UIUC Dept. of Mathematics Seminar Calendar

     August 2012           September 2012          October 2012    
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  5  6  7  8  9 10 11    2  3  4  5  6  7  8    7  8  9 10 11 12 13
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                        30

Thursday, September 27, 2012

Number Theory Seminar
11:00 am in 241 Altgeld Hall, Thursday, September 27, 2012

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Submitted by ford.

Armin Straub (UIUC Math)
Arithmetic aspects of short random walks
Abstract: We revisit a classical problem: how far does a random walk travel in a given number of steps (of length 1, each taken along a uniformly random direction)? Although such random walks are asymptotically well understood, surprisingly little is known about the exact distribution of the distance after just a few steps. For instance, the average distance after two steps is (trivially) given by $4/\pi$; but what is the average distance after three steps? In this talk, we therefore focus on the arithmetic properties of short random walks and consider both the moments of the distribution of these distances as well as the corresponding density functions. It turns out that the even moments have a rich combinatorial structure which we exploit to obtain analytic information. In particular, we find that in the case of three and four steps, the density functions can be put in hypergeometric form and may be parametrized by modular functions. Much less is known for the density in case of five random steps, but using the modularity of the four-step case we are able to deduce its exact behaviour near zero.

Math/Theoretical Physics Seminar
12:00 pm in 464 Loomis Laboratory, Thursday, September 27, 2012

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Mike Stone (Illinois Physics)
Hawking radiation in the Quantum Hall effect: special functions in Bargmann-Fock space
Abstract: I use the identification of the edge mode of the filling fraction $\nu=1$ quantum Hall phase with a 1+1 dimensional chiral Dirac fermion to construct an analogue model for a chiral fermion in a space-time geometry possessing an event horizon. By solving the model in the lowest Landau level, I show that the event horizon emits particles and holes with a thermal spectrum. The solution involves some classical hypergeometric function theory in Bargmann-Fock space.

Applied Mathematics Seminar
3:00 pm in 241 Altgeld Hall, Thursday, September 27, 2012

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Maxim Arnold (UIUC Math)
Evolution of the shape of asymptotic convex hull of the rapidly exploring random tree
Abstract: Rapidly exploring Random Trees (RRT) have become increasingly popular as a way to explore high-dimensional spaces for problems in robotics, motion planning, virtual prototyping, computational biology, and other fields. It was established that the vertex distribution converges in probability to the sampling distribution. It was also noted that there is a "Voronoi bias" in the tree growth because the probability that a vertex is selected is proportional to the volume of its Voronoi region. I shall explain the evolution of the shape of convex hull of RRT when the size of the search space is increased.

Mathematics Colloquium
4:00 pm in 245 Altgeld Hall, Thursday, September 27, 2012

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Alessio Figalli (University of Texas - Austin)
Stability results for functional inequalities and applications
Abstract: Geometric and functional inequalities play a crucial role in several problems arising in the calculus of variations, partial differential equations, geometry, etc. More recently, there has been a growing interest in studying the stability for such inequalities. The basic question one wants to address is the following: suppose we are given a functional inequality for which minimizers are known. Can we prove that if a function almost attains the equality then it is close (in some suitable sense) to one of the minimizers? The aim of this talk is to describe some ways to attack this kind of problems, and to show some applications. The talk is intended to be accessible to graduate students.