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Yuansi Chen（陈远思）：Kannan-Lovász-Simonovits (KLS) 猜想和Bourgain's slicing问题的最新进展——1

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2023-01-01 19:08:03

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https://www.youtube.com/watch?v=35NCkJBjKWc Centre de recherches mathématiques (6 octobre 2021 / October 6, 2021) Conférence Nirenberg du CRM en analyse géométrique / CRM Nirenberg Lectures in Geometrics Analysis ：http://www.crm.umontreal.ca/2021/Nirenberg-Chen-Klartag2021/index_e.php Yuansi Chen (Duke University, USA) Conférence 3 / Lecture 3: Recent progress on the Kannan-Lovasz-Simonovits (KLS) conjecture and Bourgain's slicing problem I Yuansi Chen (Duke University, USA) Conférence 4 / Lecture 4: Recent progress on the Kannan-Lovasz-Simonovits (KLS) conjecture and Bourgain's slicing problem II Abstract: Kannan, Lovász, 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.

渐进几何分析

Functional analysis

Convex geometry

Cheeger等周系数

Bourgain's slicing

Needle decomposition

Pathwise analysis

Thin-shell猜想

SIsoperimetric 发消息

微分幾何学

救命！第一天入职需要人工呼吸！

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Yuansi Chen（陈远思）：Kannan-Lovasz-Simonovits (KLS) 猜想和Bourgain's slicing问题的最新进展——1

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