Description

May 30, 2023, 9:45 AM

45m

Paul Sabatier University, Toulouse

Ramon van Handel

The concentration of measure phenomenon asserts that in a remarkably broad

range of situations, nonlinear functions of many random variables are well

concentrated around their means. A question that arises naturally in

probability theory, functional analysis, metric geometry, and geometric

group theory is whether there exist analogous phenomena for vector-valued

functions, i.e., taking values in normed spaces. While this question is

seemingly innocuous on its face, it is not even clear in first instance

how it can be meaningfully formulated or approached.

What is arguably the "correct" way to think about this problem was

discovered by Pisier in the 1980's in the setting of Gaussian measures.

The extension of Pisier's ideas beyond the Gaussian setting was a

long-standing problem. A few years ago, our work with Ivanisvili and

Volberg provided one further example: a vector concentration inequality on

the discrete cube. (There are also related works of Lafforgue and

Mendel-Naor for very carefully designed models of expander graphs.) All

these situations are rather special, and fall far short of the richness

and broad applicability of the classical concentration of measure theory.

In this talk I aim to describe the current status of a long-term effort to

discover more general principles behind the vector concentration

phenomenon. Along the way we encounter some new probabilistic questions,

unexpected phenomena, and (unfortunately) plenty of unexplained mysteries.