About

Many recent advances may be viewed as controlling the expansion of a graph, a group, or a manifold, by local quantities which depend only on small pieces of it. For instance: 

  1. Eldan’s stochastic localization process is a way to control the Poincare constant or modified log-Sobolev constant of a log-concave measure by understanding the covariance matrices of certain random tilts of the measure, which can often be understood locally.

  2. Spectral Independence, which is closely related to the above and to high dimensional expansion, does the same for certain structured Markov chains arising from combinatorics and statistical mechanics.

  3. Notions of Olivier-Ricci and Bakry-Emery curvature on graphs, inspired by similar notions on manifolds, depend only on constant radius neighborhoods of vertices and imply results about rapid mixing and cutoff in certain cases.

  4. Property (T) gives a local “sum of squares” proof of spectral gap for certain Cayley graphs, yielding infinite sequences of expanders.

  5. Aldous’ conjecture and its variants show that the spectral gap of certain group actions (i.e., Schreier graphs) is controlled by the gap of a much “smaller” action of the group.

This workshop will explore these developments and more, with the aim of discovering new connections and analogies between them.