesday, April 18, 2007

Special Stochastic Seminar: The Laplace distribution and generalizations: Fundamental properties, applications, and recent developments

by T. Kozubowski (University of Nevada, Reno) in Skiles 269

In his memoir in 1774, P.S. Laplace introduced an error distribution that now bears his name. Since then, for many years the popularity of the Laplace distribution in stochastic modeling has been by far less than that of its four-years-older "sibling", the second law of Laplace, better known as the Gaussian (normal) distribution. It is only in recent years that this distribution, together with its various generalizations, has been revived, and is now being used in a variety of fields, including archeology, biology, biostatistics, climatology, economics, environmental science, finance, geosciences, and physics. In this talk, we will review fundamental properties of the Laplace and related distributions, discuss their applications, and present some recent developments in this area.

4:30 pm Wednesday, April 18, 2007

Joint GA Tech/GSU Mathematics Colloquium: Quasihyperbolicity and strange attractors

by Leonid Shilnikov (Institute for Applied Mathematics & Cybernetics) in Skiles 255

We propose an effective criterion of dynamical chaos that is based on a quasihyperbolicity condition. The latter means the phase space of the system has an absorbing domain with such unstable trajectory behavior that lets us find a factor-system expanding volumes. We will give examples of quasihyperbolic attractors of both structurally stable, like Smale-William solenoids, and unstable types such as Lorenz and spiral ones. We single out a class of quasihyperbolic systems with a single saddle equilibrium state having a 1D unstable manifold. The periodic perturbations on an attractor of Lorenz type are also discussed.

10:00 am Thursday, April 19, 2007

QCF Seminar: Land of Addicts? An empirical investigation of habit-based asset pricing models

by Xiaohong Chen (New York University) in Skiles 269

This paper studies the ability of a general class of habit-based asset pricing models to match the conditional moment restrictions implied by asset pricing theory. We treat the functional form of the habit as unknown, and to estimate it along with the rets of the model's finite dimensional parameters. Using quarterly data on consumption growth, assets returns and instruments, our empirical results indicate that the estimated habit function is nonlinear, that habit formation is better described as internal rather than external, and the estimated time-preference parameter and the power utility parameter are sensible. In addition, the estimated habit function generates a positive stochastic discount factor(SDF) proxy and performs well in explaining a cross-sectional stock return data. We find that an internal habit SDF proxy can explain a cross-section of size and book-market sorted portfolio equity returns better than (i) the Fama and French (1993) three-factor model, (ii) the Lettau and Ludvigson (2001b) scaled consumption CAPM model, (iii) an external habit SDF proxy, (iv) the classic CAPM, and (v) the classic consumption CAPM.

4:30 pm Thursday, April 19, 2007

School of Mathematics Colloquium: Surprising patterns emerge when "bisecting" proofs of theorems

by James A. Yorke [mail] (University of Maryland) in Skiles 269

Theorem and proof are conjoined twins but I have always had a secret preference for proof, or rather for those kernels of proofs that recur. I have been developing an art form in which the goal is to take a proof of any theorem and reorganize it into two parts. Reduce it to two tasks or lemmas to be proved. Of course those parts might in turn be similarly bisected, hopefully not ad infinitum. I was led to this goal by remembering classes that presented complicated proofs requiring many disparate facts to be proved, making it impossible for me to wrap my mind around the collective entity and see it as a unity. For example I encountered the Poincare-Bendixson Theorem in college, to my considerable confusion. I often find this "bisection" task quite difficult, but when I succeed, I sometimes unearth connections with proofs of other theorems. Here I will report on three eerily connected results. 1) The Poincare-Bendixson Theorem 2) Degree theory in R^n made constructive 3) Why period doubling cascades exist in one-parameter families of maps.

4:30 pm Thursday, April 19, 2007

ACO/ARC Special Lecture: Quantum Physics and the Nature of Computation

by Umesh Vazirani (UC Berkeley) in 117 Smithgal Student Services Building

Quantum physics is a fascinating area from a computational viewpoint. The features that make quantum systems prohibitively hard to simulate classically are precisely the aspects exploited by quantum computation to obtain exponential speedups over classical computers. In this talk I will survey our current understanding of the power (and limits) of quantum computers, and prospects for experimentally realizing them in the near future. I will also touch upon insights from quantum computation that have resulted in new classical algorithms for efficient simulation of certain important quantum systems.]

This lecture is suited for a general audience. Poster: http://www.aco.gatech.edu/doc/vazirani_event.pdf

2:00 pm Friday, April 20, 2007

ARC/Networking Seminar : CANCELLED How to avoid doing research "10 years ahead of time"

by Balaji Prabhaka (Stanford University) in Klaus Advanced Computer Building, room 1116

A thread of research dating to the mid-90s is concerned with the emulation of an output-queued (OQ) switch using a combined input- and output-queued (CIOQ) switch. It showed that in order to emulate an N port OQ switch under adversarial inputs, it is necessary and sufficient for the CIOQ switch to run at a speedup of 2 - (1/N). This result validated a belief that was prevalent among implementors and researchers.

3:00 pm Friday, April 20, 2007

Combinatorics: Boundary Partitions in Trees and Dimers

by David Wilson (Microsoft Research) in Skiles 255

Abstract: We study groves on planar graphs, which are forests in which every tree contains one or more of a special set of vertices on the outer face, referred to as nodes. Each grove partitions the set of nodes. When a random grove is selected, we show how to compute the various partition probabilities as functions of the electrical properties of the graph when viewed as a resistor network. We prove that for any partition sigma, Pr[grove has type sigma] / Pr[grove is a tree] is a dyadic-coefficient polynomial in the pairwise resistances between the nodes, and Pr[grove has type sigma] / Pr[grove has maximal number of trees] is an integer-coefficient polynomial in the entries of the Dirichlet-to-Neumann matrix. We give analogous integer-coefficient polynomial formulas for the pairings of chains in the double-dimer model. We show that the distribution of pairings of contour lines in the Gaussian free field with certain natural boundary conditions is identical to the distribution of pairings in the scaling limit of the double-dimer model. These partition probabilities are relevant to multichordal SLE_2, SLE_4, and SLE_8. Joint work with Richard Kenyon.

3:35 pm Friday, April 20, 2007

Geometry-Topology Seminar: Twisted Alexander modules and Fox colorings

by Dan Silver (University of South Alabama) in Skiles 269

In this joint work with Susan Williams, we examine twisted Alexander modules of a knot from the point of view of algebraic dynamics. Generalized Fox colorings correspond to (untwisted) homology classes of cyclic coverings of the knot, and they arise as periodic points of a natural dynamical system. We extend these results in the presence of twisting.

9:00 am Saturday, April 21, 2007

Random Combinatorial Structures Conference: Various lectures

by Prominent researchers in the field in Mathematics Department, University of Nebraska-Lincoln

The conference will bring together many prominent researchers in the field of probabilistic combinatorics. The main purpose of this meeting is to create an environment in which interactions and research links between established senior faculty and younger researchers (both graduate students and new Ph.Ds) can flourish. To this end, there will be nine plenary talks and the rest of the time will be devoted to informal interactions. We encourage everyone with an interest in this field to attend. We hope to be able to provide lodging for graduate students and recent graduates, as well as some travel support, conditional on NSF funding. For more details see, http://math.unl.edu/pi/events/rcs

9:00 am Monday, April 23, 2007

Curriculum Conference: Rethinking Mathematics Curriculum for Engineering & Science Students II

by Variety of Speakers (from various affiliations) in ISyE 228

A two-day event to bring together engineers, computer scientists and mathematicians to discuss the way that recent developments in science and engineering are changing the kind of mathematics that is being used in upper level science and engineering courses and in research, the ways mathematics is used there, and the depth of understanding that is required for effective use. It follows up on a previous conference of this type held one year ago. As we hope that the discussion will be of wide interest, the proceeding of the conference will be collected, combined with contribution from last year's meeting, edited and published by the organizers. There will be no registration fee. The conference is supported by the NSF and Georgia Tech. The conference is organized by Eric Carlen, Maria Carvalho and Michael Loss. For details see, http://www.math.gatech.edu/~carlen/CurConf2

3:30 pm Monday, April 23, 2007

Geometry-Topology Seminar: Augmented Teichmueller spaces and orbifolds

by Arkady Vaintrob (University of Oregon) in Skiles 269

Augmented Teichmueller spaces were introduced by Lipman Bers by adding Riemann surfaces with nodal singularities to the classical Teichmueller space T . Unlike the space T which has a natural structure of a complex manifold, the augmented space has no complex structure (it is not even locally compact). However, its quotient by any finite index subgroup of the mapping class group has structure of a normal complex space (even of a smooth complex orbifold). This result is important for understanding the Chen-Ruan stringy orbifold cohomology and its generalizations.

4:30 pm Monday, April 23, 2007

CDSNS Colloquium: Optimal Control of Integrodifference Models

by Suzanne Lenhart [mail] (U. Tennesee) in Skiles 255

Integrodifference equations are used to model populations which have distinct growth and dispersal stages. The models are discrete in time and continuous in space with the dispersal given by integration against a kernel. We discuss a harvesting optimal control problem for these models and then give an example about a disease model in crops.

11:00 am Tuesday, April 24, 2007

Stochastic Seminar: A particle system in interaction with a rapidly varying environment: Mean field limits and applications

by David McDonald (University of Ottawa) in Room 228 Main ISyE Building

We study an interacting particle system whose dynamics depends on an interacting random environment. We apply our results to analyze the performance of communication networks where users access a channel using random distributed multi-access algorithms (like 802.11 the wifi protocol). In particular we study the throughput of users in hidden zones in a network. To understand the problem consider three people sitting on a couch following the protocol that they will start talking only if their neighbours are quiet. Then the chatterbox sitting in the middle between two chatterboxes will hardly ever get a word in. As the number of particles (users) grows large, the transition rate of the particles slows down (relative to the speed of the channel). The transition rate of a particle is determined by its state (its backoff), by the empirical distribution of all the particles (the backoffs and hence the chances of a collision with other particles) and by a rapidly varying environment (the radio conditions of the zones in the network which are either busy or silent). The transitions of the environment are determined by the empirical distribution of the particles (a single user can change the radio environment by starting a transmission). We prove the propagation of chaos on the path space of the particles thereby justifying a common engineering assumption of independence of the users when the number of users is large. The limiting trajectory of the empirical measure of the states of the particles satisfies a deterministic differential equation. This deterministic differential equation involves the time averages of the environment process. We then can explicitly calculate the steady state throughputs of hidden nodes.

3:00 pm Tuesday, April 24, 2007

PDE Seminar: Stability criteria for Reaction-Diffusion Systems with Skew-Gradient Structure

by Xijun Hu (Chinese Academy of Sciences) in Skiles 255

This talk will introduce our recent work of reaction-diffusion systems with skew-gradient structure. In connection with calculus of variations, we show that there is a close relation between the stability of a steady state and its relative Morse index. The stability criteria presented here were partially motivated by some recent works of Yanagida. This is a joint work with professor Chao-Nien Chen.

4:30 pm Tuesday, April 24, 2007

Analysis Seminar: Quantum games of Gauss sums

by Konstantin Oskolkov (University of South Carolina) in Skiles 269

The Greens function of the Cauchy initial value problem for Schroedinger equation of a free particle with the periodic initial data will be discussed. This function encodes all Gauss sums, both complete and incomplete, and provides a natural ordering of such sums, including the identities of Genocci Schaar type. Elementary operations with this Greens function, such as integration in the time parameter, provide a transparent proof of J. Gervers result on differentiability of Riemanns non-differentiable function and its extensions

3:00 pm Wednesday, April 25, 2007

Math Physics Seminar: On the Quantum Sanov Theorem

by Ruedi Seiler (Department of Mathematics, TU Berlin) in Skiles 255

Discrete stationary classical processes as well as quantum lattice states are asymptotically confined to their respective typical support, the exponential growth rate of which is given by the (maximal ergodic) entropy. In the iid case the distinguishability of typical supports can be asymptotically specified by means of the relative entropy, according to Sanov's theorem. We give an extension to the correlated case, referring to the newly introduced class of HP-states.

