Concentration of measure: from functional inequalities to optimal transport theory

26.05.2021, 14:00 - 15:45  –  Online Colloquium
Institutskolloquium

Radoslaw Adamczak (University of Warsaw) und Nathael Gozlan (Université Paris-Descartes)

2pm-2:45pm Radoslaw Adamczak (University of Warsaw): Concentration of measure and functional inequalities

3pm-3:45 pm Nathael Gozlan (Université Paris-Descartes): The two-periodic Aztec diamond

Abstracts:

Radoslaw Adamczak (University of Warsaw):  Concentration of measure and functional inequalities

Concentration of measure is a general phenomenon, according to which a sufficiently regular function depending on a large number of random parameters with very high probability takes values which are close to its median or expectation. This idea, introduced by Vitali Milman in the early 1970s in the context of local theory of Banach spaces has found numerous important applications in functional analysis, probability and statistics, combinatorics and theoretical computer science. It provides important tools, e.g., for establishing limit theorems, finding probabilistic proofs of existence of objects with extremal properties or for analysis of randomized algorithms.

I will discuss selected aspects of the theory of concentration of measure for high dimensional random vectors satisfying certain geometric or probabilistic conditions. More specifically, I will present classical results on connections between concentration for Lipschitz functions, isoperimetric inequalities and functional inequalities (with the unit sphere and the Gauss space serving as basic examples). If time permits, I will also explain some manifestations of concentration of measure in discrete situations, again with emphasis on connections with functional inequalities.

Nathael Gozlan (Université Paris-Descartes): Optimal Transport and concentration of measure

After an introduction on Monge-Kantorovich Optimal Transport Theory, the talk will explain how optimal transport techniques can be used to prove precise concentration of measure estimates in various settings. We will in particular give a detailed presentation of transport-entropy inequalities, a class of inequalities introduced by K. Marton and M. Talagrand in the nineties, and discuss their relations to the so called dimension free concentration of measure phenomenon.

If you wish to attend the talks,  please contact Sylvie Paycha paycha@math.uni-potsdam.de for the login details.