Abstract: The Kannan-Lovasz and Simonovits (KLS) conjecture considers the following isoperimetric problem on high-dimensional convex bodies: Given a convex body $K$, consider the optimal way to partition it into two pieces of equal volume so as to minimize their interface. Is it true that up to a universal constant, the minimal partition is attained via a hyperplane cut? Roughly speaking, this question can be thought of as asking "to what extent is a convex set a good expander"?
In analogy to expander graphs, such lower bounds on the capacity would imply bounds on mixing times of Markov chains associated with the convex set, and so this question has direct implications on the complexity of many computational problems on convex sets. Moreover, it was shown that a positive answer would imply Bourgain's slicing conjecture.
Very recently, Yuansi Chen obtained a striking breakthrough, nearly solving this conjecture. In this talk, we will overview some of the central ideas used in the proof. We will start with the classical concept of "localization" (a very useful tool to prove concentration inequalities) and its extension, stochastic localization - the main technique used in the proof.
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