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Title: Recent progress on the Kannan-Lovasz-Simonovits (KLS) conjecture and Bourgain's slicing problem I

Abstract: Kannan, Lov谩sz, and Simonovits (KLS) conjectured in 1995 that the Cheeger isoperimetric coefficient of any log-concave density or any convex body is achieved by half-spaces up to a universal constant factor. This conjecture now plays a central role in the field of convex geometry, unifying or implying older conjectures. In particular, it implies Bourgain's slicing conjecture (1986) and the thin-shell conjecture (2003). While it is natural to expect convex bodies to have good isoperimetry (in other words, not look like dumbbells), the progress on bringing down the Cheeger isoperimetric coefficient in the KLS conjecture has been stagnant in recent years. The previous best bound, with dimension dependency d^1/4, was established by Lee and Vempala in 2017 using Eldan's stochastic localization, and matches the best dimension dependency Klartag obtained in 2006 for Bourgain's slicing conjecture.
After becoming familiar with Eldan's stochastic localization technique in the previous lecture, first we aim to get familiar with the concept of "localization" and to view stochastic localization as an extension. Then we go through the Lee and Vempala (2017) proof to see in action a concrete application of stochastic localization.

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