Thursday, January 12, 2012
Job Candidate Talk: Counting closed loops in a stratum of quadratic differentials
Thu, 01/12/2012 - 11:05am, Skiles 006
Kasra Rafi, University of Oklohama, email Organizer: Mohammad Ghomi
In his thesis, Margulis computed the asymptotic growth rate for the number of closed geodesics of length less than R on a given closed hyperbolic surface and his argument has been emulated to many other settings. We examine the Teichmüller geodesic flow on the moduli space of a surface, or more generally any stratum of quadratic differentials in the cotangent bundle of moduli space. The flow is known to be mixing, but the spaces are not compact and the flow is not uniformly hyperbolic. We show that the random walk associated to the Teichmüller geodesic flow is biased toward the compact part of the stratum. We then use this to find asymptotic growth rate of for the number of closed loops in the stratum. (This is a joint work with Alex Eskin and Maryam Mirzakhani.)
Friday, January 13, 2012
Combinatorics Seminar: On Approximating Expansion of Small Sets in Graphs
Fri, 01/13/2012 - 3:00pm, Skiles 005
Prasad Raghavendra, School of Computer Science, Georgia Tech Organizer: Prasad Tetali
A small set expander is a graph where every set of sufficiently small size has near perfect edge expansion. This talk concerns the computational problem of distinguishing a small set-expander, from a graph containing a small non-expanding set of vertices. This problem henceforth referred to as the Small-Set Expansion problem has proven to be intimately connected to the complexity of large classes of combinatorial optimization problems. More precisely, the small set expansion problem can be shown to be directly related to the well-known Unique Games Conjecture -- a conjecture that has numerous implications in approximation algorithms. In this talk, we motivate the problem, and survey recent work consisting of algorithms and interesting connections within graph expansion, and its relation to Unique Games Conjecture.
Tuesday, January 17, 2012
Job Candidate Talk: Coupling and Upscaling of Particle Models in Multiscale Physics
Tue, 01/17/2012 - 11:00am, Skiles 006
Matthew Dobson, NSF Postdoctoral Fellow, Ecole des Ponts ParisTech Organizer: Luca Dieci
Multiscale numerical methods seek to compute approximate solutions to physical problems at a reduced computational cost compared to direct numerical simulations. This talk will cover two methods which have a fine scale atomistic model that couples to a coarse scale continuum approximation. The quasicontinuum method directly couples a continuum approximation to an atomistic model to create a coherent model for computing deformed configurations of crystalline lattices at zero temperature. The details of the interface between these two models greatly affects the model properties, and we will discuss the interface consistency, material stability, and error for energy-based and force-based quasicontinuum variants along with the implications for algorithm selection. In the case of crystalline lattices at zero temperature, the constitutive law between stress and strain is computed using the Cauchy-Born rule (the lattice deformation is locally linear and equal to the gradient). For the case of complex fluids, computing the stress-strain relation using a molecular model is more challenging since imposing a strain requires forcing the fluid out of equilibrium, the subject of nonequilibrium molecular dynamics. I will describe the derivation of a stochastic model for the simulation of a molecular system at a given strain rate and temperature.
Wednesday, January 18, 2012
Research Horizons Seminar: FINITE TIME DYNAMICS: the first steps and outlook.
Wed, 01/18/2012 - 12:05pm, Skiles 005
Leonid A. Bunimovich, Georgia Tech Organizer: Bulent Tosun
It is well known that typically equations do not have analytic (expressed by formulas) solutions. Therefore a classical approach to the analysis of dynamical systems (from abstract areas of Math, e.g. the Number theory to Applied Math.) is to study their asymptotic (when an independent variable, "time", tends to infinity) behavior. Recently, quite surprisingly, it was demonstrated a possibility to study rigorously (at least some) interesting finite time properties of dynamical systems. Most of already obtained results are surprising, although rigorously proven. Possible PhD topics range from understanding these (already proven!) surprises and finding (and proving) new ones to numerical investigation of some systems/models in various areas of Math and applications, notably for dynamical analysis of dynamical networks. I'll present some visual examples, formulate some results and explain them (when I know how).