4:00 pm Wednesday, April 25, 2007

Analysis: Minimum Riesz s-energy points on compact sets in \mathbb{R}^{3}

by Johann Brauchart (Vanderbilt University) in Skiles 269

Let K be a compact set in \mathbb{R}^{3}. We want to investigate minimum energy point charges on K that interact according to the Riesz potential 1/r^{s}, 0 < s < 1, where r is the Euclidean distance between points. Such an N-point configuration minimizes the Riesz s-energy \sum_{j\neq k}|\mathbf{x}_{j}-\mathbf{x}_{k}|^{-s} over the class of all N-point sets {\mathbf{x}_{1},\dots,\mathbf{x}_{N}}\subseteq K. In particular, we are interested in the support of the limit distribution of minimum energy points. Classical potential theory yields that this limit distribution coincides with the equilibrium measure on K which is supported on the outer boundary of K. For sets of revolution K, Hardin, Saff, and Stahl showed an even stronger remarkable result: In the logarithmic case (limit as s\to 0) the equilibrium measure on K is concentrated on the ``outer-most'' part of K. The ``outer-most'' part of a torus K, for example, is the set of revolution generated by rotating the right semi-circle about the vertical axis. We want to consider the case when 0 < s < 1. (Joint work with D. P. Hardin and E. B. Saff.)

4:30 pm Wednesday, April 25, 2007

Applied & Computational Mathematics seminar: An adaptive immersed boundary method for fluid-structure interaction with applications to cardiac fluid dynamics

by Boyce Griffith (Courant Institute, New York University) in Skiles 255

The immersed boundary method is a general approach to modeling and simulating the dynamic interaction of an elastic structure and a viscous incompressible fluid. In this talk, I will describe an adaptive version of the immersed boundary method that employs structured adaptive mesh refinement (AMR) to provide locally enhanced spatial resolution. Various applications of this adaptive methodology will be presented, including the simulation of cardiac fluid dynamics in a three-dimensional model of the human heart. Time permitting, I will also discuss work on coupling the model of cardiac mechanics to a model of the electrical activity of the heart.

1:30 pm Thursday, April 26, 2007

Graph Theory seminar: Circular choosability

by Serguei Norine (Math, GT) in Skiles 255

Circular colorings of graphs have been extensively studied in the past decade. However, a circular version of list-chromatic number, circular choosability, has been introduced only very recently by Mohar and Zhu. Many basic questions about circular choosability remain open. We will survey these open questions and present several known results. In particular, we will discuss bounds on circular choosability of planar graphs and graphs with given maximum degree. We will also discuss an algebraic method for bounding circular choosability of bipartite graphs.

3:05 pm Thursday, April 26, 2007

Stochastic Seminar: Concentration of the Spectral Measure for Matrices with Infinitely Divisible Entries

by Hua Xu (School of Mathematics, Georgia Tech) in Skiles 269

From the late 1950s, and starting with the convergence of the spectral measure to the semi--circle law, many landmark results hae been obtained in Random Matrix Theory. Most of these results assume the finiteness of the (higher) moments of the matrix entries. We study here random matrices with infinitely divisible entries, which do not necessarily have finite moments. We derive concentration inequalities for functions of the empirical spectral measure of such matrices, as well as concentration results for some other functionals, such as the largest eigenvalue or the largest singular value. Particular attention will be paid to matrices with stable (not necessarily independent) entries.

4:15 pm Thursday, April 26, 2007

Math Dept. Tea:

in Skiles 236

This will be our last math dept. tea of the semester. All math faculty, staff, and students are welcome to attend!

3:00 pm Friday, April 27, 2007

Combinatorics Seminar: Sum-product problems and incident geometry

by Jozsef Solymosi (UBC) in Skiles 255

3:30 pm Friday, April 27, 2007

Geometry-Topology Seminar: Some Recent Results in the Geometric Tomography of Convex Bodies

by Ralph Howard (University of South Carolina) in Skiles 269

Let G(n,k) be the Grassmann manifold of all k dimensional linear subspaces of R^n. Then each convex body K in R^n defines a function, V_k(K|), on G(n,k) where V_k(K|L) is the k dimensional volume of the orthogonal projection of K onto L. This is the k-th projection function, or k-brightness function, of K. Results about the extent that a convex body is determined by one or more of its projection functions will be surveyed. While there has been some recent progress in settling some long standing problems, such as Nakajima's problem from 1926 of showing that a convex body in three dimensional space with constant width and constant brightness is a Euclidean ball, there are still a large number of natural questions remaining open.

4:30 pm Monday, April 30, 2007

CDSNS Colloquium: Some recent progress in geometric methods for Arnold diffusion

by Rafael de la Llave [mail] (The University of Texas At Austin) in Skiles 255

We survey some methods to establish instability of Hamiltonian dynamical systems based on the study of invariant objects and their interesections.

3:00 pm Tuesday, May 1, 2007

PDE Seminar: Energy Scattering Theory for Nonlinear Schrodinger Equations with Exponential Growth

by Chengchun Hao (School of Mathematics, Georgia Tech) in Skiles 255

In this talk, we will show the existence of the scattering operators for the nonlinear Schr\"odinger equation with exponential nonlinearity in the whole energy spaces for one and two spatial dimensions. This is based on the joint work with B.X.Wang and H.Hudzik.

10:00 am Thursday, May 3, 2007

ACO Dissertation Defense: Network Design and Alliance Formation for Liner Shipping

by Richa Agarwal (Industrial & Systems Engineering, Georgia Tech) in 226A Groseclose Building

In the sea cargo industry, liner shipping accounts for over 60% of the value of goods shipped. In this thesis we study some of the problems related to this industry. Given a set of cargo to be transported, a set of ports and a set of ships, a common problem faced by carriers in liner shipping is the design of their service network. We develop an integrated model to design service network for the ships and to route the available cargo, simultaneously. The proposed model incorporates many relevant constraints, such as the weekly frequency constraint on the operated routes, and emerging trends, such as obtaining benefits from transshipping cargo on two or more service routes, that appear in practice but have not been considered previously in literature. Also, we design exact and heuristic algorithms to solve the integer program efficiently. The proposed algorithms integrate the ship scheduling problem, a tactical planning level decision, and the cargo routing problem, an operational planning level decision, and provide good overall solution strategy. Computational experiments indicate that larger problem instances, as compared to the literature, can be solved using these algorithms in acceptable computational time. An interesting problem that lies at the intersection of mathematics and operations research is the efficient functioning of a decentralized system composed of various individual decision makers. In this thesis we study alliance formation among liner carriers. While each carrier's individual goal is to maximize his own benefits, collaboration among carriers helps them achieve economies of scale, more extensive and regular service and better utilization of expensive assets such as ships. For the formation of a sustainable alliance, carriers need to agree on an overall service network and resolve issues concerning distribution of benefits and costs among the members of the alliance. We develop mechanisms to guide the members of an alliance to design a collaborative service network and distribute the benefits of the alliance in a fair way. The mechanism utilizes inverse optimization techniques to obtain capacity exchange costs in the network. These costs provide side payments to the members, on top of the revenue generated by them in the collaborative solution, to motivate them to act in the best interest of the alliance while satisfying their own self interests.

3:00 pm Thursday, May 3, 2007

PDE Seminar: A Nonlinear Hyperbolic System with Random Terms

by Abel Afouda (University of Abomey-Calavi, Benin) in Skiles 255

In this talk, we provide a mathematical model of the Cotonou Lagoon. The dynamics of the system is considered through the conservation laws (of mass and momentum). The resulting system is a nonlinear hyperbolic system with random source terms. The source terms correspond to the bottom slope, the friction slope and the input from flood and precipitations. The effects of random source terms on the water flow are investigated. Although the simplified model provides informations about the dynamics of the system, real data will be necessary for implementation of potential impacts of climate changes on the lagoon.

11:05 am Friday, May 4, 2007

Dissertation Defense: The Mathematical Theory of Thin Film Evolution

by Suleyman Ulusoy (School of Mathematics, Georgia Tech) in Skiles 255

We try to explain the mathematical theory of thin liquid film evolution. We start with introducing physical processes in which thin film evolution plays an important role. Derivation of the classical thin film equation and existing mathematical theory in the literature are also introduced. To explain the thin film evolution we derive a new family of degenerate parabolic equations. We prove results on existence, uniqueness, long time behavior, regularity and support properties of solutions for this equation. At the end of the thesis we consider the classical thin film Cauchy problem on the whole real line for which we use asymptotic equipartition to show H^1\mathbb&ob;R&cb; convergence of solutions to the unique self-similar solution.

9:00 am Sunday, May 6, 2007

Southeast Geometry Seminar: Various topics

by various speakers in Classroom 221, College of Management

The Southeast Geometry Seminar (SGS) is a semiannual series of one day events sponsored jointly by: The National Science Foundation, The University of Alabama at Birmingham, The Georgia Institute of Technology, and Emory University. For more details, see http://www.math.uab.edu/sgs

3:00 pm Monday, May 7, 2007

Geometry-Topology Seminar: Number of edges of capillary surfaces and first eigenvalue of minimal surfaces in S^3

by Jaigyoung Choe [mail] (Korea Institute for Advanced Study) in Skiles 269

Number of edges of capillary surfaces and first eigenvalue of minimal surfaces in S^3 Abstract: It will be proved that if the number of edges of a disk type capillary surface in a domain bounded by umbilic surfaces in R^3 is less than four, then the capillary surface itself is umbilic. Also it will be proved that if the number of edges of a disk tpe fundamental piece of an embedded minimal surface in S^3 which is invariant under a group of reflections is less than six, then the first eigenvalue of the Laplacian on the minimal surface is equal to 2. Therefore Lawson's minimal surfaces and Karcher-Pinkall-Sterling's minimal surfaces satisfy Yau's conjecture.

11:00 am Wednesday, May 9, 2007

Combinatorics Seminar: Correlation decay and applications to counting problems

by David Gamarnik (MIT) in Skiles 255

ABSTRACT: We propose new approximation algorithms for solving certain counting problems. Unlike prior algorithms which are based on Markov chain sampling technique, our algorithms are deterministic and thus do not require randomization. Our technique builds on the notion of correlation decay, which originates in statistical physics in connection with the uniqueness property of Gibbs measures on infinite lattices. This technique leads to deterministic approximation algorithms for several graph counting problems. We will illustrate it on the problem of counting the number of proper colorings of a graph. Joint work with D. Katz

3:00 pm Wednesday, May 9, 2007

Graph Theory Seminar: The Riemann-Roch theorem in the tropical world

by Michael Kerber (University of Kaiserslautern) in Skiles 269

Abstract: Tropical algebraic geometry is a recent branch of mathematics that establishes deep relations between objects of algebraic geometry and objects in combinatorics. In this talk, we consider the tropical counterpart of the Riemann-Roch theorem for divisors on curves.

10:00 am Monday, May 14, 2007

ACO Dissertation Defense: Efficient Algorithms for Market Equilibria

by Nikhil R. Devanur (College of Computing, Georgia Tech) in Klaus 2108

The mathematical modelling of a market, and the proof of existence of equilibria have been of central importance in mathematical economics. Since the existence proof is non-constructive in general, a natural question is if computation of equilibria can be done efficiently. Moreover, the emergence of Internet and e-commerce has given rise to new markets that have completely changed the traditional notions. Add to this the pervasiveness of computing resources, and an algorithmic theory of market equilibrium becomes highly desirable. The goal of this thesis is to provide polynomial time algorithms for various market models. Two basic market models are the Fisher model: one in which there is a demarcation between buyers and sellers, buyers are interested in the goods that the sellers possess, and sellers are only interested in the money that the buyers have; and the Arrow-Debreu model: everyone has an endowment of goods, and wants to exchange them for other goods. We give the first polynomial time algorithm for exactly computing an equilibrium in the Fisher model with linear utilities. We also show that the basic ideas in this algorithm can be extended to give a strongly polynomial time approximation scheme in the Arrow-Debreu model. We also give several existential, algorithmic and structural results for new market models:

# the *spending constraint* utilities (defined by Vazirani) that captures the "diminishing returns" property while generalizing the algorithm for the linear case.

