Instructor: S. Dostoglou

Description: Have you ever wished for an applied problem that would lead you to investigate partial differential equations, calculus of variations, probability theory, differential geometry AND is useful in engineering, economics, fluid mechanics, and meteorology?

The Optimal Transportation problem satisfies all these.

First posed by Monge as a problem of transferring mass with the least possible amount of work, was further developed and used by Kantorovich (Nobel prize in Economics), and continues to provide novel approaches to several current problems, such as hydrodynamics and atmospheric flows, and beautiful new results in several core fields, such as PDEs.

The story is simple: You think of the "masses" you want to transport as measures and you try to minimize some function (the "cost") over all transportation "plans." Like any decent optimization problem, this has a geometric interpretation: It leads you to define a ``distance" in the space of measures and to do geometry with respect to that distance.

With the aim or presenting solutions to the transportation problem, the course will cover in detail results in optimization, convergence of measures, the geometry of metric spaces,
convexity, and a lot more.

By the end of the course not only you will have a good understanding of such methods but you will also have had a solid introduction to Kantorovich duality, Brenier approximation,
Wasserstein distance, Boltzmann's equation, Entropy production, Caffarelli regularity, and more. (This material can easily lead to Master's projects.)

Prerequisites: A course in Real Analysis. Some familiarity with probability theory will help, but is not assumed. (The level of difficulty will not be greater than,say, our standard PDE I course. In fact, the main text for the course [V] is in the same graduate texts series as our standard PDE text.)

References

[V] Villani, C: Topics in optimal transportation. Graduate Studies in Mathematics, 58. American Mathematical Society, Providence, RI, 2003.

[AGS] Ambrosio, L.; Gigli, N.; Savaré, G.: Gradient flows in metric spaces and in the space of probability measures.Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2005.