Student Probability Seminar

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Around Eldan's Stochastic Localization process and the KLS conjecture

Speaker: Jacob Shkrob, NYU Courant

Location: Warren Weaver Hall 517

Date: Monday, November 27, 2023, 12:30 p.m.

Synopsis:

The Kannan-Lovász-Simonovits (KLS) conjecture concerns the isoperimetric of log-concave probability measures in high dimensions. In this talk, we'll begin by motivating the KLS conjecture, which arose from algorithmic questions regarding the mixing time of a random walks inside convex bodies and their volume computations. We'll then give a sketch of the "pathwise" approach of Eldan showing a way to prove 1-Lipschitz concentration of the Gaussian and the Borell noise-sensitivity inequality. Finally, we'll construct the stochastic localization process and show how to prove a weak version of the KLS conjecture in detail, namely we'll show how to get a n^{-1/4} lower bound on the Cheeger constant. If time remains, we'll talk about how SL leads to entropy-efficient measure decompositions. This talk is based off of the surveys of https://arxiv.org/abs/1807.03465 and https://www.wisdom.weizmann.ac.il/~ronene/files/Pathwise.pdf.