# the capacity allocation market (defined by Kelly), motivated by the study of fairness and stability of the Transmission Control Protocol (TCP) for the Internet, and more generally the class of Eisenberg-Gale (EG) markets (defined by Jain and Vazirani).

Finally, this line of research has given insights into the fundamental techniques in algorithm design: The primal-dual schema has been a great success in combinatorial optimization and approximation algorithms. Our algorithms use this paradigm in the enhanced setting of Karush-Kuhn-Tucker (KKT) conditions and convex programs.

2:30 pm Thursday, May 17, 2007

Geometry-Topology Seminar: The Tutte polynomial in knot theory

by Iain moffatt [mail] (University of Waterloo, Canada) in Skiles 269

In this talk I will give an overview of various connections between the Tutte polynomial and its generalizations and polynomial invariants of knots. I will begin by reviewing well known relations between the Tutte polynomial of a plane graph and the Jones and HOMFLY (a generalization of the Jones polynomial) polynomials of a knot. I will then go on to discuss several ways Bollob\'&ob;a&cb;s and Riordan's recently defined ribbon graph polynomial (which was motivated by very different knot theoretical considerations) has been used to extend these results. Finally I will outline connections between the graph polynomials and Khovanov homology.

3:00 pm Tuesday, May 22, 2007

Analysis Seminar: On constrained orthogonal polynomials

by Pierre Moussa (Service de Physique Theorique, CEA, Saclay, France) in Skiles 269

Constrained orthogonal polynomials have recently been considered in variational approach of many body quantum mechanics. We will introduce and analyse their properties, and discuss some related open problems.

3:30 pm Thursday, May 24, 2007

Geometry-Topology Seminar: Counting Lattice Points and Toric Varieties

by Jamie Pommersheim [mail] (Reed College) in Skiles 269

The problem of giving exact formulas for the number of lattice points in a convex polytope has interested mathematicians for many years. Pick's Formula (c. 1890) gives the answer in dimension two, and Ehrhart achieved interesting partial results in higher dimensions in the 1960's. In the past fifteen years, much progress has been made using the tool of toric varieties. Recent toric results of Brion, Morelli, Khovanskii, and the speaker have helped us understand the lattice point question much more clearly.

3:05 pm Thursday, June 14, 2007

Stochastic Seminar: Stable limits for transient random walks in random environment on Z

by Nathanael Enriquez (Laboratoire de Probabilite, Universite Pierre et Marie Curie) in Skiles 269

We consider transient random walks in random environment on Z with zero asymptotic speed. A classical result of Kesten, Kozlov and Spitzer says that the hitting time of the level n converges in law, after a proper normalization, towards a positive stable law, but they do not obtain a description of its parameter. A different proof of this result is presented, that leads to a complete characterization of this stable law. The case of Dirichlet environment turns out to be remarkably explicit.

3:00 pm Wednesday, June 27, 2007

Geometry Seminar: Shadow boundaries of Riemannian submanifolds

by Gabriel Ruiz Hernandez (CIMAT Mexico) in Skiles 269

Inspired by Blaschke's work on analytic convex surfaces, we study shadow boundaries of Riemannian submanifolds M, which are defined by parallel vector fields. Since a shadow boundary is just a closed subset of M, first, we will give a condition that guarantees its smoothness. This depends on the second fundamental form. It is natural to ask what kind of properties such submanifolds of M might have. Could they be totally geodesic or minimal? Answers to these and related questions are given in this work.

10:00 am Monday, August 20, 2007

Dissertation Defense: Algorithmic and topological aspects of semi-algebraic set defined by quadratic polynomials

by Michael Kettner (School of Mathematics, Georgia Tech) in GLC 129 (Global Learning Center, Tech square)

9:00 am Thursday, August 23, 2007

Princeton University Minicourse: Additive Combinatorics Minicourse

by Boaz Barak, Luca Trevisan and Avi Wigderson (Princeton, Berkeley and Inst for Advanced Study) in Princeton University

Additive combinatorics studies structural properties of subsets of numbers and other Abelian groups. It is concerned with questions such as what conditions on a set A assure that A contains long arithmetic progressions, or what conditions imply that A's sum-set (the set {x+y: x,y in A}) is small or large . Recent years saw both important advances in this field and some computer science applications. This mini course will review both of these from the perspective of theoretical computer science. The course is intended for researchers and students with background in theoretical computer science, but no prior knowledge of additive combinatorics. For details, see http://www.cs.princeton.edu/theory/index.php/Main/AdditiveCombinatoricsM...

3:30 pm Monday, August 27, 2007

Geometry Topology Seminar: Creating small Seifert fibered spaces by Dehn surgery on knots

by Ken Baker (Georgia Tech) in Skiles 269

Generically, a small Seifert fibered spaces may be viewed as a thrice-punctured sphere cross S^1 with a solid torus attached along each boundary torus. Sometimes an embedded solid torus may be excised from S^3 and then reattached along the resulting boundary torus in a different manner to produce one of these small Seifert fibered spaces; i.e. Dehn surgery on a knot in S^3 sometimes produces a small Seifert fibered space. The classification of such knots and Dehn surgeries remains open. We'll talk about the context for this problem, what's known, and some of our current on-going joint research with Cameron Gordon and John Luecke.

4:30 pm Monday, August 27, 2007

CDSNS Colloquium: Pattern recognition, state-dependent delay differential equations and analysis in locally complete spaces

by Jianhong Wu [mail] (York University) in Skiles 255

Subspace clustering arises naturally from recognition of hidden patterns in low dimensional subspaces from high dimensional data sets. Delay adaptation, a mechanism to process input information with different speeds for different components according to the similarity between the input and the stored feature of a dynamically identified cluster, provides a biologically well motivated neural network solution to the subspace clustering problem via the adaptive resonance theory. The mathematical formulation of such a delay adaptation for the computational performance of the network is a large-scale system of delay differential equations with state-dependent delay, and the convergence consideration of the relevant clustering algorithm called PART requires careful examination of the birth and continuation of Hopf bifurcation of periodic solutions. This talk presents some recent work on a functional analytic theory in locally complete spaces, and its applications to the global Hopf bifurcation theory for functional differential equations with state-dependent adaptive delays.

3:00 pm Tuesday, August 28, 2007

PDE Seminar: Single and multitransition solutions for a class of nonlinear elliptic PDE's

by Paul Rabinowitz (University of Wisconsin, Madison) in Skiles 255

The equation -(Lapacian)u + F_u(x,u) = 0, x in R^n where F is 1-periodic in its arguments, provides a simple model for the study of phase transition behavior. We will survey recent existence results for solutions displaying transition behavior, namely spatial homoclinics and heteroclinics, and will discuss the methods used to obtain these results.

3:00 pm Thursday, August 30, 2007

Special PDE Seminar: Exponentially Decaying Solutions of Schrodinger Equations

by Helena McGahagan (University of California, Santa Barbara) in Skiles 255

Are there solutions of the one-dimensional variable coefficient Schrodinger equation that are square integrable with an exponential weight? If we consider the initial value problem, there is certainly data for which the solution does not have this exponential decay at any later time. Instead, this talk will show how to construct such solutions by solving a non-standard boundary value problem, as well as discuss why we might want to do so! This construction relies strongly on a new commutator estimate for the projections onto the positive and negative frequencies.

3:05 pm Thursday, August 30, 2007

Graph theory seminar: Progress on removable path conjectures

by Paul Wollan (University of Hamburg) in Skiles Conference Room 114

Lovasz has conjectured the following: there exists a function f(k) such that in every f(k)-connected graph G and for every pair of vertices u and v in G, there exists a path P from u to v such that G - V(P) is k-connected. After quickly surveying previous partial results on the conjecture, we focus our attention on two weaker questions. In the first, one attempts to find many internally disjoint paths connecting any pair of vertices such that deleting each path leaves the graph connected. In the second, we consider a conjecture of Kriesell: that there exists a function g(k) such that in every g(k)-connected graph G and for every pair of vertices u and v in G, there exists a u-v path P such that G - E(P) is k-connected. We outline a proof showing that Kriesell's conjecture is true and conclude with a discussion of several open questions that arise from the proof of Kriesell's conjecture that may yield progress on Lovasz' conjecture.

3:05 pm Thursday, August 30, 2007

Stochastic Seminar: Some Exponential Inequalities for Bernoulli Sums

by C. Houdré (School of Mathematics, Georgia Tech) in Skiles 269

Generic lower and upper exponential tails inequalities are obtained for sums of Bernoulli random variables. These are compared to known estimates and some particular cases are further studied. Applications to random graphs and to additive combinatorics will be briefly presented.

2:00 pm Friday, August 31, 2007

Geometry Topology Working Seminar: Residually finite groups

by Igor Belegradek (Georgia Tech) in Skiles 255

A group is residually finite if can be approximated by finite groups. I shall give some examples and non-examples, explain how residually finite groups are used in geometry, and state one outstanding conjecture. This expository talk should be accessible to anyone who is fluent with the concept of a group and has some interest in geometry.

3:00 pm Friday, August 31, 2007

Combinatorics Seminar: Foster coefficients and the Jacobian of a metric graph

by Xander Faber (Math, Columbia University, New York) in Skiles 255

For a weighted graph G, let F(e) be the proportion of weighted spanning trees that fail to contain the edge e. In 1949, R.M. Foster discovered that the sum of the quantities F(e) over all edges of the graph is equal to the integer #(edges) - #(vertices) + 1, a well-known topological invariant of the graph. (His original result and proof were formulated in the language of circuit theory.) A construction inspired by the theory of Riemann surfaces and Arakelov intersection theory gives a novel interpretation of the Foster coefficients F(e). I will introduce the Jacobian of a metric graph, relate it to the more widely known Jacobian group of an unweighted graph, and describe how the Foster coefficients arise naturally in this context. This is joint work with Matt Baker.

3:00 pm Tuesday, September 4, 2007

PDE Seminar: On Contact waves for Jin-Xin Relaxation model

by Ronghua Pan (Georgia Tech) in Skiles 255

We first construct contact waves for Jin-Xin relaxation model. Such wave is approximating the coresponding contact discontiuity of equilibrium conservation laws. We then prove the contact waves are nonlinear stable under small initial perturbation.

11:00 am Wednesday, September 5, 2007

Mathematical Biology: Eggs to Die For: An Uncertain Future for an Ancient Survivor

by Douglas Peterson [mail] (Warnell School of Forest Resources, UGA) in Skiles 255

3:05 pm Thursday, September 6, 2007

Stochastic Seminar: High Resolution Space-Time Ozone Modeling for Assessing Trends

by Sujit K. Sahu (School of Mathematics, University of Southampton) in Skiles 269

This paper proposes a space-time model for daily 8-hour maximum ozone levels to provide input for regulatory activities: detection, evaluation, and analysis of spatial patterns and temporal trend in ozone summaries. The model is applied to the analysis of data from the state of Ohio which contains a mix of urban, suburban, and rural ozone monitoring sites. The proposed space-time model is auto-regressive and incorporates the most important meteorological variables observed at a collection of ozone monitoring sites as well as at several weather stations where ozone levels have not been observed. This misalignment is handled through spatial modeling. In so doing we adopt a computationally convenient approach based on the successive daily increments in meteorological variables. The resulting hierarchical model is specified within a Bayesian framework and is fitted using MCMC techniques. Full inference with regard to model unknowns as well as for predictions in time and space, evaluation of annual summaries and assessment of trends are presented. This is joint work with Alan Gelfand and David Holland.

2:00 pm Friday, September 7, 2007

Geometry Topology Working Seminar: Embedding Branch Covers I

by John Etnyre (Georgia Tech) in Skiles 269

This will be the first of (hopefully just two) talks about branched covers. In this talk I will mainly concentrate on the basics of the theory and examples. In the next talk I will discuss a simple problem I have been thinking about lately concerning embedding branched covers in vector bundles and various applications.

