Selected Papers
This is a curated list of selected papers, grouped according to the themes below:
- Optimal transport and Monge–Ampère equations
- Quantitative stability in geometric and functional inequalities
- Free boundary problems, obstacle problems, and singularities
- Elliptic PDE, stable solutions, and singular sets
- Additional selected papers
The first four sections correspond to the main themes described in Main Mathematical Contributions. The final section contains additional selected papers in related directions. For a complete list, see Published/Accepted Papers and Submitted papers and preprints.
Optimal transport and Monge–Ampère equations
$W^{2,1}$ regularity for solutions of the Monge–Ampère equation (with G. De Philippis, Invent. Math., 2013)
Proves local $W^{2,1}$ regularity for Alexandrov solutions whose Monge–Ampère density is bounded above and below. The key estimate is a higher integrability of the Hessian, which rules out a possible singular Cantor part.
Partial regularity for optimal transport maps (with G. De Philippis, Publ. Math. Inst. Hautes Études Sci., 2015)
Proves that optimal transport maps are smooth outside a closed singular set of measure zero, under natural assumptions on the source and target densities and domains. This gives a partial regularity theory for optimal transport in situations where global smoothness may fail.
Universality in several-matrix models via approximate transport maps (with A. Guionnet, Acta Math., 2016)
Constructs approximate transport maps for perturbative several-matrix models and uses them to prove universality of local eigenvalue statistics, including sine-kernel behavior in the bulk and Tracy–Widom behavior at the edge.
Quantitative stability in geometric and functional inequalities
A mass transportation approach to quantitative isoperimetric inequalities (with F. Maggi and A. Pratelli, Invent. Math., 2010)
Uses Brenier–McCann optimal transport and Gromov’s proof of the isoperimetric inequality to obtain a sharp quantitative stability theorem for the anisotropic isoperimetric inequality. As a consequence, it also gives a sharp Brunn–Minkowski stability result for convex sets.
Strong stability for the Wulff inequality with a crystalline norm (with Y. Ru-Ya Zhang, Comm. Pure Appl. Math., 2022)
Proves a linear stability estimate for crystalline surface tensions: near-minimizers are close to polyhedra with the same facet directions as the Wulff shape. It also gives a rigidity result showing that small-mass minimizers keep their crystalline structure under external potentials.
Sharp gradient stability for the Sobolev inequality (with Y. Ru-Ya Zhang, Duke Math. J., 2022)
Proves sharp quantitative stability for the $p$-Sobolev inequality in the strongest natural gradient distance to the manifold of optimizers. The paper identifies the optimal exponent and shows a different behavior depending on $p$.
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)
Gives sharp quantitative stability for Sobolev inequalities with explicit constants having the correct high-dimensional behavior. The large-dimensional limit yields an optimal dimension-free stability estimate for the Gaussian log-Sobolev inequality.
Sharp quantitative stability of the Brunn-Minkowski inequality (with P. van Hintum and M. Tiba, Preprint, 2025)
Provides a conclusive sharp stability theorem for Brunn–Minkowski for arbitrary bounded measurable sets. It gives the optimal quantitative rate for how near-equality forces two sets to be close to the same convex body.
Sharp Quantitative Stability for the Prékopa-Leindler and Borell-Brascamp-Lieb Inequalities (with P. van Hintum and M. Tiba, Preprint, 2026)
Develops a unified sharp stability framework for Borell–Brascamp–Lieb inequalities. As a consequence, it resolves the conjectured sharp stability form of the Prékopa–Leindler inequality, including the log-concave case.
Free boundary problems, obstacle problems, and singularities
On the fine structure of the free boundary for the classical obstacle problem (with J. Serra, Invent. Math., 2019)
Refines the stratification of singular free boundary points in the obstacle problem: in dimension two, singular points lie on $C^2$ curves, while in higher dimensions the singular set is described by $C^{1,1}$ manifolds up to anomalous strata.
