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Series: Other Talks

In this talk Professor Tapia identifies elementary mathematical frameworks for the study of popular drag racing beliefs. In this manner some myths are validated while others are destroyed. Tapia will explain why dragster acceleration is greater than the acceleration due to gravity, an age old inconsistency. His "Fundamental Theorem of Drag Racing" will be presented. The first part of the talk will be a historical account of the development of drag racing and will include several lively videos.

Series: Geometry Topology Seminar

After reviewing a few techniques from the theory of confoliation in dimension three we will discuss some generalizations and certain obstructions in extending these techniques to higher dimensions. We also will try to discuss a few questions regarding higher dimensional confoliations.

Series: CDSNS Colloquium

We study the behavior of the asymptotic dynamics of a dissipative reaction-diffusion equation in a dumbbell domain, which, roughly speaking, consists of two fixed domains joined by a thin channel. We analyze the behavior of the stationary solutions (solutions of the elliptic problem), their local unstable manifold and the attractor of the equation as the width of the connecting channel goes to zero.

Series: Combinatorics Seminar

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)

Friday, September 11, 2009 - 15:00 ,
Location: Skiles 269 ,
John Etnyre ,
Georgia Tech ,
Organizer:

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)

Series: Probability Working Seminar

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.

Series: SIAM Student Seminar

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.

Series: Stochastics Seminar

Given a random word of size n whose letters are drawn independently

from an ordered alphabet of size m, the fluctuations of the shape of

the corresponding random RSK Young tableaux are investigated, when both

n and m converge together to infinity. If m does not grow too fast and

if the draws are uniform, the limiting shape is the same as the

limiting spectrum of the GUE. In the non-uniform case, a control of

both highest probabilities will ensure the convergence of the first row

of the tableau, i.e., of the length of the longest increasing

subsequence of the random word, towards the Tracy-Widom distribution.

Series: Analysis Seminar

We describe how time-frequency analysis is used to analyze boundedness
and Schatten class properties of pseudodifferential operators and
Fourier integral operators.

Wednesday, September 9, 2009 - 13:00 ,
Location: Skiles 114 ,
Amy Novick-Cohen ,
Technion ,
Organizer: John McCuan

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