3:00 pm Friday, September 7, 2007

Geometry Topology Seminar: Aspects of the work of Xiao-Song Lin

by Nathan Habegger (University of Nantes, France) in Skiles 269

Abstract: I'll review my joint work with Xiao-Song Lin (2 papers) on link classification in 3-dimensional Euclidean space as well as how we were both led into Quantum Topology

3:00 pm Friday, September 7, 2007

Combinatorics Seminar: Even cycle and even cut matroids

by Paul Wollan (University of Hamburg, Germany) in Skiles 255

Abstract. Even cycle and even cut matroids are a natural class of binary matroids represented by signed graphs containing graphic and co-graphic matroids. A classic theorem of matroid theory says that every three connected graphic matroid has a unique representation as the cycle matroid of a graph. However, a single even cycle matroid can have arbitrarily many representations as a signed graph. We present a "stabilizer theorem" allowing one to control the number of representations of even cycle matroids under certain conditions. As an application, we present a theorem allowing one to calculate the minimal non-even cycle extensions of a given even cycle matroid, again under certain favorable conditions. We conclude by discussing further applications to a conjecture of Seymour on the forbidden minors for one-flowing binary matroids. This is joint work with Bertrand Guenin and Irene Pivotto

2:00 pm Monday, September 10, 2007

Algebra seminar: The moments and the norm of polynomial in non-commuting variables

by Stavros Garoufalidis [mail] (Georgia Tech) in Skiles 269

Given a polynomial with rational coefficients in non-commuting variables, we can define its moments and its C^*-algebra norm. Both are notoriously hard to compute. Our aim is to prove that the moment generating function is an algebraic function, which in particular implies that the norm of such elements are algberaic, exactly computable numbers. Our proof uses linguistics (regular vs context-free languages), and points to a number of open problems and questions. This is joint work with Jean Bellissard.

4:30 pm Monday, September 10, 2007

CDSNS Colloquium: Quasi-periodic breathers in Hamiltonian networks

by Yingfei Yi [mail] (GA Tech) in Skiles 255

11:05 am Tuesday, September 11, 2007

GaTech Seminar on Mathematical Finance and Financial Engineering: A Brief History of Arbitrage

by A. Gadidov (Kennesaw State University) in Skiles 269

The notion of arbitrage is central in the mathematics of financial markets. In addition, the Fundamental Theorem of Asset Pricing relates the notion of no arbitrage with the existence of an equivalent probability measure, called the risk-neutral measure, under which the stochastic process modeling the asset price is a martingale. In this talk I will present the evolution of this result, from its original form given by Harrison, Kreps and Pliska to the more recent results of Schachermayer and Delbaen. Slide show preview

3:00 pm Tuesday, September 11, 2007

PDE Seminar : Pressureless Euler/Euler-Poisson systems via adhesion dynamics and scalar conservation laws

by Adrian Tudorascu (Georgia Tech) in Skiles 255

We shall describe recent results on the pressureless Euler/Euler-Poisson flow in one spatial dimension. We consider the case of finite, nonnegative initial Borel measures with finite second-order moment, along with continuous initial velocities of at most quadratic growth and finite energy. We prove the time regularity of the solution for pressureless Euler and we obtain that the velocity satisfies the Oleinik entropy condition, as conjecured by previous authors. Our approach is motivated by earlier work of Brenier and Grenier who showed that one dimensional conservation laws with special initial conditions and fluxes are appropriate for studying the pressureless Euler system.

11:00 am Wednesday, September 12, 2007

Mathematical Biology: Decoding Novel Genomes: From Microbiomes to the Eukaryote

by Mark Borodovsky (Department of Biomedical Engineering at Georgia Tech andEmory University and Division of Computational Science and Engineering, College of Computing) in Skiles 255

To ensure standard initial conditions in recent gene finding competitions, the organizers specified training sets of validated eukaryotic genes, so that these sets were supposed to be used by participants for estimating parameters of statistical models, the key components of ab initio gene finding algorithms. I'll present a gene prediction algorithm that does not require a training set for the model parameter estimation. Nevertheless, this algorithm achieves the same or better level of precision in gene identification as an algorithm trained on a sufficiently large training set. With more than 600 eukaryotic genome sequencing projects currently in progress, ab initio gene finders that estimate model parameters directly from anonymous sequence will accelerate the process of identification of proteins encoded in eukaryotic genomes. Another type of challenge is presented by appeared recently in vast amounts metagenomic sequences, the mixtures of short DNA contigs originated from genomes of mostly non-cultivated organisms. Metagenomes are highly fragmented, diverse in nature, and carry larger fractions of sequence irregularities than known complete prokaryotic genomes. Finding gene starts or identifying short and partial genes in metagenomes become much more difficult than in conventional genomes. Given that the length of a individual metagenomic sequence is not sufficient for estimation of model parameters of a gene finding algorithm I'll describe an approach that allows to circumvent this difficulty and determine parameters of a model for accurate gene finding.

3:05 pm Thursday, September 13, 2007

Stochastic Seminar: Malliavin calculus in infinite dimensions

by Yuri Bakhtin (School of Mathematics, Georgia Tech) in Skiles 269

I will talk about Malliavin calculus for infinite-dimensional dynamics with polynomial nonlinearity and additive noise. I will describe conditions that guarantee existence and smoothness of densities for distributions of finite-dimensional projections of the solution. Joint work with Jonathan Mattingly.

4:30 pm Thursday, September 13, 2007

School of Mathematics Colloquium: Mechanisms of Chaos

by Leonid Bunimovich (Georgia Tech) in Skiles 269

Two major discoveries of the last century were the persistence under small perturbations of chaotic and of regular behavior in dynamical systems. In typical dynamical systems though these two types of behavior do coexist. Among the natural questions arising in the studies of such systems with mixed behavior the following ones will be addressed in this talk. 1.What are the mechanisms of chaos "compartible" with its coexistence with regular dynamics? 2.How smooth should be dynamics to make a purely chaotic motion impossible and to force coexistence? 3.What are types of coexistence of chaotic and regular dynamics? The (new) results that will be discussed deal with the studies of billiards, natural and arguably the most visual dynamical systems. These results raise new problems in the theory of dynamical systems and in geometry. The talk will be readily accessible to the students (graduate and undergraduate) who are most welcomed.

2:00 pm Friday, September 14, 2007

Geometry Topology Working Seminar: Embedding Branch Covers II

by John Etnyre (Georgia Tech) in Skiles 269

3:00 pm Friday, September 14, 2007

Combinatorics Seminar: Harmonic morphisms and hyperelliptic graphs

by Matt Baker [mail] (School of Mathematics, Georgia Tech) in Skiles 255

I will define the notion of a harmonic morphism between graphs, and give some examples. I will then present graph-theoretic analogues of the Riemann-Hurwitz formula and other classical results from algebraic geometry. Finally, I will discuss the notion of a "hyperelliptic graph" and give several equivalent characterizations (in terms of involutions, cycle spaces, Jacobians, and harmonic morphisms) of what it means for a 2-edge-connected graph to be hyperelliptic. This is all joint work with Serguei Norine.

1:00 pm Monday, September 17, 2007

ACM: Compressive Sampling for Next-Generation Signal Acquisition

by Justin Romberg (ECE, Georgia Tech) in Skiles 255

From decades of research in signal processing, we have learned that having a good signal representation is key for tasks including compression, denoising, and restoration. The new theory of Compressive Sampling (CS) shows us how a good representation can fundamentally aid us in the acquisition (or sampling) process as well. The CS paradigm can be summarized neatly: the number of measurements (e.g. samples) needed to acquire a signal or image depends more on its inherent information content than on the desired resolution (e.g. number of pixels). The CS theory suggests novel measurement schemes, where instead of making direct observations of the signal, the acquisition device encodes it as a series of random projections. In this talk, we will outline the main theoretical results in CS and discuss how the ideas can be applied in next-generation acquisition devices. We will also discuss recent work on adapting CS theory to actual hardware architectures, and fast algorithms for recovery.

2:00 pm Monday, September 17, 2007

Geometry Topology Seminar: Fibered knots and the Bennequin bound

by John Etnyre (Georgia Tech) in Skiles 269

In the early 80's Bennequin proved that the self-linking number of a transverse knot in the standard contact structure on S^3 was bounded above by minus the Euler characteristic of any Seifert surface for the knot. Eliashberg later proved the same bound in any tight contact manifold. It has been know for quite some time now that this bound is not optimal for many knot types. It turns out there is an elegant interaction between the optimality of the Bennequin inequality for fibered knots and Giroux's work on the relation between open books and contact structures. In this talk I will explain this interaction and give a precise characterization of when the Bennequin bound is optimal for fibered knots.

4:30 pm Monday, September 17, 2007

CDSNS Colloquium: Group Actions on Order-Preserving Skew-Product Semiflows with Applications

by Yi Wang (visiting GA Tech) in Skiles 255

This talk is mainly concerned with the asymptotic symmetry of the order-preserving (OP) skew-product semiflows $\Pi$ under a $G$-group action. It is shown that, for any stable minimal set $K$, there exists a residual set $Y_0$ of the base such that the fibres of $Y_0$ restricted to $K$ are symmetric w.r.t. $G$, provided that $\Pi$ is strongly order-preserving (SOP) and $G$ is compact. In particular, any uniformly stable bounded orbit is asymptotically symmetric. We further investigate the cases that $\Pi$ is OP but no SOP, and $G$ is not compact. The abstract results are then applied to the (asymptotic) symmetry of the stable solutions of nonautonomous parabolic equations on bounded or unbounded domain.

11:05 am Tuesday, September 18, 2007

GaTech Seminar on Mathematical Finance and Financial Engineering: Simple Examples of Arbitrage Opportunities in Binary Fractional Markets

by Christian Houdré (School of Mathematics, Georgia Tech) in Skiles 269

This pedestrian seminar is intended to present some results on how to explicitly construct arbitrage opportunities in fractional markets. In such markets, bond and stock price are geometric fractional Brownian motion (fBm) of index H > 1/2 (or their discretized version), and as well known fBm is not a semimartingale.

# This is part I of a two part lecture. I will start by definining fBm and recalling some of its properties.

# This seminar is based on a paper of Sottinen which appeared in Finance and Stochastics in 2001.

3:00 pm Tuesday, September 18, 2007

PDE Seminar: Initial Boundary Value Problem for Damped Euler Equations

by Kun Zhao (Georgia Tech) in Skiles 255

We construct global $L^{\infty}$ entropy weak solutions to the initial boundary value problem for the damped compressible Euler equations on bounded domain with physical boundaries. Time asymptotically, the density is conjectured to satisfy the porous medium equation and the momentum obeys to the classical Darcy's law. Based on entropy principle, we showed that the physical weak solutions converges to steady states exponentially fast in time. We also proved that the same is true for the related initial boundary value problems of porous medium equation and thus justified the validity of Darcy's law in large time.

4:30 pm Tuesday, September 18, 2007

ACO Distinguished Lecture: Games in Networks

by Eva Tardos (Cornell University) in Tannenbaum Auditorium, Instructional Center

Many large networks operate and evolve through interactions of large numbers of diverse participants. Such networks play a fundamental role in many domains, ranging from communication networks to social networks. In light of these competing forces, it is surprising how efficient these networks are. It is an exciting challenge to understand the success of these networks in game theoretic terms: what principles of interaction lead selfish participants to form such efficient networks? In this talk we present a number of network formation and routing games. We focus on simple games that have been analyzed in terms of the efficiency loss that results from selfishness. In each setting our goal is to quantify the degradation of quality of solution caused by the selfish behavior of users, comparing the selfish outcome to a centrally designed optimum, or comparing outcomes with different levels of cooperation.

10:00 am Wednesday, September 19, 2007

Web Science Lecture Series: Inaugural Lecture: Universal Access to all Human Knowledge

by Brewster Kahle (Digital Librarian, Director and Co-Founder Internet Archive) in TSRB 132 (85 5th St.)

