Convexity Seminar

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An introduction to optimal transport, part 3

Speaker: Nestor Guillen, Texas State University, NYU Courant

Location: Warren Weaver Hall 1302

Date: Tuesday, October 21, 2025, 11 a.m.

Synopsis:

In this talk we will pick up where we left in part 2. We will explain at last how optimality of transport maps means they are monotone and have a scalar potential -- for the original result of Brenier for the quadratic Euclidean cost this means the transport map must be the gradient of a convex function, for general costs (following work of McCann and Gangbo-McCann) this means the map must be the "c-gradient''  of a c-convex potential.  An important tool in our argument is a first variation formula for the dual transport problem which goes back to work of Aleksandrov on the Minkowski problem.  (After this talk, this series will move on to discuss the regularity theory of transport maps, starting with Caffarelli's theory for weak solutions of the Monge-Ampère equation in Euclidean space and Ma, Trudinger, and Wang's discovery of the fourth order tensor that now bears their name).