Abstracts and course materials

Jorge Garza Vargas
(California Institute of Technology)

Strong convergence of random matrices

Abstract: In the early 90's Voiculescu realized that certain natural tuples of random matrices converge, in some suitable sense, to certain natural tuples of bounded operators. He then leveraged this fact to tackle some, until then, intractable problems in von Neumann algebras. Since Voiculescu's discovery, researchers have realized that this sort of convergence happens in a much stronger sense (hence the term "strong convergence") than anticipated and is far more ubiquitous in mathematics than it initially appeared.
This mini-course will be about strong convergence. It will begin discussing classical examples from the theory of random graphs and will put emphasis on a new technique for establishing strong convergence. Time permitting, we will also discuss applications in spectral geometry, differential geometry, and/or operator algebras.

Saraí Hernández Torres
(Universidad Nacional Autónoma de México)

TBA

Justin Salez
(Université Paris Dauphine CEREMADE)

Modern Aspects of Markov Chains: Entropy, Curvature and the Cutoff Phenomenon

Abstract: The cutoff phenomenon is an abrupt transition from out of equilibrium to equilibrium undergone by certain Markov processes in the limit where the size of the state space tends to infinity: instead of decaying gradually over time, their distance to equilibrium remains close to its maximal value for a while and suddenly drops to zero as the time parameter reaches a critical threshold. Discovered four decades ago in the context of card shuffling, this surprising phenomenon has since then been observed in a variety of models, from random walks on groups or complex networks to interacting particle systems. It is now believed to be universal among fast-mixing high-dimensional processes. Yet, current proofs are heavily model-dependent, and identifying the general conditions that trigger a cutoff remains one of the biggest challenges in the quantitative analysis of finite Markov chains. The purpose of these lecture notes is to provide a self-contained introduction to this fascinating question, and to describe its recently-uncovered relations with entropy, curvature and concentration.

Paul Thévenin
(Université d'Angers LAREMA)

Introduction to the infinite noodle

Abstract: The infinite noodle is a discrete infinite random object, constructed by connecting together the integers of Z with arches, above and below the real axis, in a noncrossing way. It was introduced in 2019 by Curien, Kozma, Sidoravicius and Tournier, whose main motivation was to use it as a toy model to investigate the properties of large random planar graphs. In the last few years, several connections were proved between the infinite noodle and other structures such as meanders (which are configurations of two closed curves on the sphere) or non-crossing partitions of an integer.
The goal of this mini-course is to rigorously introduce the infinite noodle, discuss different structural results, and show its links to different problems arising from topology, percolation theory, random geometry or combinatorics.