Lectures on Optimal Transport

This textbook is addressed to PhD or senior undergraduate students in mathematics, with interests in analysis, calculus of variations, probability and optimal transport. It originated from the teaching experience of the first author in the Scuola Normale Superiore, where a course on optimal transport and its applications has been given many times during the last 20 years. The topics and the tools were chosen at a sufficiently general and advanced level so that the student or scholar interested in a more specific theme would gain from the book the necessary background to explore it. After a large and detailed introduction to classical theory, more specific attention is devoted to applications to geometric and functional inequalities and to partial differential equations.

This is the second edition of the book, first published in 2018. It includes refinement of proofs, an updated bibliography and a more detailed discussion of minmax principles, with the aim of giving two fully self-contained proofs of Kantorovich duality.

- 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-Ampére 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 Brézis-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.

Prof. Luigi Ambrosio is a Professor of Mathematical Analysis, a former student of the Scuola Normale Superiore and presently its Rector. His research interests include calculus of variations, geometric measure theory, optimal transport and analysis in metric spaces. For his scientific achievements, he has been awarded several prizes, in particular the Fermat prize in 2003, the Balzan Prize in 2019, the Riemann Prize in 2023 and the Nemmers Prize in 2024.

Dr. Elia Bruè is an Associate Professor at Bocconi University, Milan, Italy. He earned his PhD from the Scuola Normale Superiore in 2020. His research interests lie in the fields of Geometric Analysis and Partial Differential Equations, with a focus on Ricci curvature, metric geometry, incompressible fluid mechanics, and passive scalars with rough velocity fields.

Dr. Daniele Semola is an Assistant Professor at the University of Vienna. He was a student in Mathematics at the Scuola Normale Superiore, where he earned his PhD degree in 2020.  His research interests lie at the interface between geometric analysis and analysis on metric spaces, mainly with a focus on lower curvature bounds.