Time:

7.-11.8.2017, 10h of lectures

Abstract:

Optimal mass transportation is a powerful tool to prove geometric inequalities in various settings of metric measure spaces. In this series of lectures I will indicate such applications in the setting of Euclidean, Riemannian and sub-Riemannian spaces. As a consequence we obtain Brunn-Minkowski, Borell-Brascamp-Lieb and entropy inequalities.

Time:

14.-18.8.2017, 10h of lectures

Abstract:

An optimal transport problem consists of finding an interpolation between two probability distributions minimizing some average cost. Displacement interpolations resulting from standard optimal transport are basic tools
(i) for deriving several concentration and geometric inequalities, and
(ii) for developing the Lott-Sturm-Villani (LST) theory of curvature lower bounds of geodesic spaces.

Entropic optimal transport is a probabilistic version of the standard transport problem. It consists of finding most probable interpolations between two prescribed configurations described probability dsitributions, when one travels randomly, typically along Brownian diffusion processes on manifolds or random walks on graphs. In its original form, this optimization problem was addressed by Schrödinger in 1931 as a large deviation problem (the so-called Schrödinger problem) for a random particle system. Its solutions are called entropic interpolations.

We shall introduce both standard and entropic optimal transport problems, their interpretations in terms of large deviations of particle systems and their equations of motion. Some geometric and concentration inequalities will be proved by means of entropic interpolations. Otto's heuristic interpretation of the heat equation as the gradient flow of the entropy with respect to a transport distance will be investigated in light of Schrödinger's problem. Finally, the convergence of entropic interpolations to displacement interpolations on graphs will be investigated.