Analysis Seminar: On the behavior at infinity of solutions to difference equations in Schroedinger form
Wed, 01/18/2012 - 2:00pm, Skiles 005
Lillian Wong, Georgia Tech Organizer: Brett Wick
We offer several perspectives on the behavior at infinity of solutions of discrete Schroedinger equations. First we study pairs of discrete Schroedinger equations whose potential functions differ by a quantity that can be considered small in a suitable sense as the index n \rightarrow \infty. With simple assumptions on the growth rate of the solutions of the original system, we show that the perturbed system has a fundamental set of solutions with the same behavior at infinity, employing a variation-of-constants scheme to produce a convergent iteration for the solutions of the second equation in terms of those of the original one. We use the relations between the solution sets to derive exponential dichotomy of solutions and elucidate the structure of transfer matrices. Later, we present a sharp discrete analogue of the Liouville-Green (WKB) transformation, making it possible to derive exponential behavior at infinity of a single difference equation, by explicitly constructing a comparison equation to which our perturbation results apply. In addition, we point out an exact relationship connecting the diagonal part of the Green matrix to the asymptotic behavior of solutions. With both of these tools it is possible to identify an Agmon metric, in terms of which, in some situations, any decreasing solution must decrease exponentially.This talk is based on joint work with Evans Harrell.
Thursday, January 19, 2012
Stochastics Seminar: Asymptotic behavior for solutions of the random Schrödinger with long-range correlations.
Thu, 01/19/2012 - 3:05pm, skyles 006
Christophe Gomez, Department of Mathematics, Stanford University Organizer: Karim Lounici
In this talk we will describe the different behaviors of solutions of the random Schrödinger with long-range correlations. While in the case of arandom potential with rapidly decaying correlations nontrivial phenomenaappear on the same scale, different phenomena appear on different scalesfor a random potential with slowly decaying correlations nontrivial .
Job Candidate Talk: Recent advances on the structure of metric measure spaces with Ricci curvature bounded from below
Thu, 01/19/2012 - 4:00pm, Skiles 005
Nicola Gigli, University of Nice Organizer: Chongchun Zeng
I'll show how on metric measure spaces with Ricci curvature bounded from below in the sense of Lott-Sturm-Villani there is a well defined notion of Heat flow, and how the study of the properties of this flow leads to interesting geometric and analytic properties of the spaces themselves. A particular attention will be given to the class of spaces where the Heat flow is linear. (From a collaboration with Ambrosio and Savare')
Monday, January 23, 2012
Discrete Mathematical Biology Working Seminar
Mon, 01/23/2012 - 11:00am, Skiles 114
Shel Swenson, Georgia Tech Organizer: Christine Heitsch
A discussion of the paper "Beyond energy minimization: approaches to the kinetic folding of RNA'' by Flamm and Hofacker (2008).
CDSNS Colloquium: A numerical algorithm for the computation of periodic orbits of the Kuramoto-Sivashinsky equation.
Mon, 01/23/2012 - 11:05am, Skiles 006
Jordi Lluis Figueras, Uppsala University Organizer: Rafael de la Llave
In this talk we will present a numerical algorithm for the computation of (hyperbolic) periodic orbits of the 1-D K-S equation u_t+v*u_xxxx+u_xx+u*u_x = 0, with v>0. This numerical algorithm consists on apply a suitable Newton scheme for a given approximate solution. In order to do this, we need to rewrite the invariance equation that must satisfy a periodic orbit in a form that its linearization around an approximate solution is a bounded operator. We will show also how this methodology can be used to compute rigorous estimates of the errors of the solutions computed.
Math Physics Seminar: Parallel heat transport in reverse shear magnetic fields
Mon, 01/23/2012 - 12:05pm, Skiles 006
Daniel Blazevski, University of Texas Organizer: Michael Loss
I will discuss local and nonlocal anisotropic heat transport along magnetic field lines in a tokamak, a device used to confine plasma undergoing fusion. I will give computational results that relate certain dynamical features of the magnetic field, e.g. resonance islands, chaotic regions, transport barriers, etc. to the asymptotic temperature profiles for heat transport along the magnetic field lines.