12w5118 Optimal Transportation and Differential Geometry

Optimal mass transportation can be traced back to Gaspard Monge’s famous paper of 1781: ‘Mémoire sur la théorie des déblais et des remblais’. The problem there is to minimize the cost of transporting a given distribution of mass from one location to another. Since then, it has become a classical subject in probability theory, economics and optimization. At the end of the 80’s, the seminal work of Brenier [7, 8] paved a way to connect optimal mass transportation to partial differential equations and related areas. On the one hand, his theory was followed by McCann’s displacement convexity and Otto’s differential geometry of the space of probability measures, making the theory of mass transportation applicable to wide range of problems in differential geometry, geometric and functional inequalities, and nonlinear diffusions. On the other hand, it stimulated Caffarelli, Urbas, and many others, to develop regularity theory of Monge-Ampère equations. These two related directions have seen unexpected recent progresses: One of the highlights is the breakthrough of Lott and Villani [58] and Sturm [72, 73] who have characterized singular spaces with lower bounded Ricci curvature, solving one of the well-known open problems in Riemannian geometry. Also, Ma, Trudinger and Wang [65] extended the regularity theory for Monge-Ampère to a more general class of Monge-Ampère type equations, which surprisingly led the discovery of a new type of curvature, called Ma-Trudinger-Wang curvature, which is nowadays an object of many investigations in different directions.