L6

Given a quantum Hamiltonian, represented as an $N \times N$ Hermitian matrix $H$, we derive an expression for the largest Lyapunov exponent of the classical trajectories in the phase space appropriate for the dynamics induced by $H$. To this end we associate to $H$ a graph with $N$ vertices and derive a quantum map on functions defined on the directed edges of the graph. Using the semiclassical approach in the reverse direction we obtain the corresponding classical evolution (Liouvillian) operator. Using ergodic theory methods (Sinai, Ruelle, Bowen, Pollicott\ldots) we obtain closed expressions for the Lyapunov exponent, as well as for its variance. Applications for random matrix models will be presented.

The capacity of a set is a classical notion in potential theory and it is a measure of the size of a set as seen by a random walk or Brownian motion. Recently Zhu defined the notion of branching capacity as the analogue of capacity in the context of a branching random walk. In this talk I will describe joint work with Amine Asselah and Bruno Schapira where we introduce a notion of capacity of a set for critical bond percolation and I will explain how it shares similar properties as in the case of branching random walks. 

Dehn surgery is a method of building three-dimensional manifolds that is ubiquitous throughout low-dimensional topology. I will give an introduction to Dehn surgery and discuss recent work with M. Kegel on the uniqueness of Dehn surgery descriptions of 3-manifolds. To do this, I will discuss the reason that Dehn surgery is so prominent - namely that it interacts very well with many structures, such as the geometry and gauge theory of 3-manifolds. (I will do my very best to assume very little background knowledge.)

A fundamental phenomenon in the representation theory of finite and compact groups is that irreducible characters tend to take smaller values on elements that are far from central. Character estimates of exponential type (that is, bounds of the form |chi(g)|<chi(1)^(1-epsilon)) are particularly useful for probabilistic applications, such as bounding the mixing time of random walks supported on conjugacy classes.

In 1981, Diaconis and Shahshahani established sharp estimates for irreducible characters of the symmetric group S_n, evaluated at a transposition t = (i j). As an application, they proved that roughly n*log(n) random transpositions are required to mix a deck of n playing cards. This was extended in 2007 by Muller--Schlage-Puchta to to arbitrary permutations in S_n. Exponential character bounds for finite simple groups were subsequently developed through a series of works by Bezrukavnikov, Liebeck, Shalev, Larsen, Guralnick, Tiep, and others. 

In this talk, Itay Glazer (Technion) will present recent progress on exponential character estimates for compact Lie groups.

This is based on joint work in progress with Nir Avni, Peter Keevash, and Noam Lifshitz.

Yotam Hendel will discuss how polynomial maps can be studied by examining the analytic behavior of pushforwards of regular measures under them over finite and local fields. 

The guiding principle is that bad singularities of a map are reflected in poor analytic behavior of its pushforward measures. Yotam will present several results in this direction, as well as applications to areas such as counting points over finite rings and representation growth. 

Based on joint work with I. Glazer, R. Cluckers, J. Gordon, and S. Sodin.

to follow