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Optimal transport


Monday, 2 November 2020 to Monday, 31 May 2021
Total hours: 40
Hours of lectures: 40

Examination procedure

  • oral exam


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

Educational Goals

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.

Bibliographical references

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.