The goal of universal access to our cultural heritage is within our grasp. With current digital technology we can build comprehensive collections, and with digital networks we can make these available to students and scholars all over the world. The current challenge is establishing the roles, rights, and responsibilities of our libraries and archives in providing public access to this information. With these roles defined, our institutions will help fulfill this epic opportunity of our digital age. While digital libraries are springing up in every corner, a recent experience may serve as an example of both advances and hurdles involved in building complete collections. Over the last five years, the Internet Archive has built some of the largest text and moving images collections in existence by developing cost-effective mechanisms of collection, cataloging, and preservation. By recording approximately 40 terabytes of text, 1000 movies, and television output, much of the digital collecting technology has been successfully tested. Some of the challenges encountered involved issues of scale, funding, law, and access. Legal issues, however, remain the main area where more work is needed. To try to accommodate those concerns, three proposals for a legal structure for public access to digital material have emerged. The first proposal is the donation of intellectual property rights to preserves or conservancies in exchange for tax credits. The second proposal is to use interlibrary loan as a legal structure for the exchange of digital material between libraries. The third proposal embraces the lending library concept in which the public can get direct access to digital material from their homes for a limited period. These approaches to building a library system of digital works can achieve the far-reaching goal of universal access to our cultural heritage.

11:00 am Wednesday, September 19, 2007

Mathematical Biology: Stoichiometric producer-grazer models with daphnia-algae experimental data fitting

by Hao Wang (GA Tech) in Skiles 255

One of the simplest predator-prey models that tracks the quantity and the quality of prey is the one proposed by Loladze, Kuang and Elser (2000) (LKE model). In it, the ratio of two essential chemical elements, carbon to phosphorus, C:P, represents prey quality. However, that model does not explicitly track P neither in the prey nor in the media that supports the prey. Here, we extend the LKE model by mechanistically deriving and accounting for P in both the prey and the media. Bifurcation diagrams and simulations show that our model behaves similarly to the LKE model. However, in the intermediate range of the carrying capacity, especially near the saddle-node bifurcation point for the carrying capacity, quantitative behavior of our model is different. We analyze positive invariant region and stability of boundary steady states. We show that as the uptake rate of P by producer becomes infinite, LKE model becomes the limiting case of our model. Furthermore, our model can be readily extended to multiple producers and consumers. In the second part, the 2006 algae-daphnia experiment will be presented. We extend the mechanistical model to a competition model, which is simulated to fit the experimental data both qualitatively and quantitatively. Both the experiment and the model give us competition exclusion because of extreme light conditions. In high light treatment, daphnia lumholtzi outcompetes daphnia pulex; while, in low light treatment, the exclusion result is opposite. We hypothesize that D. lumholtzi has higher requirement for C (energy) while D. pulex has higher requirement for P (nutrient).

12:05 pm Wednesday, September 19, 2007

Research Horizons Seminar: Lieb-Thirring Inequalities

by Michael Loss (School of Mathematics, Georgia Tech) in Skiles 255

Lieb-Thirring inequalities are bounds on sums of eigenvalues of -\delta -V in terms of the potential V. More generally one can try to bound sums of powers of eigenvalues in terms of the potential. In this talk I review this topic, mention some applications, indicate some of the proofs and discuss some open problems.

4:00 pm Wednesday, September 19, 2007

Analysis Seminar: Extensions of Positive Definite Functions on Amenable and on Free Groups

by Mihaly Bakonyi (Department of Mathematics and Statistics, Georgia State University) in Skiles 269

A group for which there exists a left invariant mean is called amenable. If S is a subset of a group G such that S-1=S, then the Cayley graph \Gamma (G,S) of G relative to S has G as its vertex set and {x,y} is an edge if x-1y \in S. We prove that in case G is amenable and \Gamma (G,S) has a certain combinatorial structure, then every positive definite operator-valued function on S can be extended to a positive definite function on G. Several known extension results are obtained as corollaries. In the second part, we show that every positive definite operator-valued function on words of length <= m of the free group with n generators can be extended to a positive definite function on the whole group. Some related results will be presented, including factorization of positive polynomials in noncommutative variables.

12:05 pm Thursday, September 20, 2007

ACO Seminar: Social choice: rationality, aggregation of information, indeterminacy andchaos

by Gil Kalai (Hebrew University (Jerusalem) and Yale University (New Haven)) in Skiles 255

We will consider generalized social welfare functions (SWFs) for N individuals (voters) and M alternatives. Those are functions which associate to every profiles of individual order preference-relations on the alternatives, a social preference relation (which need not be transitive). We will discuss, power, aggregation of information, collective rationality, indeterminacy and chaos in the context of social choice. For this we will consider several properties of social welfare functions and the connections between them. The first property is well known (Condorcet and Arrow). 1) *Irrationality of the social preference*: "Undesirable outcomes can happen": When there are more than 3 alternatives and the SWF is given by the pairwise majority rule the social outcome can be cyclic: This goes back to Condorcet and a famous result by Arrow asserts that the conclusion cannot be avoided unless one voter has all the power (dictatorship). The second property also goes back to Condorcet. 2) *Information aggregation*: the voting rule enables the society to make the right choice with high probability when the voters have weak independent signals on the desirable outcome. Another famous theorem of Condorcet's asserts that this is the case for the majority rule. We prove that aggregation of information is equivalent to diminishing individual *Shapley-Shubik* power. The third property was first considered by McGarvey. 3) *Indeterminacy:* "Everything can happened". Every preference relation can occur as the social preferences, when there are sufficiently many voters. It turns out that this property follows from aggregation of information. (A slightly weaker property requires only that there are no dummies.) We also consider, 4) *Stochastic indeterminacy:* "Everything will happen." If the voters profile is random (chosen uniformly) then every social preference has a probability to occur which is bounded away from zero. It turns out that this property holds if the maximum individual * Banzhaf *power tends to zero. The next step is to describe a dichotomy between noise stable and noise sensitive SWFs. Majority is an example of a noise stable SWF. Noise sensitive SWFs can be described in several equivalent ways one of which is an extreme form of indeterminacy: 5)* **Social chaos* : "Everything is equally likely to happen." For random voter profiles when the number of voters tends to infinity all preference relations occur with approximately the same probability. The talk will be non-technical, self-contained, and with many illustrative examples.

3:05 pm Thursday, September 20, 2007

Stochastic Seminar: Convex comparison inequalities for jump-diffusion processes

by J. C. Breton (School of Mathematics & Universite de La Rochelle) in Skiles 269

In this talk, we gives convex ordering results for random variables admitting a predictable representation in terms of a Brownian motion and a non-necessarily independent jump component. The method uses forward-backward stochastic calculus. We deal also with exponential jump-diffusion processes when the local characteristics of the stochastic logarithms satisfy some comparison inequalities. As an application, we deduce bounds on option prices in markets with jumps.

4:30 pm Thursday, September 20, 2007

ACO Colloquium: Fourier Analysis of Boolean Functions

by Gil Kalai (Hebrew University and Yale University ) in Skiles 269

Boolean functions namely functions f(x_1,x_2,...x_n) where each variable as well as the value of the function itself attain the value 0 or 1. Boolean functions are fundamental objects in combinatorics, complexity theory, probability and other areas. Fourier analysis of Boolean functions plays important role in these areas since the mid eighties. Fourier analysis is related to discrete isoperimetric results, threshold phenomena for probabilistic models such as random graphs and percolations, low complexity classes, hardness of approximation and noise sensitivity. Hypercontractive estimates, namely results asserting that certain operators contract even when considered from 2-norm to p-norm for p>2 play (rather mysteriously) a crucial role. The talk, which will be self-contained, we will discuss some of the developments in this area in a friendly way. Students are encouraged to attend.

1:00 pm Friday, September 21, 2007

Special ACM seminar: Minimal molecular surface

by Shan Zhao (University of Alabama) in Skile 255

In this talk, I will present a novel concept, the minimal molecular surface (MMS), as a new paradigm for the theoretical modeling of biomolecule-solvent interfaces. When a less polar macromolecule is immersed in a polar environment, the surface free energy minimization occurs naturally to stabilize the system, and leads to an MMS separating the macromolecule from the solvent. For a given set of atomic constraints (as obstacles), the MMS is defined as one whose mean curvature vanishes away from the obstacles. Based on the theory of differential geometry, an iterative procedure is proposed to compute the MMS via the mean curvature minimization of molecular hypersurface functions. Extensive numerical experiments, including those with internal and open cavities, are carried out to demonstrate the proposed concept and algorithms. Comparison is given to other space filling models, such as the molecular surface. Unlike the molecular surface, the proposed MMS is typically free of singularities. The application of the MMS to a protein-DNA complex shows that the MMS could provide an indication to DNA-binding specificity.

3:00 pm Friday, September 21, 2007

Algebra-Combinatorics seminar: G-functions and a counterexample to a conjecture of Zeilberger

by Stavros Garoufalidis [mail] (Georgia Tech) in Skiles 255

A holonomic (i.e., $D$-finite, or $P$-recursive) sequence is one that satisfies a linear recursion relation with polynomial coefficients. A multisum sequence is one that is given by a multisum of a proper hypergeometric term. A fundamental theorem of Wilf-Zeilberger states that every multisum sequence is holonomic. For over 15 years, it was accepted as a reasonable conjecture that the converse holds. Our main result is to prove and to explain why the converse does not hold, i.e., that there exist plenty holonomic sequences that are not balanced multisums. Our proof uses $G$-function theory and the quasi-unipotence of the local monodromy around the singularities. As a side bonus, we define a class of holonomic $G$-functions that come from enumerative combinatorics that complement the holonomic $G$-functions that appear in geometry and arithmetic. Time permitting, we will discuss an efficient ansatz for computing the singularities of the holonomic $G$-functions that come from enumerative combinatorics.

2:00 pm Monday, September 24, 2007

Geometry Topology Seminar: New Symplectic 4-Manifolds with Nonnegative Signature.

by Anar Ahmadov (Georgia Tech) in Skiles 269

The geography problem for simply-connected symplectic 4-manifolds with nonnegative signature is poorly understood. In this talk, we construct new families of symplectic 4-manifolds with nonnegative signature that are interesting with respect to the geography problem. In particular, we construct an irreducible symplectic 4-manifold that is homeomorphic to mCP^2#m(-CP^2) for each odd integer m \geq 49.

3:30 pm Monday, September 24, 2007

ARC Theory of Computation Colloquium: New Locally Decodable Codes and Private Information Retrieval Schemes

by Sergey Yekhanin (Institute for Advanced Studies, Princeton, NJ) in Klaus Room 1116E

A q-query Locally Decodable Code (LDC) is an error-correcting code that encodes an n-bit message x as a codeword C(x), such that one can probabilistically recover any bit x_i of the message by querying only q bits of the codeword C (x), even after some constant fraction of codeword bits has been corrupted. The goal of LDC related research is to minimize the length of such codes. A q-server private information retrieval (PIR) scheme is a cryptographic protocol that allows a user to retrieve the i-th bit of an n-bit string x replicated between q servers while each server individually learns no information about i. The goal of PIR related research is to minimize the communication complexity of such schemes. We present a novel algebraic approach to LDCs and PIRs and obtain vast improvements upon the earlier work. Specifically, given any Mersenne prime p = 2^t - 1, we design three query LDCs of length Exp(n^{1/t}), for every n. Based on the largest known Mersenne prime, this translates to a length of less than Exp(n^{10^{-7}}), compared to Exp(n^{1/2}) in the previous constructions. We also design 3-server PIR schemes with communication complexity of O(n^{10^{-7}}) to access an n-bit database, compared to the previous best scheme with complexity O(n^{1/5.25}). It has often been conjectured that there are infinitely many Mersenne primes. Under this conjecture, our constructions yield three query locally decodable codes of subexponential length and three server private information retrieval schemes with subpolynomial communication complexity.

