Lectures by Cyril Letrouit, winner of the Peccot Lecture Series for 2024-2025, nominated by Prof. Nalini Anantharaman.
The optimal transport problem, introduced by Monge in 1781, aims to determine the most efficient way of moving a quantity of resources from one place to another. Mathematically, it involves finding an optimal way, for a certain cost, to transport a so-called "source" measure to a so-called "target" measure. Because of its simplicity and generality, optimal transport is now at the heart of an extremely wide range of mathematics, in analysis, geometry, probability, statistics, optimization and machine learning.
The lecture will focus on the following stability question: how sensitive is the optimal transport application to perturbations of the target measure? The aim of this lecture is to explain both the computational and fundamental interest of this problem, and to present the many theoretical advances made recently on this subject, mixing partial differential equations, spectral theory and functional inequalities. Several open problems and research directions will be indicated during the lecture, concerning the optimal transport application and other applications of transport between measures.
