PhD students, undergraduates with a basic knowledge of measure theory and functional analysis.
The optimal transport problem: Monge and Kantorovich formulations
Existence and stability of optimal plans
Necessary and sufficient optimality conditions
Existence of optimal transport maps and applications
The metric side of optimal transportation
The differentiable side of optimal transportation
Gradient flows in Hilbert spaces and in metric spaces
Heat flow and diffusion equations
The aim of the course is to provide a quite detailed introduction to the theory of
optimal mass transportation, from the classical facts up to the most recent developments, including the connections with the theory of gradient flows and Ricci curvature.
L.Ambrosio, N.Gigli, G.Savaré: Gradient flows in metric spaces and
in the spaces of probability measures. ETH Lectures in
Mathematics, Birkh"auser, 2005 (II edition, 2008).
C.~Villani, Topics in optimal transportation, vol. 58 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2003.
C.~Villani, Optimal transport. Old and new, vol. 338 of Grundlehren
der Mathematischen Wissenschaften, Springer-Verlag, Berlin, 2009.
Lecture notes given by the lecturer.