Broadly speaking, my research interests are in nonlinear partial differential equations and geometric analysis. My work largely falls within the following themes:
Conformal geometry (existence and uniqueness for fully nonlinear Yamabe-type problems, including the study of conformally compact metrics of negative curvature type)
Regularity theory (regularity of viscosity solutions and strong solutions to fully nonlinear augmented Hessian equations, including fully nonlinear Yamabe-type equations)
Spectral geometry (eigenvalue estimates on compact manifolds, e.g. Laplace eigenvalue estimates on minimal surfaces and Steklov eigenvalue estimates on manifolds with nonnegative Ricci curvature)
(10) The sigma_k-Loewner-Nirenberg problem on Riemannian manifolds for k=n/2 and beyond (with Luc Nguyen)
Submitted (2025)
(9) The fully nonlinear Loewner-Nirenberg problem: Liouville theorems and counterexamples to local boundary estimates (with Luc Nguyen)
Submitted (2025)
(8) The first Steklov eigenvalue on manifolds with non-negative Ricci curvature and convex boundary (with Aditya Kumar)
Journal of Geometric Analysis, vol. 35, no. 95 (2025)
(7) Fully nonlinear Yamabe-type problems on non-compact manifolds (with Yi Wang)
Preprint 2024, to appear in Communications in Analysis and Geometry
(6) The sigma_k-Loewner-Nirenberg problem on Riemannian manifolds for k<n/2 (with Luc Nguyen)
Preprint 2023, to appear in Analysis & PDE
(5) An improved eigenvalue estimate for embedded minimal hypersurfaces in the sphere (with Yannick Sire, Joel Spruck)
International Mathematics Research Notices, vol. 2024, no. 18 (2024)
(4) Augmented Hessian equations on Riemannian manifolds: from integral to pointwise local second derivative estimates
Nonlinear Analysis, vol. 238 (2024)
(3) Interior regularity for strong solutions to a class of fully nonlinear elliptic equations
Proceedings of the American Mathematical Society, vol. 151, no. 10 (2023)
(2) Differential inclusions for the Schouten tensor and nonlinear eigenvalue problems in conformal geometry (with Luc Nguyen)
Advances in Mathematics, vol. 432 (2023)
(1) Local pointwise second derivative estimates for strong solutions to the sigma_k Yamabe problem on Euclidean domains (with Luc Nguyen)
Calculus of Variations and Partial Differential Equations, vol. 60, no. 5 (2021)
Geometry Seminar, University College London (upcoming, 15th October 2025)
Analysis Seminar, University of Bath (upcoming, 9th October 2025)
Perspectives in Geometric Analysis, Politecnico di Milano (upcoming, 8th-11th September 2025)
Analysis Seminar, University of Warwick (May 2025)
AIMS Conference on Dynamical Systems, Differential Equations and Applications, Abu Dhabi National Exhibition Centre (Dec 2024)
Conformal and CR Geometry Workshop, Banff International Research Station (Dec 2024)
AMS Sectional Meeting: Recent Developments in Geometric Analysis, Howard University (Apr 2024)
Geometric Analysis seminar, University of Chicago (Feb 2024)
Differential Geometry seminar, University of Iowa (Oct 2023)
AMS Sectional Meeting: Geometric and Analytic Methods in PDE, University of Cincinnati (Apr 2023)
AMS Sectional Meeting: Nonlinear PDEs from Variational Problems & Fluid Dynamics, University of Cincinnati (Apr 2023)
Nonlinear Analysis Seminar, Rutgers University (Dec 2022)
Differential Geometry & Geometric Analysis seminar, Princeton University (Dec 2022)
PDE & Geometric Analysis seminar, UW Madison (Feb 2022)
Felix Klein Seminar in Geometry, University of Notre Dame (Jan 2022)
Analysis & PDEs seminar, Johns Hopkins University (Mar 2021)
PDE CDT Lunchtime Seminar, University of Oxford (Jan 2021)