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Optimal transport

Schedule

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

Examination procedure

  • oral exam

Prerequisites

PhD students, undergraduates with a basic knowledge of measure theory and functional analysis.

Syllabus

The optimal transport problem: Monge and Kantorovich formulations

 

Existence and stability of optimal plans

 

Duality

 

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.