(Abstract: Following the book of Villani (Topics in Optimal Transportation, Graduate Studies in Mathematics, American Mathematical Society, Providence, 2003), we will give a short presentation of the theory of optimal transportation and some of its connections with partial differential equations.

Our plan is to continue with a series of short lectures (taking place during the DIFTA seminar when no seminar is scheduled) where we will go in more details through some selected results.)

(Abstract: We start with the first chapter of Villani's book, which deals with duality. We will prove the existence of a minimizer for the optimal transportation problem and then start the proof of the Kantorovich duality. Some of the ingredients: a compactness theorem on the space of probability measure (Prokhorov theorem), and a “minimax” principle (“inf sup = sup inf”) from convex analysis.)

(Abstract: We finish the proof of the existence of a minimizer and go through the duality theory (Theorem 1.3 in the book).)

(Abstract: We start the second chapter of Villani's book, which deals with a characterization of optimal transportation plans. We will introduce a few concepts from convex analysis and prove that, for measures with finite second moments, any optimal transportation plan is related to the subdifferential of a convex function. Finally, we will focus on optimal plans in R, trying to make the characterization a bit clearer.)

(Abstract: To conclude the second chapter of Villani's book, we will focus on optimal plans in R, trying to make the characterization of optimal transportation plans a bit clearer, and we will show what is known (at the present stage of the research) about the general case of Polish spaces X, Y and lower semicontinuous costs c(x,y).)

(Abstract: Strichartz estimates are spacetime estimates on homogeneous and inhomogeneous linear dispersive and wave equations. They are particularly useful for solving semilinear perturbations of such equations, in which no derivatives are present in the nonlinearity.
In these talks I will first discuss briefly the local existence theory for a model nonlinear wave equation, as a motivation. I will then present the proof of the Strichartz type estimate for the homogenous wave equation.)