After successful completion of the course, students are able to:

• Formulate the optimal transportation problem according to Monge resp. Kantorovich.
• Solve the 1-dimensional optimal transportation problem.
• Formulate the dual problem according to Kantorovich and prove the existence of maximizers.
• Define the Wasserstein space and Wasserstein distance.
• Prove classical geometric inequalities using optimal transportation.

Optimal Transport formulation according to Monge resp. Kantorovich; Kantorovich Duality; solution of the 1-dimensional problem; theory of convex functions, in particular the Legendre transform; Wasserstein distances and Wasserstein spaces; geometric inequalites (Brunn-Minkowski, Prékopa-Leindler,...)

Mathematical definitions and proves.

preliminary lecture (also to dertermine the dates): Tuesday March 7th, 11 a.m. (Freihaus 7. Floor, green section, Besprechungszimmer)

Additional literature:

• Cédric Villani, "Topics in Optimal Transportation", Graduate Studies in Mathematics Volume 58, American Mathematical Society, 2003.
• Cédric Villani, "Optimal transport: old and new",  Grundlehren der mathematischen Wissenschaften 338, Springer-Verlag, 2009.

oral exam

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