Light refreshments will be served before the talk at 2:30 in Klaus 2222

4:30 pm Monday, September 24, 2007

CDSNS Colloquium: Stochastic Cellular Automata as a Model for HIV Virus Dynamics

by Jane Hawkins [mail] (UNC) in Skiles 255

Cellular automata (CA) are discrete symbolic dynamical systems used to model many complex types of behavior. They are in their own right of great interest in dynamical systems and were first studied by von Neumann. We give an overview of the basic properties of CA's with intrinsic randomness, called stochastic CA's. These provide models for some systems that can also be modeled on ODE's and as a particular example we look at a CA model used by biologists to study the spread of HIV virus in the lymph node. We focus our discussion on the mathematical properties of the model but refer back to primary biological sources for supporting evidence.

11:05 am Tuesday, September 25, 2007

POSTPONEDGaTech Seminar on Mathematical Finance and Financial Engineering: Simple Examples of Arbitrage Opportunities in Binary Fractional Markets II

by Christian Houdré ((School of Mathematics, Georgia Tech) in Skiles 269

This pedestrian seminar is intended to present some results on how to explicitly construct arbitrage opportunities in fractional markets. In such markets, bond and stock price are geometric fractional Brownian motion (fBm) of index H > 1/2 (or their discretized version), and as well known fBm is not a semimartingale. This is part II of a two part lecture.

3:00 pm Tuesday, September 25, 2007

PDE Seminar: Existence and regularity of solutions to shock reflection problem

by Mikhail Feldman (University of Wisconsin, Madison) in Skiles 255

We show existence of global solutions to regular shock reflection problem for potential flow. We reduce the shock reflection problem to a free boundary problem for a nonlinear elliptic equation, with ellipticity degenerate near a part of the fixed boundary (the sonic line), and discuss the methods and estimates used to solve this problem. We also discuss optimal regularity of solutions near the sonic line.

10:30 am Wednesday, September 26, 2007

Discrete Optimization Seminar: Mingling: Mixed-Integer Rounding with Bounds

by Alper Atamturk (University of California-Berkeley) in Executive Classroom, main ISyE building

Mixed-integer rounding (MIR) is a simple, yet powerful procedure for generating valid cuts for mixed-integer programming. MIR cuts tend to work better for problems with unbounded integer variables. Computational experience suggests that for problems with 0-1 variables, cuts based on lifting techniques are more effective. This is not surprising because lifting techniques make explicit use of the bounds on the variables, whereas the MIR procedure does not. In this talk we will describe a simple procedure, which we call mingling, for incorporating variable bound information into mixed- integer rounding. By explicitly using the variable bounds, the mingling procedure leads to strong inequalities for mixed-integer sets with bounded variables. Joint work with Oktay Gunluk.

11:00 am Wednesday, September 26, 2007

Mathematical Biology & Ecology Seminar: A Large Deviation Principle for random trees with applications to RNA secondary structure

by Yuri Bakhtin (School of Mathematics, Georgia Tech) in Skiles 255

RNA secondary structure is often modeled by plane trees. In this talk the set of plane trees will be endowed with the Gibbs distribution associated to a certain energy function. For this model, I will give a Large Deviation Principle with an explicit rate function. I will show that a notion of typical configuration makes sense and discuss the implications for RNA. In particular, a typical configuration is not the one that minimizes the additive free energy. Joint work with Christine Heitsch.

12:00 pm Wednesday, September 26, 2007

Research Horizons Seminar: Music, Time-Frequency Shifts, and Linear Independence

by Chris Heil (Georgia Institute of Technology) in Skiles 255

Fourier series provide a way of writing almost any signal as a superposition of pure tones, or musical notes. But this representation is not local, and does not reflect the way that music is actually generated by instruments playing individual notes at different times. We will discuss Fourier series, and then present time-frequency representations, which are a type of local Fourier representation of signals. This gives us a mathematical model for representing music. While the model is crude for music, it is in fact a powerful mathematical representation that has appeared widely throughout mathematics (e.g., partial differential equations), physics (e.g., quantum mechanics), and engineering (e.g., time-varying filtering). We ask one very basic question: are the notes in this representation linearly independent? This seemingly trivial question leads to surprising mathematical difficulties.

3:00 pm Wednesday, September 26, 2007

Physics Colloquium: Icicles, washboard road and meandering syrup

by Stephen Morris (Physics, U. Toronto) in Physics Bldg., Lecture Room 5

This talk will describe three recent experiments on emergent patterns in three diverse physical systems. The overall shape and subsequent rippling instability of icicles is an interesting free-boundary growth problem. It has been linked theoretically to similar phenomena in stalactites. We attempted (with limited success) to grow icicles and determine the motion of their ripples. Washboard road is the result of the instability of a flat granular surface under the action of rolling wheels. The rippling of the road, which is a major annoyance to drivers, sets in above a threshold speed and leads to waves which travel down the road. We studied these waves, which have their own interesting dynamics, both in the laboratory and using 2D molecular dynamics simulation. A viscous fluid, like syrup, falling onto a moving belt creates an novel device called a "fluid mechanical sewing machine". The belt breaks the rotational symmetry of the rope-coiling instability, leading to a rich zoo of states as a function of the belt speed and nozzle height. (Work done with Stuart Dalziel, Nicolas Taberlet, Jim McElwaine, Jon Dawes, John Lister, Sunny Chiu-Webster and various other people at DAMTP, Cambridge University.)

4:30 pm Wednesday, September 26, 2007

Analysis Seminar: Discrete Radon transforms and applications to additive number theory

by Alex Iosevich (U Missouri) in Skiles 269

The rotating planes operator in ${\Bbb R}^d$, $$ \int_{x \cdot y=t} f(y) \psi(y)dy$$ is a classical example of an operator satisfying the Phong-Stein rotational curvature condition. We will show that an analog of this operator is still "smoothing", in an appropriate sense, in vector spaces over finite fields. We shall then apply these estimates to sum-product type results in finite fields, improving earlier results due to Glibichuk, Konyagin and others.

12:05 pm Thursday, September 27, 2007

Graph theory seminar: Bounding Minimum Feedback Arc Sets by Girth

by Blair D. Sullivan (Princeton University) in Skiles 255

Given a directed graph G with girth at least m+1 (and no parallel edges), let b(G) denote the size of the smallest subset X of E(G) so that G-X has no directed cycles, and let g(G) be the number of non-edges. Prior joint work with Maria Chudnovsky and Paul Seymour showed that when m=3, b(G)<=g(G), and we conjectured b(G)<=g(G)/2. Can one say anything stronger if m>3? In this talk, I will discuss a new conjecture giving a ratio between b(G) and g(G), namely b(G)<=2g(G)/(m^2-m-1)$, for m>2. The talk will also cover two new results in this direction: the bound b(G)<=g(G)/3 when m=4, and for circular interval graphs, a generalization of previous methods which gives a new bound for all m.

3:05 pm Thursday, September 27, 2007

Stochastic Seminar: Brownian Motion on Manifolds and Morse Theory

by I. Popescu (School of Mathematics, Georgia Tech) in Skiles 269

Given a Morse function on a manifold, Morse theory constructs a complex associated to the function which has the same cohomology as the De Rham complex. Witten introduced a deformation (depending a parameter in [0,infinity)) of the De Rham complex which at 0 is just De Rham complex and at infinity is the Morse complex. Using probability theory we can interpret the heat kernels of the so called Witten Laplacian and show how one can recover and eventually extend some of the results in Morse theory.

4:30 pm Thursday, September 27, 2007

Stelson Lecture: Energy-Driven Pattern Formation

by Robert V. Kohn (Courant Institute, New York University) in Student Success Center (Clary Theatre)

Energy-driven pattern formation is difficult to define, but easy to recognize. I'll discuss two examples: (a) cross-tie wall patterns in magnetic thin films, and (b) surface-energy-driven coarsening of two-phase mixtures. The two problems are rather different -- the first is static, the second dynamic. But they share certain features: in each case nature forms complex patterns as it attempts to minimize a suitable "free energy". The task of modeling and analyzing such patterns is a rich source of challenges -- many still open -- in the multidimensional calculus of variations.

Reception will follow.

2:00 pm Friday, September 28, 2007

Stelson Lecture: Parabolic PDE's and deterministic games

by Robert V. Kohn (Courant Institute, New York University) in Skiles 255

We usually think of parabolic partial differential equations and first-order Hamilton-Jacobi equations as being quite different. Parabolic equations are linked to random walks, and often arise as steepest-descents; Hamilton-Jacobi equations have characteristics, and often arise from optimal control problems.

In truth, these equations are not so different. I will discuss recent work with Sylvia Serfaty, which provides deterministic optimal-control interpretations of many parabolic PDE. In some cases -- for example motion by curvature -- the optimal control viewpoint is very natural, geometric, and easy to understand. In other cases -- for example the linear heat equation -- it seems a bit less natural, and therefore even more surprising.

4:00 pm Friday, September 28, 2007

Special Seminar (Combinatorics, Algebra, Number theory): Polynomial families that meet the Davenport-Mason bound

by Mark Watkins in Skiles 255

Abstract: Hall's conjecture is a famous number theory problem that asks how small $|x^3-y^2|$ can be (if nonzero), with the conjecture being that it at least size roughly $\sqrt x$. The polynomial version of this is a theorem --- given polynomials $(F,G)$ of degrees $(2m,3m)$, the degree of $F^3-G^2$ must be at least $m+1$. We can then ask how often this minimal behaviour occurs, a question that was largely answered by Stothers using Belyi functions and relating simultaneous conjugacy classes in $S_n$ to the Catalan numbers. For instance, for $m=10$ we get $442={18!\over 9!\,11!}$ solutions, which we expect to be conjugate over a number field of the same degree. Our work (joint with N. D. Elkies) has two main directions. First we try to exhibit these solutions explicitly. This has already been done for many small cases, and using multidimensional $p$-adic Newton iteration, we are able to find a few more, reaching a field of degree 32 in one case. The fields we obtain are possibly of interest because they are only ramified at small primes (via a theorem of Beckmann). Secondly, we investigate the "Pell" version of this problem, where $F^3-QG^2$ is of degree $m$ with $(F,G,Q)$ of degrees $(2m,3m-1,2)$.

2:00 pm Monday, October 1, 2007

Geometry Topology Seminar: Knotted surfaces in 4-manifolds

by Tom Mark (University of Virginia ) in Skiles 269

Some time ago Fintushel and Stern introduced a technique they called "rim surgery" for changing the embedding of a smooth surface S in a closed 4-manifold X. Using Seiberg-Witten invariants they proved that when X is symplectic, S is a symplectic surface representing a homology class of nonnegative square, and X and X-S are simply connected, rim surgery produces infinite families of smooth surfaces S' that are smoothly knotted in the sense that the pairs (X,S') and (X,S) are topologically but not smoothly equivalent. We describe a new proof of this fact that uses Ozsvath-Szabo invariants, and has the advantage of applicability in somewhat broader circumstances. In particular, the assumption that S represents a class of nonnegative square may be removed.

4:30 pm Monday, October 1, 2007

ACO Colloquium: Iterative Methods in Combinatorial Optimization

by Mohit Singh (Tepper School of Business, Carneigie Mellon University) in Klaus 1116E

Linear programming has been a successful tool in combinatorial optimization to achieve polynomial time algorithms for problems in P and also to achieve good approximation algorithms for problems which are NP-hard. We demonstrate that iterative methods give a general framework to analyze linear programming formulations of combinatorial optimization problems. We show that iterative methods are well-suited for problems in P and lead to new proofs of integrality of linear programming formulations for various problems in P. This understanding provides us the basic groundwork to address various problems that are NP-hard and to achieve good approximation algorithms. In this talk, we focus on degree bounded network design problems. The most well-studied problem in this class is the Minimum Bounded Degree Spanning Tree problem. We present a polynomial time algorithm that returns a spanning tree of optimal cost such that the degree of any vertex in the tree exceeds its degree bound by at most an additive one. This generalizes a result of Furer and Raghavachari to weighted graphs, and thus settles a 15-year-old conjecture of Goemans affirmatively. This is essentially the best possible result for this problem. For degree constrained versions of more general network design problems, we obtain strong bi-criteria approximation algorithms using the iterative method.

