Optimal transport

The theory of optimal transport began in the eighteenth century with the Monge problem (1781), which is to minimize the cost of transporting an amount of material from the given set of origins to the given set of destinations. In the fourties, Kantorovitch gave an important reformulation of the problem and, since then, the Monge-Kantorovitch problem has been a classical subject in probability theory, economics and optimization. More recently, the interplay between optimal transport and various fields such as PDEs (Ricci flow, Euler equations¿), fluid mechanics, geometric analysis (isoperimetric and Sobolev inequalities, curvature-dimension conditions), functional analysis, urban planning and economics has been deeply investigated.

The first part of the course will be devoted to Monge and Kantorovitch's problems, discussing the existence and the properties of the optimal plan under different conditions on the cost. We will exploit the relation with Kantorovitch's duality theorem, with Brenier's polar decomposition theorem, and with the Monge-Ampere equation, a PDE which arises naturally in this context. The second part of the course will be centered on the applications of optimal transport to different problems: after introducing the Wasserstein distance, we will see the connection with some PDEs and functional/geometric inequalities.