Seminars and Colloquia Schedule

On asymptotics, structure and stability for multicomponent reactive flows

Series
PDE Seminar
Time
Tuesday, September 8, 2009 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Konstantina Trivisa – University of Maryland, College Park
Multicomponent reactive flows arise in many practical applicationssuch as combustion, atmospheric modelling, astrophysics, chemicalreactions, mathematical biology etc. The objective of this work isto develop a rigorous mathematical theory based on the principles ofcontinuum mechanics. Results on existence, stability, asymptotics as wellas singular limits will be discussed.

Some Problems and Results in Additive Combinatorics.

Series
Research Horizons Seminar
Time
Wednesday, September 9, 2009 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 171
Speaker
Ernie Croot – School of Mathematics, Georgia Tech –
Additive combinatorics is a relatively new field, with many diverse and exciting research programmes. In this talk I will discuss two of these programmes -- the continuing development of sum-product inequalities, and the unfolding progress on arithmetic progressions -- along with some new results proved by me and my collaborators. Hopefully I will have time to mention some nice research problems as well.

A polyhedral study of the mixed integer cut

Series
ACO Student Seminar
Time
Wednesday, September 9, 2009 - 12:00 for 1 hour (actually 50 minutes)
Location
ISyE Executive Classroom
Speaker
Steve Tyber – ISyE, Georgia Tech
In 1969, Gomory introduced the master group polyhedron for pure integer programs and derives the mixed integer cut (MIC) as a facet of a special family of these polyhedra. We study the MIC in this framework, characterizing both its facets and extreme points; next, we extend our results under mappings between group polyhedra; and finally, we conclude with related open problems. No prior knowledge of algebra or the group relaxation is assumed. Terminology will be introduced as needed. Joint work with Ellis Johnson.

Introduction to Sheaf Theory

Series
Other Talks
Time
Wednesday, September 9, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
John Etnyre – Ga Tech
In these talks we will introduced the basic definitions and examples of presheaves, sheaves and sheaf spaces. We will also explore various constructions and properties of these objects.

Grain boundary motion in thin films

Series
Applied and Computational Mathematics Seminar
Time
Wednesday, September 9, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 114
Speaker
Amy Novick-Cohen – Technion
Grain boundaries within polycrystalline materials are known to be governed by motion by mean curvature. However, when the polycrystalline specimen is thin, such as in thin films, then the effects of the exterior surfaces start to play an important role. We consider two particularly simple geometries, an axi-symmetric geometry, and a half loop geometry which is often employed in making measurements of the kinetic coefficient in the motion by mean curvature equation, obtaining corrective terms which arise due to the coupling of grain boundaries to the exterior surface.   Joint work with Anna Rotman, Arkady Vilenkin & Olga Zelekman-Smirin

Simultaneous Asymptotics for the Shape of Young Tableaux: Tracy-Widom and beyond.

Series
Stochastics Seminar
Time
Thursday, September 10, 2009 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Christian Houdré – Georgia Tech

Given a random word of size n whose letters are drawn independently<br />
from an ordered alphabet of size m, the fluctuations of the shape of<br />
the corresponding random RSK Young tableaux are investigated, when both<br />
n and m converge together to infinity. If m does not grow too fast and<br />
if the draws are uniform, the limiting shape is the same as the<br />
limiting spectrum of the GUE. In the non-uniform case, a control of<br />
both highest probabilities will ensure the convergence of the first row<br />
of the tableau, i.e., of the length of the longest increasing<br />
subsequence of the random word, towards the Tracy-Widom distribution.

Viscosity and Principal-Agnet Problem

Series
SIAM Student Seminar
Time
Friday, September 11, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Ruoting Gong – Georgia Tech –
We develop a stochastic control system from a continuous-time Principal-Agent model in which both the principal and the agent have imperfect information and different beliefs about the project. We attempt to optimize the agent’s utility function under the agent’s belief. Via the corresponding Hamilton-Jacobi-Bellman equation we prove that the value function is jointly continuous and satisfies the Dynamic Programming Principle. These properties directly lead to the conclusion that the value function is a viscosity solution of the HJB equation. Uniqueness is then also established.

