Isoperimetric inequalities in high-dimensional convex sets

25 September - 11 December 2025

Thursdays, 10:15 - 12:00

Location: HG G 43

First lecture: 25 September

No lectures on 4 and 18 December

Abstract

We will discuss recent progress in understanding the uniform measure on high-dimensional convex bodies, focusing on advances toward the Kannan-Lovász-Simonovits (KLS) isoperimetric conjecture, as well as the resolution of Bourgain’s slicing problem and the thin-shell conjecture.

The study of uniform measures on high-dimensional convex bodies provides a testing ground for powerful analytic methods with applications in broader mathematical contexts. These techniques include spherical and Gaussian concentration of measure, convex localization, optimal transport with the Monge cost, Bochner identities and curvature, heat flow, and Eldan's stochastic localization.

We will begin the first part of the course by examining the high-dimensional cube and Euclidean ball, and by proving the isoperimetric inequality on the sphere. This inequality is the cornerstone of spherical and Gaussian concentration of measure, and we will discuss some of its applications, such as the Johnson-Lindenstrauss lemma and the phenomenon of approximately Gaussian marginals. We will then turn to log-concavity and the Bochner technique, the Bourgain-Milman inequality (with its many elegant proofs), exponential tilts, the existence of Milman ellipsoids, the slicing problem, and its relation to the thin-shell phenomenon.

In the second part, we will study in detail the heat evolution of log-concave measures using the Brownian interpretation and pathwise methods. As an application beyond convex geometry, we will also discuss high-dimensional sphere packing.