Convexity Seminar

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Regularity theory for optimal transport maps I

Speaker: Nestor Guillen, Texas State University, NYU Courant

Location: Warren Weaver Hall 1302

Date: Tuesday, October 28, 2025, 11 a.m.

Synopsis:

(This talk will be self contained, attendance of previous lectures is not required) In previous talks we covered Brenier's theory (and Gangbo-McCann's generalizations) for optimal transport maps. The maps produced are a priori not continuous, even if they are given by the gradient map of a convex function. For the next couple of lectures we will discuss Caffarelli's regularity theory for optimal transport maps. We will take a modern point of view following works of Figalli-Kim-McCann and work of myself with Kitagawa that see Caffarelli's estimates as arising from a combination of the Blashke-Santaló and reverse Blashke-Santaló inequalities. These inequalities constraint the eccentricity of level sets of weak solutions to the Monge-Ampère equation, ultimately leading to strict convexity and differentiability.