Week 03.12.2023 – 09.12.2023

Part I, Introduction and theoretical basis: how time-reversing Langevin dynamics one can create images from white noise.

Given a graph G = (V,E), consider the set of all discrete-time, reversible Markov chains with equilibrium distribution uniform on V and transitions only across edges E of the graph. We establish a Cheeger-type inequality for the fastest mixing time using the vertex conductance of G. We also consider chains with almost-uniform invariant distribution. Time permitting, we also discuss a construction of a continuous-time chain with exactly-uniform invariant distribution and average jump-rate 1, and mixing time bounded by the d^2 log(n), where d is the graph diameter and n is the number of vertices.

Mutations of quivers were introduced by Fomin and Zelevinsky in the context of cluster algebras. Since then, mutations appear (sometimes completely unexpectedly) in various domains of mathematics and physics. Using mutations of quivers, Barot and Marsh constructed a series of presentations of finite Coxeter groups as quotients of infinite Coxeter groups. I will discuss a geometric interpretation of this construction: these presentations give rise to a construction of geometric manifolds with large symmetry groups, in particular to some hyperbolic manifolds of relatively small volume with proper actions of Coxeter groups. If time permits, I will discuss a generalization of the construction of Barot and Marsh leading to a new invariant of bordered marked surfaces, and relation to extended affine Weyl groups. The talk is based on joint works with Anna Felikson, John Lawson and Michael Shapiro.

Part II. A statistical physics analysis using random matrix theory and mean-field methods.

In this talk, we revisit several results on exponential integrability in probability spaces and derive some new ones. In particular, we give a quantitative form of recent results by Cianchi, Musil, and Pick in the framework of Moser-Trudinger-type inequalities, and recover Ivanisvili-Russell’s inequality for the Gaussian measure. One key ingredient is the use of a dual argument, which is new in this context, that we also implement in the discrete setting of the Poisson measure on integers. This is a joint work with Ali Barki, Sergey Bobkov, and Cyril Roberto.