Full Professor of Mathematical Analysis at the Scuola Normale since 1998. As a student of the Scuola Normale, he graduated in Mathematics from the University of Pisa and subsequently held positions as a Professor of Mathematical Analysis at the University of Pisa, the University of Salerno, and then at the University of Pavia.
He was a member of the editorial committee of "Archive for Rational Mechanics and Analysis", "Calculus of Variations and Partial Differential Equations" and the "Journal of the European Mathematical Society", and was an invited speaker at the II European Congress in Budapest (1996), the International Congress of Mathematics in Beijing (2002), and a Plenary Speaker at the International Congress of Mathematics in Rio De Janeiro (2018). He is a corresponding member of the Accademia Nazionale dei Lincei and the Istituto Lombardo, Accademia di Scienze e Lettere.
He has been awarded several prizes for his research activities, including the Premio Nazionale per la Matematica e la Meccanica (National Prize for Mathematics and Mechanics) from the Italian Ministry of Research and University (1996), the Caccioppoli Prize from the Unione Matematica Italiana (Italian Mathematical Union) (1999), the Fermat Prize from the Université de Toulouse (2003), the Gold Medal for Mathematics, conferred by the Accademia Nazionale dei XL (2015), the Balzan Prize (2019), and the Riemann Prize (2023).
His main research interests are in Calculus of Variations, Geometric Measure Theory, Optimal Transport Theory, and their applications to Partial Differential Equations.
Since 2019, he has been the Director of the Scuola Normale Superiore.
Among his most significant publications, we can mention:
 Existence theory for a new class of variational problems, Archive for Rational Mechanics & Analysis, 111, 291-322, 1990.
 Approximation of functionals depending on jumps by elliptic functionals via Γ-convergence (with V.M.Tortorelli), Communications on Pure and Applied Mathematics, 43, 999-1036, 1990.
 On the propagation of singularities of semi-convex functions, (with P.Cannarsa, H.M.Soner), Ann. Sc. Norm. Sup., 20, 597-616, 1993.
 A level set approach for the evolution of surfaces of any codimension, (with H.M.Soner), J. Diff. Geom. 43 (1996), 693-737.
 Partial regularity of free discontinuity sets I, II, (with N.Fusco, D.Pallara), Annali della Scuola Normale Superiore di Pisa, 24 (1997), 1-38 and 39-62.
 Currents in metric spaces, (with B.Kirchheim), Acta Math., 185 (2000), 1-80.
 Transport equation and Cauchy problem for BV vector fields, Inventiones Math. 158 (2004), 227-260.
 Existence of solutions for a class of hyperbolic systems of conservation laws in several space dimensions, (with C.De Lellis) IMRN, 41 (2003), 2205-2220.
 Calculus and heat flow in metric measure spaces and applications to spaces with ricci bounds from below, (with N.Gigli, G.Savaré), Inventiones Mathematicae, 195 (2014), 289-391.
 Metric measure spaces with Riemannian Ricci bounds from below, (with N.Gigli, G.Savaré) Duke Math. J., 163 (2014), 1405-1490.
 Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds, (with N.Gigli, G.Savaré) Annals of Probability, 43 (2015), 339-404.
 On the duality between p-modulus and probability measures, (with S.Di Marino, G.Savaré) Journal of the European Mathematical Society, 17 (2015), 1817-1853.
 Well posedness of Lagrangian flows and continuity equations in metric measure spaces, (with D.Trevisan) Analysis & PDE, 7 (2014), 1179-1234.
 A BMO-type seminorm related to the perimeter of sets, (with J.Bourgain, H.Brezis, A.Figalli) Communications in Pure and Applied Mathematics, LXIX (2016), 1062-1086.
 A PDE approach to a 2-dimensional matching problem, (with F.Stra, D.Trevisan) Probability theory and Related Fields, 173 (2019), 443-477.