On Chen’s recent breakthrough on the Kannan-Lovasz-Simonovits conjecture and Bourgain's slicing problem - Part II

Ronen Eldan

This is the second talk in a 3-lecture series whose goal is to give background as well as a self contained proof of Chen's recent breakthrough on the KLS conjecture and slicing problem (a video of the first lecture can be found here: https://www.ias.edu/video/chens-recent-breakthrough-kannan-lovasz-simonovits-conjecture-and-bourgains-slicing-problem). After becoming familiar with the concept of localization in the first talk, in this talk we will focus on stochastic localization, the central ingredient used in Chen's result. Stochastic localization is a process driven by a Brownian motion which gives rise to a decomposition of a given measure to "well-behaved" measures. We will describe the construction of the process, establish some of its basic properties, and then show how it can be used to prove concentration and isoperimetric inequalities. In the third and last talk, we'll finally be in a position to use this construction to establish Chen's result. A note on required background: The technique that we discuss uses stochastic calculus and in particular Ito's formula.  Since this talk is aimed for the CS and discrete math communities, I will do my best to explain the formulas in a way that requires no prior knowledge in stochastic calculus. However, a pretty short read that may be very helpful in following the formulas is found in Section 1 here: http://www.wisdom.weizmann.ac.il/~ronene/GFANotes.pdf.

https://www.ias.edu/video/chens-recent-breakthrough-kannan-lovasz-simonovits-conjecture-and-bourgains-slicing-problem-0