2024 Hayden-Howard Lecture

Felix Otto, Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany

A Variational Regularity Theory for Optimal Transportation and Its Applications to Matching

Optimal transportation, which identifies an optimal coupling between two probability measures, is a simple to state variational problem with surprisingly diverse connections.  A couple of years ago, with M. Goldman, we devised a new approach to the regularity theory for the coupling. It mimics De Giorgi's approach to the regularity theory of minimal surfaces in the sense that a harmonic approximation result is at its center: Under a non-dimensional smallness condition, the displacement is close to the gradient of a harmonic function. The main advantage of this variational regularity theory over the one based on maximum principle for the Monge-Ampere equation is that it does not require any regularity of the involved measures. Hence it can be applied to the popular matching problem, where it provides regularity on large scales.

About the Speaker:

Felix Otto has been the director at the Max Planck Institute for Mathematics in the Sciences in Leipzig, Germany, since 2010. His main expertise lies in the applied analysis of partial differential equations and in the calculus of variations, with a recent focus on randomness. He is well known for his work on gradient flow structures of dissipative evolution equations, pattern formation in ferromagnets, and stochastic homogenization. His scientific honors include the A.P. Sloan Research Fellowship, the Max Planck Research Prize, the Leibniz Prize from the German Science Foundation. He was an invited speaker at the 2002 International Congress of Mathematicians in Beijing, China.