Research area: Differential Geometry, Partial Differential Equations, Calculus of Variations
My research is in the field of geometric analysis—the study of geometric partial differential equations and variational problems motivated by questions in geometry, topology, and mathematical physics. My work combines local analytical tools and global variational methods to study solutions of partial differential equations related to minimal submanifolds, harmonic maps, gauge theory, and geometric measure theory, with applications to the study of scalar curvature and spectral theory, among other areas.
Existence of harmonic maps and eigenvalue optimization in higher dimensions (with M. Karpukhin), arXiv preprint, arXiv:2207.13635.
Quantization and non-quantization of energy for higher-dimensional Ginzburg—Landau vortices (with A. Pigati), Ars Inveniendi Analytica (2023), Paper No. 3, 55p.
From Steklov to Laplace: free boundary minimal surfaces with many boundary components (with M. Karpukhin), arXiv preprint, arXiv:2109.11029.
Min-max harmonic maps and a new characterization of conformal eigenvalues (with M. Karpukhin), arXiv preprint, arXiv:2004.04086.
Harmonic functions and the mass of 3-dimensional asymptotically flat Riemannian manifolds (with H. Bray, D. Kazaras, and M. Khuri), Journal of Geometric Analysis, vol. 32 (2022).
Scalar curvature and harmonic maps to S^1, Journal of Differential Geometry, vol. 122 no. 2 (2022), 259—269.
Minimal submanifolds from the abelian Higgs model (with A. Pigati), Inventiones Mathematicae, vol. 223 (2021), 1027—1095.
MATH Courses - Fall 2023
- MATH 3210 : Manifolds and Differential Forms
- MATH 4900 : Supervised Research
- MATH 4901 : Supervised Reading