Four Lectures on Mass Transport and Gradient Flows

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These lectures have been given in the framework of the PDE seminar organized by N. Alikakos at the University of Athens, in room A31, Departement of Mathematics.

Friday May 9th, 3pm-4.15pm
Friday May 16th, 3pm-4.15pm
Friday May 30th, 3pm-3.50pm
Friday June 6th, 3pm-4.15pm

During the weeks between Lectures 1 and 2 and between Lectures 3 and 4 I was kindly hosted by the Departement (office 110, first floor, or in Nick's office).

Program

Lecture 1 The formulations by Monge and Kantorovich. Existence and duality for the Kantorovich problem. Existence of an optimal transport in the strictly convex case. Brenier's Theorem (the quadratic optimal transport is the graident of a convex function), assumptions and counterexample. Application to the isoperimetric inequality.

Lecture 2 Definition, interpretation and properties of the Wasserstein distances Wp from the Lp transport costs (triangle inequality, equivalence with weak convergence). Link between absolutely continuous curves in the Wasserstein space and the continuity equation.

Lecture 3 Characterization of the geodesics in the Wasserstein spaces. The Benamou-Brenier method for the numerical computation of the geodesic and of the optimal map. Geodesically convex functionals and applications.

Lecture 4 Short introduction to Gradient Flows ODEs in Rn and in metric spaces. The minimizing movement scheme. How to get some evolution PDEs (heat, Fokker-Plank, porous media, non-local interactions...) as gradient flows for the distance W2. Structure of the technique in the case of Fokker Planck.

References

Two classical references are the first book by Cédric Villani Topics in Optimal Transportation (Am. Math. Soc., GSM, 2003) on the general theory, and Gradient Flows in Metric Spaces and in the Space of Probabiliy Measures, by Luigi Ambrosio, Nicola Gigli and Giuseppe Savaré (Birkhäuser, 2005). Yet, the approach we use for gradient flows will not exactly be the same as in this book.

A short reference, with almost no proofs, for Lectures 1-3 is given by the Lecture Notes for a Summer School held in Grenoble in 2009 (to be published now by Cambridge Univ. Press).

I also have a project of a book, an evolution from the above lecture notes and from some notes I wrote for a course (20h) that I gave at Orsay in 2011 and 2012. You can have a look here, it is still in progress. Lecture 1 corresponds to a large part of Chapter 1, and Lecture 2 of Chapter 5. The last two lectures discussed part of Chapters 7 and 8.