4:30 pm Monday, October 1, 2007

CDSNS Colloquium: Quenching of Unsteady Vortex Breakdown

by John Lopez [mail] (Arizona State University) in Skiles 255

Vortex breakdown is a phenomenon inherent to many practical problems, such as leading-edge vortices on aircrafts, atmospheric tornadoes, and flame-holders in combustion devices. The breakdown of these vortices is associated with the stagnation of the axial velocity on the vortex axis and the development of a near-axis recirculation zone. For large enough Reynolds number, the breakdown can be time-dependent. The unsteadiness can have serious consequences in some applications, such as tail-buffeting in aircrafts flying at high angles of attack. There has been much interest in controlling the vortex breakdown phenomenon, but most efforts have focused on either shifting the threshold for the onset of steady breakdown or on altering the spatial location of the recirculation zone. There has been much less attention paid to the problem of controlling unsteady vortex breakdown. In this talk, recent results from a combined experimental and numerical investigation of vortex breakdown in an enclosed cylinder will be present in which low amplitude modulations of the rotating endwall that sets up the vortex are used as an open-loop control. As expected, for very low amplitudes of the modulation, variations of the modulation frequency reveals typical resonance tongues and frequency lockings, so that the open-loop control allows us to drive the unsteady vortex breakdown to a prescribed periodicity within the resonance regions. For modulation amplitudes above a critical level that depends on the modulation frequency (but still very low), the result is a periodic state synchronous with the forcing frequency over an extensive range of forcing frequencies. But what is particularly of interest is the spatial form of this forced periodic state: for modulation frequencies less than about twice the natural frequency of the unsteady breakdown, the oscillations of the near-axis recirculation zone are amplified, whereas for modulation frequencies larger than about twice the natural frequency the oscillations of the recirculation zone are quenched, and the near-axis flow is driven to the steady axisymmetric state.

11:05 am Tuesday, October 2, 2007

GaTech Seminar on Mathematical Finance and Financial Engineering: Simple Examples of Arbitrage Opportunities in Binary Fractional Markets II

by Christian Houdré (School of Mathematics, Georgia Tech) in Skiles 269

This pedestrian seminar is intended to present some results on how to explicitly construct arbitrage opportunities in fractional markets. In such markets, bond and stock price are geometric fractional Brownian motion (fBm) of index H > 1/2 (or their discretized version), and as well known fBm is not a semimartingale. This is part II of a two part lecture.

3:00 pm Tuesday, October 2, 2007

PDE Seminar: Asymptotic behavior of infinity harmonic functions near an isolated singularity

by Changyou Wang (University of Kentucky, Lexington) in Skiles 255

I will discuss an asymptotic behavior of the infinity harmonic function near a non-removable isolated singular point. More precisely, it is asymptotically a cone near the singularity. This is a joint work with Yifeng Yu and Ovidiu Savin.

4:30 pm Tuesday, October 2, 2007

ACM seminar (Special time and location for this week): Inverse Boundary Value Problems for Maxwell's Equations

by Gang Bao (Michigan State University) in Skiles 269

Since A. P. Calderon's ground-breaking paper in 1980, inverse boundary value problems have received ever growing attention because of broad industrial, medical, and military applications, such as nondestructive testing, seismic imaging, submarine detections, near-field or subsurface imaging, and medical imaging. Lots of exciting new theorems have been proved about the uniqueness, stability, and range of the inverse problems. However, numerical solution of the inverse problems remains to be challenging since the problems are nonlinear, large-scale, and most of all ill-posed! The severe ill-posedness has thus far limited in many ways the scope of inverse problem methods in practical applications. For instance, on the best mathematically studied inverse conductivity problem, the optimal stability result is a logarithm type estimate. Roughly speaking, in order to obtain one digit numerical reconstruction of the coefficient function, at least ten digit accurate boundary data would be required. In this talk, recent progress of our research group over the past several years in mathematical analysis and computational studies of the inverse boundary value problems for the Helmholtz and Maxwell equations will be reported. I will present a continuation approach based on the uncertainty principle. By using multi-frequency or multi-spatial frequency boundary data, our approach is shown to overcome the ill-posedness for the inverse medium scattering problems. I will also discuss convergence issues for the continuation algorithm and highlight ongoing projects in limited aperture imaging, breast cancer imaging (dispersive medium), and nano optics modeling.

11:00 am Wednesday, October 3, 2007

Mathematical Biology: Biodiversity and Ecosystem functioning: beyond complementarity and positive selection effects

by Lin Jiang [mail] (GA Tech, Biology) in Skiles 255

Human activities have dramatically impacted most natural ecosystems on Earth, leading to unprecedented rates of species extinction around the globe. Driven by concerns that biodiversity loss may impair services provided by ecosystems to the human society, ecologists have recently devoted considerable efforts into understanding the relationship between biodiversity and ecosystem functioning (BEF). Much of this research has focused on the effects of changing species diversity on aggregate community biomass. Many experimental studies on this subject have shown that aggregate community biomass tends to increase as the number of species increases in the community, a phenomenon that can be explained by two mechanisms -- niche complementarity and positive selection effects. Here I argue that it is premature to conclude that there exists a general positive BEF relationship. In particular, strong density compensation and negative selection effects may result in the magnitude of ecosystem functioning being decoupled from the level of species diversity. While the emergence of density compensation often requires multiple generations? competitive interactions, existing BEF experiments, conducted mostly in perennial plant communities, rarely lasted for one full generation. Also, given the common presence of keystone species which, despite their low abundance or biomass, can impose large influences on ecosystem functions, negative selection effects are likely to be common when examining ecosystem functions other than aggregate community biomass. Results from experiments and meta-analyses are used to support these arguments.

11:30 am Wednesday, October 3, 2007

ARC ThinkTank Seminar: Which Graphs are Extremal?

by Láci Lovász (Eotvos Loránd University) in TSRB Ballroom, 85 5th St.

Consider a problem in extremal graph theory of the following type: find the maximum density of a subgraph F in a graph, where the density of one or more other subgraphs are fixed. More generally, we may want to maximize some linear combination of densities of various graphs. In almost all cases when the answer is known, the extremal graph has a finite structure, at least asymptotically: the nodes can be partitioned into subsets with given proportions, and the subgraphs between these classes are quasirandom with given densities. Is this always so? To get a cleaner formulation of this, we formulate the question in terms of limits of growing graph sequences, which can be described by 2-variable measurable symmetric functions from [0,1]^2 to [0,1]. The density of any finite simple graph in a such a function can be defined in a natural way. Then the above problem leads to the following: which functions are "finitely forcible", i.e., uniquely determined by a finite number of subgraph densities? With Vera Sos we proved that every stepfunction is finitely forcible. Somewhat surprisingly, the converse is not true, and one can find quite general classes of finitely forcible functions (each of which describes the asymptotic structure of an extremal graph). We offer some necessary and some sufficient conditions. A complete characterization seems difficult (but perhaps not hopeless). This is joint work with Balazs Szegedy. For details see, http://www.arc.gatech.edu/firstanniversary.php

4:30 pm Wednesday, October 3, 2007

Analysis Seminar: Lame Operator and its generalizations

by Oleg Chalykh (University of Leeds) in Skiles 269

It is well known since the work of Hermite that the Lam\'e operator $L=-\frac{d^2}{dx^2}+m(m+1)\wp(x)$ has remarkable properties when $m$ is integer; for instance, $L$ has explicit Bloch eigenfunctions parameterized by the points of an algebraic variety. In the 70-s it was discovered that $L$ can be generalized to a large family of the so-called 'finite-gap' operators having similar properties and closely related to the KdV equation. I will discuss how one can generalize $L$ in another direction, replacing it by a Schr\"odinger operator $L=-\Delta+u(x)$ in higher dimension. Our approach uses monodromy and differential Galois theory and it is already useful in dimension one. This is based on my joint work with Pavel Etingof and Alexei Oblomkov.

3:00 pm Thursday, October 4, 2007

Stochastic Seminar: Approximation of Haar Distributed Matrices and Limiting Distributions of Eigenvalues of Jacobi Ensembles

by Tiefeng Jiang (University of Minnesota) in Skiles 269

I will first present tools to approximate the entries of a large dimensional real and complex Jacobi ensembles with independent complex Gaussian random variables. Based on this, we obtain the Tracy-Widom law of the largest singular values of the Jacobi emsemble. Moreover, the circular law, the Marchenko-Pastur law, the central limit theorem, and the laws of large numbers for the spectral norms are also obtained.

2:00 pm Friday, October 5, 2007

Working Topology seminar: Introduction to the Kontsevich integral

by Thang Le (Georgia Tech) in Skiles 269

We will discuss the Gauss linking number formula and its generalization -- the Kontsevich integral.

4:30 pm Monday, October 8, 2007

CDSNS Colloquium: No meeting - Fall recess

11:00 am Wednesday, October 10, 2007

Mathematical Biology: You say pertussis, I say petussis: the epidemiology of whooping cough in the USA and the UK

by Pej Rohani [mail] (UGA) in Skiles 255

In this talk, I will use whooping cough as a case study in order to demonstrate (i) how models and ecological perspectives can inform epidemiology and (ii) how case notification data can provide invaluable ecological insights. Whooping cough is caused by a bacterium and remains a significant source of childhood mortality responsible for an estimated 300-500,000 annual infant deaths. While pediatric whooping cough immunizations have substantially reduced incidence in the developed world, there is widespread belief that vaccines prevent disease and not transmission, with important implications for potential eradication. The issue is further complicated by documented loss of infection- and vaccine-derived immunity. Using mathematical models and time-series data, I will first examine the epidemiological consequences of vaccination and will attempt to obtain bounds for the period of immunity. I will then briefly explore the spatial pattern of transmission cycles in the US.

12:00 pm Wednesday, October 10, 2007

Research Horizons Seminar: Symplectic Geometry --- the language of classical mechanics

by John Etnyre (Georgia Institute of Technology) in Skiles 255

I will begin this talk by discussing a classical question concerning periodic motions of particles in classical physics. In trying to better understand this question we will develop the notion of a symplectic structure. This is a fundamental geometric object that provides the "right way" to think about classical mechanics, and many many other things too. I will then indicate some recent results (just a few months old) relating to our initial naive questions about periodic motions.

4:00 pm Wednesday, October 10, 2007

Analysis Seminar: Smooth transition densities for Stochastic PDEs. Malliavin calculus in infinite dimensions

by Yuri Bakhtin (School of Mathematics, Georgia Tech) in Skiles 269

Abstract: I will talk about Malliavin calculus for infinite-dimensional dynamics with polynomial nonlinearity and additive noise. I will describe conditions that guarantee existence and smoothness of densities for distributions of finite-dimensional projections of the solution. This is the same talk that I gave at Stochastics Seminar in September. Joint work with Jonathan Mattingly.

