Research

I work in calculus of variations and partial differential equations, with a focus on regularity, stability, and geometric structure in nonlinear problems. A central theme of my research is to understand when solutions, minimizers, interfaces, or transport maps are more regular, more stable, or more rigid than expected. My work connects optimal transport and Monge–Ampère equations, free boundary problems, geometric and functional inequalities, elliptic PDEs, evolution equations, transport equations, and geometric measure theory.

For a broad description of the research areas, questions, and methods, see Research Description

For a result-oriented summary of the main mathematical achievements, see Main Contributions

For a curated list of representative papers, see Selected Papers

Publication lists

For a complete list of publications, see Published/Accepted Papers and Submitted papers and preprints. See also Surveys and lecture notes and Books.

Research areas

The following pages list publications grouped by research topic.

Free boundary problems
Regularity and singularities of interfaces determined by solutions of PDEs, including the obstacle problem, the Stefan problem, Bernoulli and free boundary Allen–Cahn problems, constraint maps, and nonlocal obstacle problems, with emphasis on generic regularity, stratification of singular sets, and classification of global solutions.

Functional and geometric inequalities
Sharp quantitative forms of classical inequalities, measuring how close near-optimizers are to exact ones, including isoperimetric and Wulff inequalities, Brunn–Minkowski, Prékopa–Leindler, Borell–Brascamp–Lieb, Sobolev, and log-Sobolev stability.

Elliptic PDEs
Regularity and rigidity for nonlinear elliptic equations, especially stable and finite Morse index solutions, Serrin-type overdetermined problems, boundary reactions, degenerate equations, and convex-envelope regularity.

Regularity of optimal maps and Monge–Ampère equations
Regularity and partial regularity for optimal maps and Monge–Ampère equations, including Sobolev estimates, boundary ε-regularity, stability under perturbations, and optimal maps between measures with rough or singular densities.

General optimal transport theory
Results on the existence, uniqueness, and structure of optimal maps, including partial transport, non-compact manifolds, sub-Riemannian geometries, and economic screening models.

Optimal transport and Riemannian geometry
How the geometry of a manifold controls optimal transport maps, through cut loci, injectivity domains, Ma–Trudinger–Wang curvature, and the convexity properties needed for continuity and regularity.

Evolution equations and gradient flows
Analysis of nonlinear time-dependent PDEs through variational and transport methods, including fast diffusion, porous medium equations, aggregation, Keller–Segel, kinetic alignment models, and Wasserstein-type gradient flows.

Non-local energies and elliptic operators
Regularity and rigidity for problems governed by nonlocal interactions, including fractional and integro-differential operators, nonlocal minimal surfaces, fractional perimeter, nonlocal mean curvature, and interaction energies.

Calculus of Variations
Minimization and geometric structure in variational models, from liquid drops, crystals, and thin elastic sheets to generalized Euler flows, measure-preserving maps, and singular minimizing geodesics.

Sets of finite perimeter and geometric measure theory
Regularity and structure of sets and currents under minimal assumptions, including perimeter in Euclidean and Wiener spaces, BMO-type perimeter norms, recognition of convexity from marginals, and codimension-one minimizing currents.

Transport equations
Well-posedness and Lagrangian structure for transport and continuity equations with nonsmooth vector fields, with applications to Vlasov–Poisson, semigeostrophic equations, semiclassical limits, and chromatography systems.

Stochastic analysis and random matrices
Transport-based approaches to random matrix universality, together with stochastic equations with rough or degenerate coefficients and propagation-of-chaos limits.

Dynamical systems, weak KAM, and symplectic geometry
Hamilton–Jacobi and weak KAM questions connected with Aubry–Mather sets, generic hyperbolicity and closing results, together with selected work on symplectic capacities and integrable systems.

Applied Mathematics
Selected collaborations where analytical, variational, PDE, and optimal-transport methods are used in concrete models outside pure analysis, including sensitivity analysis, finance, physics experiment design, and selected problems in data analysis.