Surveys and lecture notes

  1. An introduction to optimal transport and Wasserstein gradient flows
    Preprint 2023.
  2. The continuous formulation of shallow neural networks as wasserstein-type gradient flows (with X. Fernández-Real)
    Preprint 2020.
  3. Free boundary regularity in obstacle problems
    Journées EDP 2018, to appear.
  4. On the Monge-Ampère equation
    Séminaire Bourbaki. Vol. 2017/2018. Exposé 1136-1150 (2019), Exp. No. 1148, 477–504.
  5. Regularity of interfaces in phase transitions via obstacle problems
    Proceedings of the International Congress of Mathematicians, 2018.
    Vol. I. Plenary lectures, 225–247, World Sci. Publ., Hackensack, NJ, 2018.
  6. Global existence for the semigeostrophic equations via Sobolev estimates for Monge-Ampère
    Partial differential equations and geometric measure theory, 1-42,
    Lecture Notes in Math., 2211, Fond. CIME/CIME Found. Subser., Springer, Cham, 2018.
  7. Regularity theory for local and nonlocal minimal surfaces: an overview (with M. Cozzi)
    Nonlocal and nonlinear diffusions and interactions: new methods and directions, 117-158,
    Lecture Notes in Math., 2186, Fond. CIME/CIME Found. Subser., Springer, Cham, 2017.
  8. A transportation approach to universality in random matrix theory
    Boll. Unione Mat. Ital. 10 (2017), no. 1, 55-74.
  9. An overview of unconstrained free boundary problems (with H. Shahgholian)
    Philos. Trans. A 373 (2015), no. 2050, 20140281, 11 pp.
  10. Perimeter of sets and BMO-type norms (with L. Ambrosio, J. Bourgain, and H. Brezis)
    C. R. Math. Acad. Sci. Paris 352 (2014), no. 9, 697-698.
  11. Stability results for the Brunn-Minkowski inequality
    Colloquium De Giorgi 2013 and 2014, 119-127, Colloquia 5, Ed. Norm., Pisa, 2014.
  12. Quantitative stability results for the Brunn-Minkowski inequality
    Proceedings of the International Congress of Mathematicians, 2014. Vol. III, 237-256.
  13. Partial regularity results in optimal transportation (with G. De Philippis)
    Trends in Contemporary Mathematics, Springer INdAM Series Vol. 8, (2014), 293-307
  14. The Monge-Ampère equation and its link to optimal transportation (with G. De Philippis)
    Bull. Amer. Math. Soc. (N.S.) 51 (2014), no. 4, 527-580.
  15. Sobolev regularity for the Monge-Ampère equation, with application to the semigeostrophic equations
    Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 411 (2013), Teoriya Predstavlenii Dinamicheskie Sistemy, Kombinatornye Metody. XXII, 103-118, 242; translation in J. Math. Sci. (N.Y.) 196 (2014), no. 2, 175-183.
  16. Lecture notes on variational models for incompressible Euler equations (with L. Ambrosio)
    Optimal transportation, 58-71, London Math. Soc. Lecture Note Ser. 413, Cambridge Univ. Press, Cambridge, 2014.
  17. Aubry sets, Hamilton-Jacobi equations, and the Mañé Conjecture (with L. Rifford)
    Geometric analysis, mathematical relativity, and nonlinear partial differential equations, 83-104, Contemp. Math. 599, Amer. Math. Soc., Providence, RI, 2013.
  18. Variational models for the incompressible Euler equations (with S. Daneri)
    HCDTE lecture notes. Part II. Nonlinear hyperbolic PDEs, dispersive and transport equations, 51 pp., AIMS Ser. Appl. Math. 7, Am. Inst. Math. Sci. (AIMS), Springfield, MO, 2013.
  19. Stability in geometric and functional inequalities
    European Congress of Mathematics, 585-599, Eur. Math. Soc., Zurich, 2013.
  20. Existence and uniqueness results for the continuity equation and applications to the chromatography system (with L. Ambrosio, G. Crippa, and L. V. Spinolo)
    Nonlinear conservation laws and applications, 195-204, IMA Vol. Math. Appl. 153, Springer, New York, 2011.
  21. Quantitative isoperimetric inequalities, with applications to the stability of liquid drops and crystals
    Concentration, functional inequalities and isoperimetry, 77-87, Contemp. Math. 545, Amer. Math. Soc., Providence, RI, 2011.
  22. Optimal Transport and Curvature (with C. Villani)
    Nonlinear PDE's and applications, 171-217, Lecture Notes in Math. 2028, Springer, Heidelberg, 2011.
  23. Almost everywhere well-posedness of continuity equations with measure initial data (with L. Ambrosio)
    C. R. Math. Acad. Sci. Paris 348 (2010), no. 5-6, 249-252.
  24. Optimal Transport. Old and New. [book review]
    Bull. Amer. Math. Soc. (N.S.) 47 (2010), no. 4, 723-727.
  25. Regularity of optimal transport maps [after Ma-Trudinger-Wang and Loeper]
    Séminaire Bourbaki. Vol. 2008/2009. Exposés 997-1011. Astérisque 332 (2010), Exp. No. 1009, ix, 341-368.
  26. Cédric Villani reçoit un prix de la Société Mathématique Européenne
    (French) [Cédric Villani, 2008 EMS Prize] (with L. Desvillettes), Gaz. Math. no. 120 (2009), 76-81.
  27. Optimal transport, Euler equations, Mather and DiPerna-Lions theories.
    Mémoire d'Habilitation à Diriger de Recherche (HDR). Nice, 2009.