# Probability and Mathematical Physics Seminar

#### Matrix displacement convexity and intrinsic dimensional functional inequalities

**Speaker:**
Yair Shenfeld, Brown University

**Location:**
Warren Weaver Hall 109

**Date:**
Friday, November 3, 2023, 11:10 a.m.

**Synopsis:**

The discovery by McCann of displacement convexity had a significant impact on probability, analysis, and geometry. I will introduce a new and stronger notion of displacement convexity which operates on the matrix level. I will then show that a broad class of flows satisfy matrix displacement convexity: heat flow, optimal transport, entropic interpolation, mean-field games, and semiclassical limits of non-linear Schrödinger equations. Consequently, the ambient dimensions of functional inequalities describing the behavior of these flows can be replaced by their intrinsic dimensions, capturing the behavior of the flows along different directions in space. This leads to intrinsic dimensional functional inequalities which provide a systematic improvement on numerous classical functional inequalities.