Introduction to Optimal Transport (2 ECTS)

In this course we will provide an introduction to Optimal Transport Theory.
Optimal Transport theory deals with the optimal transportation/allocation of resources, and was first formalized by G. Monge in 1781 and developed by L. Kantorovich in 1942-1948.
More recently, optimal transport methods have earned an increasing importance, both from the point of view of the applications and from a theoretical point of view. From a modeling point of view, optimal transport theory is useful to model complex systems, where the number of particles is so large that only a statistical description is viable (as in statistical mechanics).

Summary
1. Review on measure theory
2. The optimal transport problem in the Monge's formulation
3. Relaxation of the optimal transport problem: Kantorovich's formulation
4. Kantorovich Duality and its consequences
5. Special costs:|x-y|, h(|x-y|) with h strictly convex.
6. Benamou-Brenier's Dynamical formulation of the optimal transport problem
7. The Wasserstein space and its differential structure.
8. Applications.

References
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L. Ambrosio, N. Gigli and G. Savaré, Gradient flows in metric spaces and in the spaces of
probability measures. Lectures in Mathematics, ETH Zurich, Birkh ̈auser, 2005

F. Santambrogio. Optimal Transport for Applied Mathematicians, Progress in Nonlinear Differential Equations and Their Applications, Vol. 87, Birkhäuser Basel, 2015.

C. Villani. Topics in Optimal Transportation. Graduate Studies in Mathematics, AMS, 2003.

C. Villani, Optimal transport: Old and New, Springer Verlag (Grundlehren der mathematischen
Wissenschaften), 2008

Schedule
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- 28/02/2019, 13:30-16:30 Room M - Borgo Roma - Ca' Vignal 2 - Verona
- 07/03/2019, 13:30-16:30 Room M - Borgo Roma - Ca' Vignal 2 - Verona
- 14/03/2019, 12:30-15:30 Room M - Borgo Roma - Ca' Vignal 2 - Verona
- 21/03/2019, 12:30-15:30 Room M - Borgo Roma - Ca' Vignal 2 - Verona
- 28/03/2019, 12:30-15:30 Room M - Borgo Roma - Ca' Vignal 2 - Verona