1:00 pm Thursday, October 11, 2007

Announcement: Clemson Mini-Conference: Discrete Mathematics and Algorithms

in Clemson University, Clemson, SC

The mini-conference will include about a dozen 40-minute talks from invited speakers talking on any topic they like. About 100 people attend, mostly from the Southeast, but a few come from just about anywhere, including Canada, Europe and even South Africa. The initial list of speakers includes: John Bartholdi (Georgia Tech); Bert Hartnell (St. Mary's University); Kevin Hutson (Furman); Reinhard Laubenbacher (Virginia Tech); Colm Mulcahy (Spelman); Kieka Mynhardt (Victoria); Prasad Tetali (Georgia Tech) http://www.cs.clemson.edu/~goddard/MINI/

12:05 pm Friday, October 12, 2007

Civil and Environmental Engineering: A Novel Approach to the Development of Linear Elliptic Solvers

by Sylvain Nintcheu Fata (Oak Ridge National Laboratory Computer Science and Mathematics Division ) in Mason 142A

This study is concerned with the development of accurate and efficient techniques for solving linear elliptic systems such as the Laplace and Lam equations in three-dimensional domains. The methodology is investigated in the context a Galerkin boundary integral equation approach with both singular and hypersingular formulations. Taking the 3D Laplace equation as the example to discuss the algorithm, surface integrals are defined as limits to the boundary and linear surface elements are employed to approximate the geometry and boundary functions. In the inner integration procedure, it is shown that all singular and non-singular integrals over a triangular boundary element can be expressed exactly as a sum of analytic formulae over the sides of the designated element. It is also established that weakly-singular, strongly-singular and hypersingular integrals are simply special cases of non-singular integrals. In the outer integration scheme, closed-form expressions are obtained for the coincident case, wherein the divergent hypersingular terms are identified explicitly, and shown to cancel with corresponding terms from the edge-adjacent case. The remaining edge-adjacent, vertex-adjacent and non-coincident integrals contain only weak singularities and can be carried out easily by use of suitable numerical cubature. This semi-exact treatment does not seem to suffer from the usual inaccuracies associated with near-singular (or quasi-singular) integrals; i.e., integrals over a pair of triangular elements that are "very close". The semi-analytic method is further accelerated by a Fast-Fourier-Transform-based matrix compression technique to achieve a significant reduction in execution time and memory requirements over standard approaches. The performance of the proposed algorithm is illustrated with several numerical examples.

2:00 pm Friday, October 12, 2007

Working Geometry/Topology seminar: Introduction to the Kontsevich integral

by Thang Le (GaTech) in Skiles 269

I will recall the definition of the Kontsevich integral, give some calculation examples, and explain why the Kontsevich integral is the holonomy of a ``flat non-commutative connection".

1:30 pm Monday, October 15, 2007

ARC Colloquium: Algorithmic and topological complexity of semi-algebraic sets

by Saugata Basu (School of Mathematics, Georgia Tech) in Klaus 1116E

Semi-algebraic sets are subsets of {\mathbb R}^n defined in terms of a finite number of real polynomial equalities and inequalities. They have important applications in many areas of science and engineering. Moreover, they satisfy important finiteness properties. A semi-algebraic set can have only finitely many connected components, finite Betti numbers etc. It is in fact possible to bound the Betti numbers, as well as the number of topological types, of semi-algebraic sets in terms of the number and the degrees of the polynomials defining them. In this talk I will give a survey of recent results on obtaining tight bounds on the Betti numbers, as well as on the number of homotopy types, of semi-algebraic sets in terms of the parameters mentioned above. I will also describe how some of these results generalize to the more abstract setting of definable sets in any o-minimal structure. Finally, time permitting, I will describe recent progress in developing efficient algorithms for computing the Betti numbers of semi-algebraic sets. No background in semi-algebraic geometry or topology will be assumed. Certain parts of this work are joint with R. Pollack, M-F. Roy and (separately) with N. Vorobjov.

Light refreshments will be served after the talk in Room 2222

2:00 pm Monday, October 15, 2007

Topology Seminar: Renormalized quantum invariants

by Nathan Geer (GaTech) in Skiles 269

In this talk I will discuss a renormalization of the Reshetikhin-Turaev quantum invariants, by "fake quantum dimensions." In the case of simple Lie algebras these "fake quantum dimensions" are proportional to the genuine quantum dimensions. More interestingly I will discuss two examples where the genuine quantum dimensions vanish but the "fake quantum dimensions" are non-zero and lead to non-trivial link invariants. The first of these examples is a class of invariants arising from Lie superalgebras defined by Patureau-Mirand and myself. These invariants are multivariable and generalize the multivariable Alexander polynomial. The second example, is a hierarchy of invariants arising from nilpotent representations of quantized sl(2) at a root of unity. These invariants contain Kashaev's quantum dilogarithm invariants of knots. This is joint work with Bertrand Patureau-Mirand and Vladimir Turaev.

3:00 pm Monday, October 15, 2007

Center for Biologically Inspired Design Seminar: Optimal flexibility in flapping appendages

by Silas Alben (School of Mathematics, Georgia Tech) in ES&T L1205

When oscillated in a fluid, appendages such as insect wings and fish fins can produce large thrust forces while undergoing considerable bending. Can we understand these bending patterns by comparing them with the patterns which produce maximum thrust, or a given thrust at maximum efficiency? We present a general model for how flexible surfaces produce vorticity and bend actively and passively in a fluid. We solve the model numerically, and discuss results for moderate deflections (relevant for large thrust), and for small deflections (relevant for high efficiency). We'll then consider how a fish-fin-like structure might be designed for optimal performance.

A reception will follow.

4:30 pm Monday, October 15, 2007

CDSNS Colloquium: A KAM Theorem with Applications to Partial Differential Equations of Higher Dimensions

by Xiaoping Yuan (Fudan University) in Skiles 255

There have been many results on periodic solutions of nonlinear partial differential equations. As an example, let us consider nonlinear wave equation (NLW): $$u_{tt}-u_{xx}+V(x)u+h.o.t.=0\qquad u(t,0)=u(t,\pi)=0.$$ Using variational method, Robinowitz showed that there was a periodic solution of period $T$ with $T/\pi\in\Bbb Q.$ Here are two questions. {\it (1). A natural question is that what happens when $T/\pi\in\Bbb R\setminus\Bbb Q$.} In this case, the compactness condition is not available any more. (2). In geometric view of point, the periodic solution can be regarded as an invariant closed curve, or invariant torus of Dimension $1$. Therefore, {\it another question is whether or not there is a or many invariant tori of dimension $N$ with $N>1$.} Here arises the small divisor problem. The KAM theory of infinite dimension, due to Kuksin, can answer those questions in some sense. If we consider PDEs of higher spatial dimensions, new difficult arises. In this talk, I will give out a brief history of infinitely dimensional KAM theory with application to PDEs, and present a recent KAM theorem, due to myself, which can be applied to higher spatial dimensional PDEs, such nonlinear wave equations and Schr\"odinger equations.

3:00 pm Tuesday, October 16, 2007

PDE Seminar: Steady Water Waves: Theory and Computation

by Walter Strauss (Brown University) in Skiles 255

Consider a classical 2D gravity wave (studied by Euler, Poisson, Cauchy, Airy, Stokes, Levi-Civita,...) with an arbitrary vorticity function. Assume such a wave is traveling at a constant speed over a flat bed. Using local and global bifurcation theory, one can prove that there exist many such waves of large amplitude. I will outline the existence proof and also exhibit some recent computations of the waves using numerical continuation. The computations illustrate certain relationships between the amplitude, energy and mass flux of the waves. If the vorticity is sufficiently large, the first stagnation point of the wave occurs not at the crest (as with the much-studied irrotational flows) but on the bed directly below the crest or else in the interior of the fluid.

4:30 pm Tuesday, October 16, 2007

ACO Colloquium: Van der Waerden/Schrijver-Valiant like Conjectures and Stable Homogeneous Polynomials: One Theorem for All

by Leonid Gurvits (Los Alamos National Labs) in Skiles 255

http://www.math.gatech.edu/news/seminars/abstracts/gurvits.pdf

11:00 am Wednesday, October 17, 2007

Mathematical Biology: Mechanistic models for the scaling of plant form and funtion

by Chuck Price [mail] (GA Tech (Biology)) in Skiles 255

Recent advances in metabolic scaling theory have highlighted the importance of vascular network geometry in understanding the integration and scaling of whole-plant form and function. Additional work has demonstrated general scaling relationships for many leaf traits. However, it is unclear if a common theoretical framework can reveal the general rules underlying much of the variation observed in scaling relationships at the whole-plant and leaf level. Here we present an extension of the general model introduced by West, Brown and Enquist, that predicts scaling relationships in whole plants and leaves. Specifically, the model shows how the exponents describing how whole plant and leaf dimensions (surface area, length and diameter) should change with increasing mass (or with one another) as plants and leaves meet the demands of network flow in varying environments. As a result, our expanded network model predicts a highly constrained continuum of biological exponents that covary according to specific quantitative functions. Compilations of allometric data for a wide variety of plant taxa strongly support the extended models predictions. Together, our results (i) underscore the importance of network geometry in generating the diversity of allometric scaling relationships in biology, (ii) support the hypothesis that while selection has primarily acted to minimize the scaling of hydrodynamic resistance and, (iii) additional selection pressures for alternative branching geometries apparently govern much of the observed co-variation in intraspecific and interspecific scaling exponents. Further, our model provides a general theory to understand the origin and variation of many allometric trait correlations within complex integrated phenotypes.

12:00 pm Wednesday, October 17, 2007

Research Horizons Seminar: Rare Events and Action Minimization

by M. G. Westdickenberg (Georgia Institute of Technology) in Skiles 255

When a system is perturbed by random noise, unexpected things can happen. Even a small noise term will, given enough time, lead to behavior that would never be seen in the deterministic setting. Large deviation theory is a mathematical theory that answers some questions about stochastically-driven rare events in the limit of small noise. At its heart is the large deviations action functional, through which questions about a stochastic equation reduce to solving a deterministic calculus of variations problem. As a particular example, we will consider the action functional associated to the Allen-Cahn PDE, where an interesting sharp-interface limit emerges.

3:00 pm Thursday, October 18, 2007

Analysis Seminar: Moment inequalities for equilibrium measures in the plane.

by Al Baernstein (Washington University, Saint Louis) in Skiles 269

Abstract: The equilibrium measure of a compact plane set gives the steady state distribution of charges on the conductor. We show that certain moments of this equilibrium measure, when taken about the electrostatic centroid and depending only on the real coordinate, are extremal for an interval centered at the origin. This has consequences for means of zeros of polynomials and for means of critical points of Green functions. We also study moments depending only on the distance from the centroid, such as the electrostatic moment of inertia.

4:30 pm Thursday, October 18, 2007

ACO Colloquium: The power of quantum systems on a line

by Sandy Irani (Computer Science Department, UC Irvine) in Skiles 269

In this talk, we discuss the computational strength of finite dimensional quantum particles arranged on a line. We prove that adiabatic computation is equivalent to standard quantum computation even when the adiabatic quantum system is restricted to 2-local interactions of nearest neighbors on a line. The particles in this construction require 9 states per particle. We then adapt this construction to show that the 2-local Hamiltonian for 12 state particles on a line is QMA-complete. QMA is the quantum analog of NP. This result contrasts with the classical analog in which one-dimensional Max-2-SAT with nearest neighbor constraints is in known to be in P. Similar results were obtained independently by Aharonov, Gottesman and Kempe. The work appears jointly in FOCS 2007.

***Refreshments in Skiles 236 at 4PM***

2:00 pm Friday, October 19, 2007

Working Geometry-Topology Seminar: Solenoids as Inverse Limits of Branched Manifolds

by Ian Palmer (Ga Tech) in Skiles 269

A tiling space is the orbit closure of a collection of tilings of R^n. Under certain conditions (finite local complexity and repetitivity) these spaces are compact foliated spaces (known as solenoids) whose leaves are a copy of R^n but with Cantor set transversals. They inherit the translation action of R^n and all the leaves are dense in the solenoid, so they are minimal dynamical systems. The tilings themselves can be approximated by branched manifolds - loosely speaking, CW-complexes with a well-defined smooth structure. The tiling space solenoid will be shown to be the inverse limit of a sequence of branched manifold approximants in a way that is consistent with the dynamics.

1:00 pm Monday, October 22, 2007

ACM: Hierarchical reconstruction for DG, Central DG and finite volume schemes

by Yingjie Liu (Georgia Tech, School of Mathematics) in Skiles 255

Motivated by the moment limiter of Biswas, Devine and Flaherty [Appl. Numer. Math. 14(1994)], we develop a general non-oscillatory hierarchical reconstruction (HR) procedure for removing the spurious oscillations in the high degree polynomial of a cell computed by the central DG scheme. This procedure is fully multidimensional and can be applied to any shape of cells at least in theory. HR uses the most compact stencil and thus fits very well with DG. Furthermore, it doesn't need any characteristic decomposition even for very high order,