# Lecture notes and reviews

- The continuous formulation of shallow neural networks as wasserstein-type gradient flows (with X. Fernández-Real)

*Preprint*2020. - Free boundary regularity in obstacle problems

*Journées EDP*2018, to appear. - On the Monge-Ampère equation

*Séminaire Bourbaki.*Vol. 2017/2018. Exposé 1136-1150 (2019), Exp. No. 1148, 477–504. - 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. - 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. - 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. - A transportation approach to universality in random matrix theory

*Boll. Unione Mat. Ital.*10 (2017), no. 1, 55-74. - An overview of unconstrained free boundary problems (with H. Shahgholian)

*Philos. Trans. A*373 (2015), no. 2050, 20140281, 11 pp. - 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. - Stability results for the Brunn-Minkowski inequality

Colloquium De Giorgi 2013 and 2014, 119-127,*Colloquia*5, Ed. Norm., Pisa, 2014. - Quantitative stability results for the Brunn-Minkowski inequality

Proceedings of the International Congress of Mathematicians, 2014. Vol. III, 237-256. - Partial regularity results in optimal transportation (with G. De Philippis)

Trends in Contemporary Mathematics,*Springer INdAM Series*Vol. 8, (2014), 293-307 - 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. - 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. - 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. - 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. - 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. - Stability in geometric and functional inequalities

European Congress of Mathematics, 585-599,*Eur. Math. Soc.*, Zurich, 2013. - 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. - 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. - Optimal Transport and Curvature (with C. Villani)

Nonlinear PDE's and applications, 171-217,*Lecture Notes in Math.*2028, Springer, Heidelberg, 2011. - 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. - Optimal Transport. Old and New. [book review]

*Bull. Amer. Math. Soc. (N.S.)*47 (2010), no. 4, 723-727. - 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. - 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. - Optimal transport, Euler equations, Mather and DiPerna-Lions theories.

Mémoire d'Habilitation à Diriger de Recherche (HDR). Nice, 2009.