Arbeitsgemeinschaft: Optimal Transport and Geometry

Abstract

This Arbeitsgemeinschaft was devoted to recent developments in optimal transport with emphasis on links and applications to geometry. The topics reached from the origin of optimal transport as a variational problem, where one minimizes a transportation cost when transporting one density into another, over the introduction of a metric on the space of probability measures, leading to the Wasserstein-space and convex funtionals on it, to the connection between Ricci-curvature and the optimal mass transport problem. Mathematics Subject Classification (2000): 49-02 (28Axx 37J50 49Q20 53Cxx 58Cxx 82C70). Introduction by the Organisers In its origin, optimal transportation is a variational problem where one minimizes a transportation cost when transporting one density into another (Monge). Via its relaxed version (Kantorovich), the solution of this problem (Brenier) connects with convex analysis. Entire classes of inequalities in analysis can be easily proven with this tool. Even for the simplest transportation cost, i. e. the square of the Euclidean distance, the regularity theory for the minimizers is subtle (Caffarelli and others): Its Euler-Lagrange equation is the role model for a fully nonlinear elliptic equation in non-divergence form, the Monge-Ampere equation. The existence and the elements of a theory for more subtle transportation costs, like the Euclidean distance itself or the square of a Riemannian distance, are areas of current research. Optimal transportation can be used to introduce a metric (distance function) on the space of probability measures which metrizes the weak topology. If the transportation cost is the square of a Euclidean or Riemannian distance, this metric can be seen as induced from a formal, infinite-dimensional Riemannian structure 984 Oberwolfach Report 18 on the space of probability measures (Otto). Loosely speaking this geometry is the “complement” (in the sense of polar decomposition of vector fields) of the one of the space of volume-preserving diffeomorphisms (Arnold), which is motivated from fluid mechanics. Like for the space of volume-preserving diffeomorphisms, the space of probability measures has interesting geometrical properties itself. For instance, in the Aleksandrov sense, this space has non-positive sectional curvature, if the underlying space has this property. Certain entropy functionals (including the usual entropy) turn out to be convex with respect to this geometry (McCann). Moreover, the convexity properties of these functionals can be used to characterize lower bounds on the Ricci curvature and the dimension of the underlying space — and can be used to define Ricci curvature bounds in the absence of a smooth structure (Sturm, Lott-Villani). This relation between geodesic convexity and Ricci curvature can be assimilated to the longer-known relation between the logarithmic Sobolev inequality and Ricci curvature (Bakry-Emery). Closely related to this property is the fact that the gradient flow (steepest descent) of the entropy functional is a contraction if the Ricci curvature is non-negative. In fact it is always a contraction if the underlying geometry evolves by Ricci flow (McCann-Topping). This brief tour d’horizon shows that over the past 15 years, many connections between optimal transportation and seemingly unrelated fields have been discovered. Three monographs [1, 2, 3] and several lecture notes address these recent developments. The Arbeitsgemeinschaft “Optimal transport and geometry”, being held from March 29th to April 4th 2009, attracted more than 30 participants from various mathematical fields like partial differential equations, probability theory or geometry, who shared their different views about the topics presented in the 17 well-prepared talks in lively discussions. We would like to thank the staff of Oberwolfach for providing such perfect and pleasant working conditions, resulting in a stimulating atmosphere that gave the highly motivated participants extraordinary possibilities for learning and exchanging ideas.