Figalli has worked in the theory of
optimal transport, with particular emphasis on the regularity theory of optimal transport maps and its connections to
Monge–Ampère equations. Amongst the results he obtained in this direction, there stand out an important higher integrability property of the second derivatives of solutions to the Monge–Ampère equation and a partial regularity result for Monge–Ampère type equations, both proved together with
Guido de Philippis. He used optimal transport techniques to get improved versions of the anisotropic
isoperimetric inequality, and obtained several other important results on the stability of functional and geometric inequalities. In particular, together with Francesco Maggi and Aldo Pratelli, he proved a sharp quantitative version of the anisotropic
isoperimetric inequality. Then, in a joint work with
Eric Carlen, he addressed the stability analysis of some
Gagliardo–Nirenberg and logarithmic
Hardy–Littlewood–Sobolev inequalities to obtain a quantitative rate of convergence for the critical mass Keller–Segel equation. He also worked on
Hamilton–Jacobi equations and their connections to weak
Kolmogorov–Arnold–Moser theory. In a paper with Gonzalo Contreras and Ludovic Rifford, he proved generic hyperbolicity of Aubry sets on compact surfaces. In addition, he has given several contributions to the Di Perna–Lions' theory, applying it both to the understanding of
semiclassical limits of the
Schrödinger equation with very rough potentials, and to study the Lagrangian structure of weak solutions to the
Vlasov–Poisson equation. More recently, in collaboration with
Alice Guionnet, he introduced and developed new transportation techniques in the topic of
random matrices to prove universality results in several-matrix models. Also, together with Joaquim Serra, he proved the
De Giorgi's conjecture for boundary reaction terms in dimension lower than five, and he improved the classical results by
Luis Caffarelli on the structure of singular points in the
obstacle problem. ==Books==