Vortrag: "Optimal Transport in Geometry, Analysis and Probability"

Ein Vortrag im Rahmen des Seminars "Gruppen, Geometrie, Dynamik".

Abstract

We give a brief survey on recent progress in optimal transportation on manifolds
and metric spaces.

In particular, we will introduce the Riemannian structure on the space P(M)
of probability measures induced by optimal transports on a given Euclidean
or Riemannian space M and we give several applications to equilibration and
concentration phenomena.

Moreover, we present the concept of synthetic Ricci curvature bounds for
metric measure spaces, introduced by Lott & Villani and the author. Based on
this concept, in recent years a powerful analysis on singular spaces has been de-
veloped with deep results and far reaching applications (heat kernel comparison,
Li-Yau estimates, splitting theorem, maximal diameter theorem, fine properties
of Brownian motions).