Generic regularity of free boundaries for the obstacle problem (with X. Ros-Oton and J. Serra, Publ. Math. Inst. Hautes Études Sci., 2020)
Shows that singular free-boundary points disappear generically in low dimensions. More precisely, the singular set has codimension at least three in the free boundary, implying generic smoothness for $n \leq 4$ and settling Schaeffer's conjecture in that range.
The singular set in the Stefan problem (with X. Ros-Oton and J. Serra, J. Amer. Math. Soc., 2024)
Develops a fine stratification theory for singularities in the Stefan problem. The paper bounds the parabolic Hausdorff dimension of the singular set, proves smooth expansions outside a lower-dimensional exceptional set, and gives strong regularity consequences in three space dimensions.
Complete classification of global solutions to the obstacle problem (with S. Eberle and G.S. Weiss, Ann. of Math., 2025)
Classifies all global solutions of the classical obstacle problem. Equivalently, it gives a conclusive answer to the long-standing problem of null quadrature domains, showing that only the classical model geometries can occur.
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)
Proves a Bernstein-type rigidity theorem for global stable solutions of the one-phase Bernoulli problem in dimension three and applies it to the free boundary Allen–Cahn problem. The result also yields universal curvature estimates for local stable Bernoulli free boundaries.
Elliptic PDE, stable solutions, and singular sets
Stable solutions to semilinear elliptic equations are smooth up to dimension 9 (with X. Cabré, X. Ros-Oton, and J. Serra, Acta Math., 2020)
Proves that stable solutions of semilinear elliptic equations are bounded, hence smooth, in dimensions $n \leq 9$, and that this threshold is optimal. The proof gives universal estimates independent of the nonlinearity and answers problems of Brezis and Brezis–Vázquez.
On stable solutions for boundary reactions: a De Giorgi type result in dimension $4+1$ (with J. Serra, Invent. Math., 2020)
Establishes a one-dimensional symmetry theorem for stable solutions with nonlinear boundary reactions. This gives a De Giorgi-type classification result for the half-Laplacian/boundary-reaction setting.
Stable Semilinear Elliptic Equations: Brezis-Type $\varepsilon$-Regularity and Dimensional Bounds for the Singular Set (with F. Franceschini, Preprint, 2026)
Proves an $\varepsilon$-regularity criterion near singular points, settling a conjecture of Brezis. It also gives explicit Hausdorff-dimension bounds for the singular set in terms of the nonlinearity, with optimal bounds for standard model nonlinearities.
Additional selected papers
Generic hyperbolicity of Aubry sets on surfaces (with G. Contreras and L. Rifford, Invent. Math., 2015)
Proves that, for an open dense set of $C^2$ potentials on a compact surface, the Aubry set of a Tonelli Hamiltonian is hyperbolic in its energy level. This settles a generic hyperbolicity problem in weak KAM/Aubry–Mather theory for surfaces.
Strong Sard Conjecture and regularity of singular minimizing geodesics for analytic sub-Riemannian structures in dimension 3 (with A. Belotto da Silva, A. Parusiński, and L. Rifford, Invent. Math., 2022)
Proves the Strong Sard conjecture for analytic three-dimensional sub-Riemannian structures: endpoints reached by singular horizontal paths form a set of Hausdorff dimension at most one. It also implies strong regularity of minimizing geodesics, which are proven to be $C^1$, and analytic away from finitely many points.
Constraint maps: singularities vs free boundaries (with A. Guerra, S. Kim, and H. Shahgholian, Preprint, 2024)
Studies energy-minimizing maps constrained to avoid an obstacle in the target, a vectorial analogue of the obstacle problem where harmonic-map singularities, branch points, and free boundaries may coexist. The paper separates these phenomena showing that, for uniformly convex constraints, topological singularities are confined to the interior of the contact set, while branch points can generate genuinely new free-boundary behavior.