Seminars and Colloquia by Series

Knots in contact 3-manifolds

Series
Geometry Topology Seminar
Time
Friday, October 10, 2008 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Vera Vertesi – School of Mathematics, Georgia Tech
In this talk I will give a purely combinatorial description of Knot Floer Homology for knots in the three-sphere (Manolescu-Ozsvath-Szabo- Thurston). In this homology there is a naturally associated invariant for transverse knots. This invariant gives a combinatorial but still an effective way to distinguish transverse knots (Ng-Ozsvath-Thurston). Moreover it leads to the construction of an infinite family of non-transversely simple knot-types (Vertesi).

The giant component in a random subgraph of a given graph

Series
Combinatorics Seminar
Time
Thursday, October 9, 2008 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Lincoln Lu – University of South Carolina
We consider a random subgraph G_p of a host graph G formed by retaining each edge of G with probability p. We address the question of determining the critical value p (as a function of G) for which a giant component emerges. Suppose G satisfies some (mild) conditions depending on its spectral gap and higher moments of its degree sequence. We define the second order average degree \tilde{d} to be \tilde{d}=\sum_v d_v^2/(\sum_v d_v) where d_v denotes the degree of v. We prove that for any \epsilon > 0, if p > (1+ \epsilon)/\tilde{d} then almost surely the percolated subgraph G_p has a giant component. In the other direction, if p < (1-\epsilon)/\tilde{d} then almost surely the percolated subgraph G_p contains no giant component. (Joint work with Fan Chung Graham and Paul Horn)

Hyperbolic geometry and Jones polynomial of embedded graphs

Series
Graph Theory Seminar
Time
Thursday, October 9, 2008 - 12:05 for 1.5 hours (actually 80 minutes)
Location
Skiles 255
Speaker
Roland van der Veen – University of Amsterdam
The aim of this talk is to introduce techniques from knot theory into the study of graphs embedded in 3-space. The main characters are hyperbolic geometry and the Jones polynomial. Both have proven to be very successful in studying knots and conjecturally they are intimately related. We show how to extend these techniques to graphs and discuss possible applications. No prior knowledge of knot theory or geometry will be assumed.

Estimating PageRank on Graph Streams

Series
ACO Student Seminar
Time
Wednesday, October 8, 2008 - 13:30 for 2 hours
Location
Skiles 269
Speaker
Atish Das Sarma – CS/ACO, Georgia Tech
This study focuses on computations on large graphs (e.g., the web-graph) where the edges of the graph are presented as a stream. The objective in the streaming model is to maintain small amount of memory and perform few passes over the data. In the streaming model, we show how to perform several graph computations including estimating the probability distribution after a random walk of certain length l, estimate the mixing time, and the conductance. We can compute the approximate PageRank values in O(nM^{-1/4}) space and O(M^{3/4}) passes (where n is the number of nodes and M is the mixing time of the graph). In comparison, a standard (matrix-vector multiplication) implementation of the PageRank algorithm will take O(n) space and O(M) passes. The main ingredient in all our algorithms is to explicitly perform several random walks of certain length efficiently in the streaming model. I shall define and motivate the streaming model and the notion of PageRank, and describe our results and techniques. Joint work with Sreenivas Gollapudi and Rina Panigrahy from Microsoft Research.

A population model of influenza designed to evaluate projected pandemic vaccine production in Taiwan

Series
Mathematical Biology Seminar
Time
Wednesday, October 8, 2008 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Dr. John Glasser – CDC/CCID/NCIRD

Background: We endeavor to reproduce historical observations and to identify and remedy the cause of any disparate predictions before using models to inform public policy-making. We have no finely age- and time-stratified observations from historical pandemics, but prior exposure of older adults to a related strain is among the more compelling hypotheses for the w-shaped age-specific mortality characterizing the 1918 pandemic, blurring the distinction between annual and pandemic influenza.

Methods: We are attempting to reproduce patterns in annual influenza morbidity and mortality via a cross-classified compartmental model whose age class sojourns approximate the longevity of clusters of closely-related strains. In this population model, we represent effective inter-personal contacts via a generalization of Hethcote's formulation of mixing as a convex combination of contacts within and between age groups. Information about mixing has been sought in face-to-face conversations, a surrogate for contacts by which respiratory diseases might be transmitted, but could also be obtained from household and community transmission studies. We reanalyzed observations from several such studies to learn about age-specific preferences, proportions of contacts with others the same age. And we obtained age-specific forces of infection from proportions reporting illness in a prospective study of household transmission during the 1957 influenza pandemic, which we gamma distributed to correct for misclassification. Then we fit our model to weekly age-specific hospitalizations from Taiwan's National Health Insurance Program, 2000-07, by adjusting a) age-specific coefficients of harmonic functions by which we model seasonality and b) probabilities of hospitalization given influenza.

