Alessio Figalli: Quantitative stability in geometric and functional inequalities

Abstract

Geometric and functional inequalities play a crucial role in several problems arising in analysis and geometry. The issue of the sharpness of a constant, as well as the characterization of minimizers, is a classical and important question. More recently, there has been a growing interest in studying the stability of such inequalities. The basic question one wants to address is the following:

Suppose we are given a functional inequality for which minimizers are known. Can we quantitatively show that if a function “almost attains the equality,” then it is close to one of the minimizers?

In this series of lectures, I will first give an overview of this beautiful topic and then discuss some recent results concerning the Sobolev, isoperimetric, and Brunn–Minkowski inequalities.