Diffusion, Entropy, and the Geometry of Sets in Gaussian Space
Thursday, March 19, 2015 - 4:00pm - 4:40pm
It is a well-studied and very useful phenomenon that functions on product spaces (e.g., Gaussian space, the discrete cube) become smoother under the corresponding diffusion operator (Ornstein-Uhlenbeck and the noise operator, respectively). Hypercontractivity of these operators is a tool with many applications across mathematics and the study of computation. In 1989, Talagrand conjectured that if we start with an initial probability density and allow it to diffuse for a small amount of time, then the height of the resulting density must be anti-concentrated. In joint work with Ronen Eldan, we prove the Gaussian case of Talagrand's conjecture.