## Optimal transport and the Monge-Ampère equation

The optimal transport problem is finding the cheapest way of transporting a distribution of mass from one location to another given some cost measure. Apart from their economic applications, optimal transportation theory and techniques have become increasingly common and power tools in tackling some of the most challenging questions in partial differential equations, fluid mechanics, geometry, probability, and functional analysis. However, depending on the cost function and geometry of the ambient space, the existence, uniqueness, and regularity of optimal maps are non-trivial issues.

Figalli’s contributions are numerous. For instance, with Fathi, he obtained the existence and uniqueness of optimal maps on non-compact Riemannian manifolds for general Lagrangian-action costs. Also, he has proved the existence and uniqueness of optimal maps in the partial transport problem and studied their regularity properties.

The regularity of optimal transport maps for quadratic cost, where the cost is given by the squared Euclidean distance, is intimately related to the regularity of solutions to the Monge-Ampère equation.
In collaboration with De Philippis, Figalli has proved second-order Sobolev regularity for the Monge-Ampère equation in the borderline case when the right-hand side is only bounded away from zero and infinity. This regularity result led to understanding the existence and regularity of solutions to the semigeostrophic equations, a classical system of partial differential equations used in meteorology.

More generally, the regularity of solutions to the optimal transport problem for arbitrary costs is tied to the regularity of solutions to Monge-Ampère-type equations. Together, Figalli and De Philippis have obtained a general partial regularity result for optimal transport maps, which is also valid in situations where the structure and data of the problem often rule out the possibility of local and global regularity estimates.

Also, in joint works with Rifford and Villani, Figalli has used regularity properties of optimal transports to understand the geometric structure of the underlying space and the stability of the cut locus.

## Geometric and functional inequalities

Geometric and functional inequalities play a crucial role in many problems arising in, among other areas, the calculus of variations, partial differential equations, and geometry.

With Maggi and Pratelli, Figalli has used optimal transportation techniques to obtain a sharp quantitative stability theorem for the Wulff inequality. In the crystalline setting, this result has then been improved with Zhang.

Figalli also obtained several sharp quantitative stability results for other geometric inequalities and functional inequalities of Sobolev-type, regarding the stability of minimizers and critical points. Furthermore, he investigated the quantitative stability for the Brunn-Minkowski inequality. In particular, together with Maggi and Pratelli, he obtained a sharp stability result for the Brunn-Minkowski inequality on convex sets. Then, with Jerison, Figalli combined tools from additive combinatorics, affine geometry, and harmonic analysis to obtain the first quantitative stability result for the Brunn-Minkowski inequality in arbitrary dimension on generic sets. More recently, he has also obtained stability results in the more general context of the Prekopa-Leindler inequality.

## Free boundary problems

Free boundary problems naturally arise when studying many physical phenomena. Typically, paired with the solution is a domain determined by the solution and the problem’s specific data, whose boundary is called the free boundary. The main questions in free boundary problems are the solution’s regularity and the free boundary’s structure and regularity, i.e., the boundary of the solution determined domain.

Figalli has studied free boundary problems both for local and nonlocal operators. In particular, in collaboration with Caffarelli and with Barrios and Ros-Oton, he proved several general results regarding the regularity of the solution and the free boundary in the parabolic and elliptic fractional obstacle problems.

Also, together with Serra, he proved a very fine result on the structure of the singular set for the classical obstacle problem, improving previous contributions by Caffarelli and Weiss. The techniques introduced with Serra have been further developed in a paper with Ros-Oton and Serra, where they proved the generic regularity of the free boundary for the obstacle problem, solving a conjecture of Schaeffer in dimensions n≤4. These ideas also played a crucial role in understanding the free boundary behavior in the Stefan problem.

## Local and nonlocal elliptic operators

Both local and nonlocal equations have their roots in different physical phenomena. They appear in, among other areas, the dislocation dynamics in crystals, conformal geometry, probability (considering either the Brownian motion or general jump processes), and phase transition theory. Because of their many appearances, their study has received increasing attention.

Figalli’s contributions to this setting include the development of a Schauder theory in the nonlocal setting and several regularity and rigidity results and estimates for nonlocal minimal surfaces, a natural nonlocal variant of classical minimal surfaces that arises in the study of phase transitions.

With Cabré, Ros-Oton, and Serra, Figalli proved the smoothness of stable solutions to semilinear elliptic equations with convex nonlinearities n dimensions n≤9, solving a celebrated conjecture of Brezis. Also, with Zhang, he extended this result to the case of finite Morse index solutions.

In addition, Figalli has also studied degenerate equations of porous medium type together with Bonforte and Vazquez, developing a complete study of nonnegative solutions to these types of equations driven by local or nonlocal diffusion.