Results: While our model accounts for only 30% of the temporal variation in hospitalizations, estimated conditional probabilities resemble official health resource utilization statistics. Moreover, younger and older people are most likely to be hospitalized and elderly ones to die of influenza, with modeled deaths 10.6% of encoded influenza or pneumonia mortality.

Conclusions: Having satisfactorily reproduced recent patterns in influenza morbidity and mortality in Taiwan via a deterministic model, we will switch to a discrete event-time simulator and - possibly with different initial conditions and selected parameters - evaluate the sufficiency of projected pandemic vaccine production.

Joint work with Denis Taneri, and Jen-Hsiang Chuang

Navier-Stokes evolutions as self-dual variational problems

Series
PDE Seminar
Time
Tuesday, October 7, 2008 - 15:15 for 1.5 hours (actually 80 minutes)
Location
Skiles 255
Speaker
Nassif Ghoussoub – University of British Columbia, Canada
We describe how several nonlinear PDEs and evolutions ­including stationary and dynamic Navier-Stokes equations­ can be formulated and resolved variationally by minimizing energy functionalsof the form I(u) = L(u, -\Lambda u) + \langle \Lambda u, u\rangle and I(u) = \Int^T_0 [L(t, u(t), -\dot u(t) - \Lambda u(t)) + \langle\Lambda u(t), u(t)\rangle]dt + \ell (u(0) - u(T) \frac{u(T) + u(0)}{2} where L is a time-dependent "selfdual Lagrangian" on state space, is another selfdual "boundary Lagrangian", and is a nonlinear operator (such as \Lambda u = div(u \otimes u) in the Navier-Stokes case). However, just like the selfdual Yang-Mills equations, the equations are not obtained via Euler-Lagrange theory, but from the fact that a natural infimum is attained. In dimension 2, we recover the well known solutions for the corresponding initial-value problem as well as periodic and anti-periodic ones, while in dimension 3 we get Leray solutions for the initial-value problems, but also solutions satisfying u(0) = \alpha u(T ) for any given in (-1, 1). It is worth noting that our variational principles translate into Leray's energy identity in dimension 2 (resp., inequality in dimension 3). Our approach is quite general and does apply to many other situations.

3-manifolds and lagrangian subspaces of character varieties

Series
Geometry Topology Seminar
Time
Monday, October 6, 2008 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
A. Sikora – SUNY Buffalo
W. Goldman proved that the SL(2)-character variety X(F) of a closed surface F is a holonomic symplectic manifold. He also showed that the Sl(2)-characters of every 3-manifold with boundary F form an isotropic subspace of X(F). In fact, for all 3-manifolds whose SL(2)-representations are easy to analyze, these representations form a Lagrangian space. In this talk, we are going to construct explicit examples of 3-manifolds M bounding surfaces of arbitrary genus, whose spaces of SL(2)-characters have dimension as small as possible. We discuss relevance of this problem to quantum and classical low-dimensional topology.

Conformal dimension of self-affine sets and modulus of a system of measures

Series
Analysis Seminar
Time
Monday, October 6, 2008 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Hrant Hakobyan – University of Toronto
A mapping F between metric spaces is called quasisymmetric (QS) if for every triple of points it distorts their relative distances in a controlled fashion. This is a natural generalization of conformality from the plane to metric spaces. In recent times much work has been devoted to the classification of metric spaces up to quasisymmetries. One of the main QS invariants of a space X is the conformal dimension, i.e the infimum of the Hausdorff dimensions of all spaces QS isomorphic to X. This invariant is hard to find and there are many classical fractals such as the standard Sierpinski carpet for which conformal dimension is not known. Tyson proved that if a metric space has sufficiently many curves then there is a lower bound for the conformal dimension. We will show that if there are sufficiently many thick Cantor sets in the space then there is a lower bound as well. "Sufficiently many" here is in terms of a modulus of a system of measures due to Fuglede, which is a generalization of the classical conformal modulus of Ahlfors and Beurling. As an application we obtain a new lower bound for the conformal dimension of self affine McMullen carpets.

The existence of three-dimensional generalized solitary waves

Series
Applied and Computational Mathematics Seminar
Time
Monday, October 6, 2008 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Shengfu Deng – School of Mathematics, Georgia Tech
We consider the three-dimensional gravity-capillary waves on water of finite-depth which are uniformly translating in a horizontal propagating direction and periodic in a transverse direction. The exact Euler equations are formulated as a spatial dynamical system in stead of using Hamiltonian formulation method. A center-manifold reduction technique and a normal form analysis are applied to show that the dynamical system can be reduced to a system of ordinary differential equations. Using the existence of a homoclinic orbit connecting to a two-dimensional periodic solution for the reduced system, it is shown that such a generalized solitary-wave solution persists for the original system by applying a perturbation method and adjusting some appropriate constants.

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