## Optimal transport and applications

The optimal transport problem is to find the cheapest way of
transporting a distribution of mass from one location to another given
some measure of cost. Apart from their applications in economics, the
theory and techniques of optimal transportation have become
increasingly common and power tools in tackling the some of the most
challenging questions in partial differential equations, fluid
mechanics, geometry, probability, and functional analysis. 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.
He has proved the existence and uniqueness of optimal maps on non-compact Riemannian manifolds for general Lagrangian-action costs and
also on a wide class of sub-Riemannian manifolds. Figalli has proved
the existence and uniqueness of optimal maps in the partial transport
problem, as well as studied their regularity properties. In addition,
he has used optimal transportation techniques to prove new and improve
classical functional and geometric inequalities.
Figalli has also studied the regularity of optimal transports for
general cost functions and on Riemannian manifolds, and used his
understanding of the regularity of these maps to deduce strong
geometric properties of the underlying manifolds. Similarly, Figalli
has used his some of his regularity results to prove the existence and
regularity of solutions to evolution equations.

## Regularity of optimal maps and the Monge-Ampère equation

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.

The regularity of solutions to the optimal transport problem for
general cost is similarly tied to the regularity of solutions to Monge-Ampère-type equations. Together, Figalli and De Philippis have obtained
a general partial regularity result on Riemannain manifolds, a
situation where the structure and data of the problem often rule out
the possibility of local and global regularity.
Also, in joint work 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.

## 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 diffeomophisms, in the 1980’s, 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 problems, and
as a result, find and classify generalized geodesics.

## 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. Together with Maggi and Pratelli, Figalli has used optimal transportation techniques to obtain a sharp quantitative stability theorem for the Wulff inequality. In the cristalline 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, both regarding the stability of minimizers and the one of critical points. Furthermore, he investigated the quantitative stability for the Brunn-Minkowski inequality. 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.

## Evolution equations

A major discovery of Otto was that many evolution equations can be
interpreted as gradient flows in the space of probability measures
endowed with the Wasserstein distance. In a joint work with Carrillo,
Di Francesco, Laurent, and Slepcev, Figalli were 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. 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.

## 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. This was
then extended by Ambrosio to the case of BV vector fields.

Figalli has
developed this theory in many settings.
He extended this duality theory to the cases of stochastic differential
equations with rough coefficients and drifts. With Ambrosio, Figalli
extended DiPerna-Lions’s theory to the case of rough vector fields in
infinite dimensional spaces, when the reference measure is different
from Lebesgue. As a consequence, 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.

## 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 specific data of the problem. For instance, the contact set of the solution in the obstacle problem. The main questions in free boundary problems are the regularity of the solution and the structure and regularity of the free boundary, i.e., the boundary of the solution determined domain.

Figalli has studied free boundary problems both for local and nonlocal operator. In particular, in collaboration with Caffarelli and also 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 results on the structure of the singular set for the classical obstacle problem, improving previous results 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.

## Local and non-local elliptic operators

Both local and non-local 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 non-local elliptic setting, and several regularity and rigidity results and estimates for non-local minimal surfaces, a natural non-local variant of classical minimal surfaces that arises in the study of phase transitions. Concerning local elliptic equations, together with Cabré, Ros-Oton, and Serra, he proved the smoothness of stable solutions to semilinear elliptic equations with convex nonlinearities n dimensions n≤9, thus solving a conjecture of Brezis.

## 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 1950’s, and Wigner proved
that the distribution of the spectrum converges, under very mild
assumptions, to the so-called semi-circle law. However, 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 one with 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
important features of 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 establish 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 1998 ICM on the generic hyperbolicitiy of the
Aubry set on surfaces.