Geometry of Spaces of Measures

Abstract: Motivated by geometric (and also physical) problems, Misha Gromov advocated the loosening of the relation between a metric and a measure that is familiar in Riemannian geometry where the metric is obtained by integrating the length of shortest paths and the volume element is induced by the metric. This lead to the study of Metric Measured Spaces.

Motivated by optimal transport, Felix Otto and Cedric Villani made evident the importance of the geometry of spaces of measures. Their efforts met earlier ones made by probabilists such as Dominique Bakry and Michel Emery who looked for conditions ensuring the validity of log-Sobolev inequalities. Maxim Kontsevitch and John Lott rediscovered some of these facts in their attempts to generalize the notion of Ricci curvature to non smooth metrics.

This very rich circle of ideas has become a very active area of research allowing to revisit some classical domains of Analysis, such as solving Monge Ampere equations. The intricate mixing of ideas coming from diverse fields makes it quite remarkable.

Refreshments will be served beforehand over at PIMS.