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 stochastic localization..
The first part of the course will be dedicated to background on geometric phenomena in high-dimensions. In the second part, we will study in detail the heat evolution of log-concave measures using the Brownian interpretation and pathwise methods.
Feb 5-6, 2026, at Institut de Mathématique d'Orsay
Lecturer: Bo'az Klartag, Tel Aviv University and Weizmann institute for science
