Convexity Seminar

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An Introduction to Optimal Transport, Part 1

Speaker: Nestor Guillen, Texas State, Courant

Location: Warren Weaver Hall 1302

Date: Tuesday, September 16, 2025, 11 a.m.

Synopsis:

The optimal transport (OT) problem is at the simplest level a problem of mapping one mass density into another in a way that minimizes a given cost. At the broadest level, it is a framework to understand optimal rearrangements or matchings between various geometric and analytical objects. This broad perspective goes a long way to explain the fast and dramatic spread of OT methods throughout mathematics (spanning uses in geometry, statistics, and PDE) starting from  Yann Brenier's foundational polar factorization paper in 1991. I will present the general framework for optimal transport, review Brenier's result for the quadratic cost in Euclidean space, as well as subsequent extensions to general costs in works of Caffarelli, Gangbo-McCann, and Ma-Trudinger-Wang. We will follow with some exploratory discussion of how these ideas have been used and might be pushed to solve Monge-Ampère equations that don't fall strictly within OT, such as Minkowski problems and Generated Jacobian equations.