Counting Independent Sets using the Bethe Approximation

Series
Combinatorics Seminar
Time
Friday, September 11, 2009 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Jinwoo Shin – MIT
We consider the #P complete problem of counting the number of independent sets in a given graph. Our interest is in understanding the effectiveness of the popular Belief Propagation (BP) heuristic. BP is a simple and iterative algorithm that is known to have at least one fixed point. Each fixed point corresponds to a stationary point of the Bethe free energy (introduced by Yedidia, Freeman and Weiss (2004) in recognition of Hans Bethe's earlier work (1935)). The evaluation of the Bethe Free Energy at such a stationary point (or BP fixed point) leads to the Bethe approximation to the number of independent sets of the given graph. In general BP is not known to converge nor is an efficient, convergent procedure for finding stationary points of the Bethe free energy known. Further, effectiveness of Bethe approximation is not well understood. As the first result of this paper, we propose a BP-like algorithm that always converges to a BP fixed point for any graph. Further, it finds an \epsilon approximate fixed point in poly(n, 2^d, 1/\epsilon) iterations for a graph of n nodes with max-degree d. As the next step, we study the quality of this approximation. Using the recently developed 'loop series' approach by Chertkov and Chernyak, we establish that for any graph of n nodes with max-degree d and girth larger than 8d log n, the multiplicative error decays as 1 + O(n^-\gamma) for some \gamma > 0. This provides a deterministic counting algorithm that leads to strictly different results compared to a recent result of Weitz (2006). Finally as a consequence of our results, we prove that the Bethe approximation is exceedingly good for a random 3-regular graph conditioned on the Shortest Cycle Cover Conjecture of Alon and Tarsi (1985) being true. (Joint work with Venkat Chandrasekaran, Michael Chertkov, David Gamarnik and Devavrat Shah)

Hyperbolic structures on surfaces and 3-manifolds

Series
Geometry Topology Working Seminar
Time
Friday, September 11, 2009 - 15:00 for 2 hours
Location
Skiles 269
Speaker
John Etnyre – Georgia Tech
We will discuss how to put a hyperbolic structure on various surface and 3-manifolds. We will being by discussing isometries of hyperbolic space in dimension 2 and 3. Using our understanding of these isometries we will explicitly construct hyperbolic structures on all close surfaces of genus greater than one and a complete finite volume hyperbolic structure on the punctured torus. We will then consider the three dimensional case where we will concentrate on putting hyperbolic structures on knot complements. (Note: this is a 2 hr seminar)

Asymptotic coupling and a weak form of Harris' theorem with applications to stochastic delay equations

Series
Probability Working Seminar
Time
Friday, September 11, 2009 - 15:00 for 2 hours
Location
Skiles 154
Speaker
Sergio Almada – Georgia Tech
The talk is based on the recent paper by M.Hairer, J.Mattingly, and M.Scheutzow with the same title.There are many Markov chains on infinite dimensional spaces whose one-step transition kernels are mutually singular when starting from different initial conditions. We give results which prove unique ergodicity under minimal assumptions on one hand and the existence of a spectral gap under conditions reminiscent of Harris' theorem. The first uses the existence of couplings which draw the solutions together as time goes to infinity. Such "asymptotic couplings" were central to recent work on SPDEs on which this work builds. The emphasis here is on stochastic differential delay equations.Harris' celebrated theorem states that if a Markov chain admits a Lyapunov function whose level sets are "small" (in the sense that transition probabilities are uniformly bounded from below), then it admits a unique invariant measure and transition probabilities converge towards it at exponential speed. This convergence takes place in a total variation norm, weighted by the Lyapunov function. A second aim of this article is to replace the notion of a "small set" by the much weaker notion of a "d-small set," which takes the topology of the underlying space into account via a distance-like function d. With this notion at hand, we prove an analogue to Harris' theorem, where the convergence takes place in a Wasserstein-like distance weighted again by the Lyapunov function. This abstract result is then applied to the framework of stochastic delay equations.