**"KLS conjecture and log-Sobolev constant" **

Yin-Tat Lee (University of Washington)

Monday, December 4, 2017, 2:00 pm

EBU3B, Room 4258

**Abstract:**

Kannan-Lovasz-Simonovitz (KLS) conjecture asserts that the isoperimetric constant of any isotropic convex set is uniformly bounded below. It turns out that this conjecture implies several well-known conjectures from multiple fields:

- (Convex Geometry) Each unit-volume convex set contains a constant area cross-section.
- (Information Theory) Each isotropic logconcave distribution has O(d) KL distance to standard Gaussian distribution.
- (Statistics) A random marginal of a convex set is approximately a Gaussian distribution with 1/sqrt(d) error in total variation distance.
- (Measure Theory) Any function with Lipschitz constant 1 on an isotropic logconcave distribution is concentrated to its median by O(1).

In this talk, we will discuss the latest development on the KLS conjecture and give the tight bound of log-Sobolev constant on logconcave distributions (which in turn gives the best bound on the isoperimetric constant).

Joint work with Santosh Vempala.