Lectures on Optimal Transport (Paperback, 2, Second 2024)

- 1.?Lecture I. Preliminary notions and the Monge problem.- 2. Lecture II. The Kantorovich problem.- 3.?Lecture III. The Kantorovich - Rubinstein duality.- 4.?Lecture IV. Necessary and sufficient optimality conditions.- 5.?Lecture V. Existence of optimal maps and applications.- 6.?Lecture VI. A proof of the isoperimetric inequality and stability in Optimal Transport.- 7.?Lecture VII. The Monge-Ampere equation and Optimal Transport on Riemannian manifolds.- 8.?Lecture VIII. The metric side of Optimal Transport.- 9.?Lecture IX. Analysis on metric spaces and the dynamic formulation of Optimal Transport.- 10.?Lecture X.Wasserstein geodesics, nonbranching and curvature.- 11.?Lecture XI. Gradient flows: an introduction.- 12.?Lecture XII. Gradient flows: the Brezis-Komura theorem.- 13.?Lecture XIII. Examples of gradient flows in PDEs.- 14.?Lecture XIV. Gradient flows: the EDE and EDI formulations.- 15.?Lecture XV. Semicontinuity and convexity of energies in the Wasserstein space.- 16.?Lecture XVI. The Continuity Equation and the Hopf-Lax semigroup.- 17.?Lecture XVII. The Benamou-Brenier formula.- 18.?Lecture XVIII. An introduction to Otto's calculus.- 19.?Lecture XIX. Heat flow, Optimal Transport and Ricci curvature.