Host Department: Rice University-Mathematics
The optimal transport problem consists in finding the "most efficient" way of moving a distribution of mass from one place to another. This problem, formulated by Monge in 1781, has been solved only recently: in 1987 Brenier showed existence and uniqueness of optimal transports on Euclidean spaces when the cost for moving a unit mass from a point x to a point y is |x-y|^2.
This result has been the starting point for many developments in different directions: for instance, optimal transport has revealed to be a powerful tool for proving functional and geometric inequalities (like isoperimetric or Sobolev inequalities) or for studying the asymptotic of evolution equations (like heat or porous medium equations). More recently, optimal transport has found applications in Riemannian geometry: first of all, after the solution of the optimal transport problem on Riemannian manifolds (McCann 2001), people started to realize that optimal transport could be used to study Ricci curvature bounds. Moreover, in these last 2-3 years, it was discovered that optimal transport techniques could even be used to study the geometry of the cut-locus of the manifold.
In these lectures, which are supposed to be accessible to a broad audience, I will start from the very beginning, explaining the strategy to solve the optimal transport problem both in Euclidean spaces and on Riemannian manifolds. Then, to show some of the possible applications, I'll apply optimal transport techniques to prove functional inequalities. Finally, if time permits, I'll explain the link between optimal transport on Riemannian manifolds, Monge-Ampere equations, and the geometry of the cut-locus of the manifold.