## Evolution equations

A major discovery of Otto was that one could interpret many evolution equations as gradient flows in the space of probability measures endowed with the Wasserstein distance.

In joint work with Carrillo, Di Francesco, Laurent, and Slepcev, Figalli was able to take advantage of this interpretation to study interaction equations modeling the collective behavior of individuals where finite time blow-up is expected. In particular, they developed a global well-posedness theory for measure-valued solutions.

Together with Carlen, Figalli has studied the asymptotic behavior for the critical mass Keller-Segel equation, a classical model of the macroscopic description of chemotaxis. In their work, Carlen and Figalli leverage quantitative stability results of some Gagliardo-Nirenberg and logarithmic Hardy-Littlewood-Sobolev inequalities to show a quantitative rate of convergence to a steady-state.

## Transport equations

Transport equations with rough coefficients play a prominent role in mathematical physics and continuum mechanics. In their seminal paper, DiPerna and Lions proved a precise correspondence between global well-posedness of transport equations with Sobolev vector fields and almost everywhere existence and uniqueness for the associated ODE. Ambrosio then extended this to the case of BV vector fields.

Figalli has developed this theory in many settings. First, he extended this duality theory to the cases of stochastic differential equations with rough coefficients and drifts. Then, with Ambrosio, Figalli developed DiPerna-Lions’s theory for rough vector fields in infinite-dimensional spaces, when the reference measure is different from Lebesgue. Consequently, in collaboration with Ambrosio, Friesecke, Giannoulis, and Paul, Figalli was able to study the semiclassical limit of the Schrödinger equation with very rough potentials. Later on, with Ambrosio and Colombo, Figalli has developed a local version of the DiPerna-Lions theory, corresponding to the classical local Cauchy-Lipschitz theory, to the case of rough vector fields. This general theory has, for example, allowed Ambrosio, Colombo, and Figalli to provide new insight on physical systems like the Vlasov-Poisson equation.

## Sub-Riemannian manifolds

Sub-Riemannian manifolds are a generalization of Riemannian manifolds where to measure distances, one can only go along curves tangent to so-called horizontal subspaces.

Sub-Riemannian manifolds often occur in the study of constrained systems in classical mechanics, such as the motion of vehicles on a surface and of robot arms, or the orbital dynamics of satellites. The Heisenberg group provides a fundamental example of a sub-Riemannian manifold.

Together with Rifford, Figalli has studied the existence and uniqueness of optimal transport maps on sub-Riemannian manifolds.Very recently, in collaboration with Belotto da Silva, Parusiński, and Rifford, he proved the strong Sard conjecture for sub-Riemannian structures on three-dimensional analytic manifolds and the C^1 regularity of minimizing geodesics.

## Random matrices

Large random matrices arise as natural models in diverse fields such as quantum mechanics, quantum chaos, telecommunications, finance, and statistics. The central mathematical question in this area is: how much do the asymptotic properties of the spectrum depend on the fine detail of the model? This question dates back to the 1950s, and Wigner proved that the distribution of the spectrum converges, under very mild assumptions, to the so-called semi-circle law. However, the local properties of the spectrum are still not fully understood.

In collaboration with Guionnet and Bekelman, Figalli introduced a new approach to this question, considering suitable approximate transport maps to provide effective changes of variables between different laws. Then, with Guionnet, he pushed this approach further to obtain universality in perturbative several-matrix models.

## Weak-KAM theory

Given an autonomous Hamiltonian system on a smooth, compact Riemannian manifold without boundary, the so-called Aubry set captures many essential features of the system’s Hamiltonian dynamics. Moreover, its structure is strongly related to the regularity of viscosity solutions to the Hamilton-Jacobi equation. A celebrated conjecture due to Mañé states that generically the Aubry set is either a fixed point or a periodic orbit.

In two papers with Rifford, Figalli proved Mañé’s conjecture in several cases. In particular, they established that Mañé’s conjecture is equivalent to the generic existence of smooth critical viscosity subsolutions to the Hamilton-Jacobi equation. Then, building on these techniques, Contreras, Figalli, and Rifford, solved a problem proposed by Herman during the ICM-1998 on the generic hyperbolicity of the Aubry set on surfaces.

## Variational methods for the incompressible Euler equations

Starting from Arnold’s seminal interpretation of solutions to the incompressible Euler equations as geodesics in the space of measure-preserving diffeomorphisms, in the 1980s Brenier introduced a relaxed model to study the problem of finding minimizing geodesics in this infinite dimensional space.

Exploiting tools and ideas from geometric measure theory and optimal transportation, Ambrosio and Figalli have obtained fine regularity estimates on the pressure in the incompressible Euler equations. These estimates, in turn, have allowed them to find sharp necessary and sufficient optimality conditions to the relaxed geodesic problem.

This has become a powerful tool to find and classify generalized geodesics.