Main Contributions
My research is centered on calculus of variations and partial differential equations, with a particular emphasis on regularity, stability, singularities, and geometric structure. A recurring theme in my work is the development of methods that reveal hidden structure in nonlinear problems: proving regularity from weak assumptions, understanding the geometry of singular sets, quantifying stability near optimal configurations, and classifying the global models that arise in blow-up analysis.
The following is a summary of some of my main mathematical contributions.
Optimal transport and Monge–Ampère equations
A major part of my work concerns optimal transport and the Monge–Ampère equation. Optimal transport asks how to move one mass distribution to another in an optimal way, but it has also become a powerful tool in partial differential equations, geometry, probability, and mathematical physics.
The papers listed below reflect two complementary aspects of this work. The first is regularity theory: I have studied Sobolev regularity for Alexandrov solutions of the Monge–Ampère equation and partial regularity for optimal transport maps, showing that even when singularities cannot be excluded globally, the regular part of the solution has a strong and robust structure. The second is the use of optimal-transport ideas beyond their original setting, for instance through approximate transport maps in random matrix theory.
Together, these works illustrate one of the central roles of optimal transport in my research: it is both a subject of regularity theory and a method for solving problems in other areas of analysis and probability.
Representative papers:
$W^{2,1}$ regularity for solutions of the Monge–Ampère equation (with G. De Philippis, Invent. Math., 2013)
Partial regularity for optimal transport maps (with G. De Philippis, Publ. Math. Inst. Hautes Études Sci., 2015)
Universality in several-matrix models via approximate transport maps (with A. Guionnet, Acta Math., 2016)
Quantitative stability in geometric and functional inequalities
Another central direction of my research is quantitative stability. Many sharp geometric and functional inequalities identify the optimal constant and characterize the equality cases. Stability asks a finer question: if a set or function almost attains equality, must it be quantitatively close to an optimizer?
The papers listed below develop this question in several fundamental settings, proving sharp quantitative stability for isoperimetric inequalities, Wulff inequalities with crystalline norms, Brunn–Minkowski inequalities, Prékopa–Leindler and Borell–Brascamp–Lieb inequalities, and Sobolev and log-Sobolev inequalities.
A common goal in these works is to go beyond the identification of optimizers and understand the precise geometry of near-optimizers. Depending on the problem, this means finding the correct notion of distance to the family of optimizers, the optimal exponent in terms of the deficit, or the optimal dependence on the dimension.
Representative papers:
A mass transportation approach to quantitative isoperimetric inequalities (with F. Maggi and A. Pratelli, Invent. Math., 2010)
Strong stability for the Wulff inequality with a crystalline norm (with Y. Ru-Ya Zhang, Comm. Pure Appl. Math., 2022)
Sharp gradient stability for the Sobolev inequality (with Y. Ru-Ya Zhang, Duke Math. J., 2022)
Sharp stability for Sobolev and log-Sobolev inequalities, with optimal dimensional dependence (with J. Dolbeault, M. J. Esteban, R. L. Frank, and M. Loss, Camb. J. Math., 2025)
Sharp quantitative stability of the Brunn-Minkowski inequality (with P. van Hintum and M. Tiba, Preprint, 2025)
Sharp Quantitative Stability for the Prékopa-Leindler and Borell-Brascamp-Lieb Inequalities (with P. van Hintum and M. Tiba, Preprint, 2026)
Free boundary problems, obstacle problems, and singularities
In recent years, a central part of my research has concerned free boundary problems. These are problems in which the unknown is not only a function, but also an interface determined by the solution itself. The main questions are the regularity of this interface, the structure of its singularities, and the classification of global solutions that arise as blow-up limits near singular points.
A particularly important part of this work concerns the classical obstacle problem. This problem models, for example, the equilibrium shape of an elastic membrane constrained to lie above an obstacle. The solution is harmonic away from the contact set, while the interface between the contact and non-contact regions forms the free boundary. Since the 1960s, understanding the geometry and regularity of this free boundary has been a central question in nonlinear analysis.
The papers listed below develop a broad program around this problem. One line of work gives a fine description of the singular set in the classical obstacle problem. Another proves generic regularity of free boundaries in low dimensions, showing that singular free-boundary points disappear under generic perturbations and addressing Schaeffer’s conjecture in that range. A more recent result gives a complete classification of global solutions to the classical obstacle problem and resolves the related problem of characterizing null quadrature domains.
This program also extends to other central free boundary problems. For the Stefan problem, a classical model for phase transitions such as melting ice, my work gives a refined description of the singular set, including stratification and dimension estimates. For Bernoulli-type and free boundary Allen–Cahn problems, my work proves a Bernstein-type rigidity theorem showing that global stable solutions in dimension three are one-dimensional, with consequences for curvature estimates for local stable free boundaries.
Together, these works aim to understand free boundaries at several levels: the fine structure of singular points, the generic absence of singularities, the classification of global models, and the rigidity of stable solutions.
Representative papers:
On the fine structure of the free boundary for the classical obstacle problem (with J. Serra, Invent. Math., 2019)
Generic regularity of free boundaries for the obstacle problem (with X. Ros-Oton and J. Serra, Publ. Math. Inst. Hautes Études Sci., 2020)
The singular set in the Stefan problem (with X. Ros-Oton and J. Serra, J. Amer. Math. Soc., 2024)
Complete classification of global solutions to the obstacle problem (with S. Eberle and G.S. Weiss, Ann. of Math., 2025),
Global stable solutions to the free boundary Allen–Cahn and Bernoulli problems in 3D are one-dimensional (with H. Chan, X. Fernández-Real, and J. Serra, J. Amer. Math. Soc., to appear.)
Elliptic PDE, stable solutions, and singular sets
I have also worked on regularity and classification problems for nonlinear elliptic PDE, especially stable solutions of semilinear elliptic equations. Stability is a variational condition: it means that the second variation is nonnegative. This condition strongly restricts the possible behavior of solutions, but in high dimensions singularities may still occur.
The papers listed below address several aspects of this problem. One result proves that stable solutions to semilinear elliptic equations are bounded, and hence smooth, up to dimension 9 for a broad class of nonlinearities; this dimension threshold is optimal. Another result proves a De Giorgi-type classification theorem for stable solutions with nonlinear boundary reactions, or equivalently in a half-Laplacian setting.
More recent work studies stable semilinear elliptic equations near singular points. It proves an ε-regularity criterion à la Brezis and gives quantitative bounds for the Hausdorff dimension of the singular set. These results are part of a broader effort to understand how stability controls regularity, rigidity, and the possible formation of singularities in nonlinear elliptic equations.
Representative papers:
Stable solutions to semilinear elliptic equations are smooth up to dimension 9 (with X. Cabré, X. Ros-Oton, and J. Serra, Acta Math., 2020)
On stable solutions for boundary reactions: a De Giorgi type result in dimension $4+1$ (with J. Serra, Invent. Math., 2020)
Stable semilinear elliptic equations: $\varepsilon$-regularity à la Brezis and dimensional bounds for the singular set (with F. Franceschini), Preprint, 2026)