Research interest:

Geometric analysis and geometric PDEs. I am interested in understanding the formation of singularities and developing/improving regularity theory of geometric partial differential equations such as minimal surfaces, mean curvature flows, Allen-Cahn equations, Ginzburg-Landau and Yang-Mills-Higgs equations. These problems exhibit a rich interplay between geometry, topology and physics.

Publications:

• Some properties of closed hypersurfaces of small entropy and the topology of hypersurfaces through singularities of mean curvature flow. ETD - Doctoral Dissertations. Johns Hopkins University (2018).
• Round spheres are Hausdorff stable under small perturbation of entropy. J. Reine Angew. Math. 758, 261-280 (2020).
• On the topological rigidity of self shrinkers in R^3. (Joint with Alexander Mramor). Int. Math. Res. Not. 2020, 1933-1941 (2020).
• The level set flow of a hypersurface in R^4 of low entropy does not disconnect. (Joint with Jacob Bernstein). Comm. Anal. Geom. 29, 1523-1543 (2021).
• Warped tori with almost non-negative scalar curvature. (Joint with Brian Allen, Lisandra Hernandez-Vazquez, Davide Parise, Alec Payne). Geometriae Dedicata 200, 153-171 (2019).
• Integrability of scalar curvature and normal metric on conformally flat manifolds. (Joint with Yi Wang). J. Differential Equations 265, 1353-1370 (2018).
• Low entropy and the mean curvature flow with surgery. (Joint with Alexander Mramor). Calc. Var. Partial Differential Equations. 60, (2021).
• Extended abstract "The level set flow of a hypersurface in R^4 of low entropy does not disconnect" in the Proceedings of the John H. Barrett Memorial Lectures at the University of Tennessee, Knoxville, May 29 - June 1, 2018 (edited by Theodora Bourni and Mat Langford). De Gruyter Proc. Math. (2020). 
• Precise asymptotics near a generic S^1 × R^3 singularity of mean curvature flow. (Joint with Zhou Gang). Nonlinear Anal. 251, (2025)
• Second order estimates for transition layers and a curvature estimate for the parabolic Allen-Cahn. (Joint with Huy Nguyen). Int. Math. Res. Not. 2024, 6749-6789 (2024).
• Brakke regularity for the Allen-Cahn flow. (Joint with Huy Nguyen). Preprint. 
• Quantization of the energy for the inhomogeneous Allen-Cahn mean curvature. (Joint with Huy Nguyen). Math. Ann. 390, 6399-6457 (2024)
• Allard Regularity for Abelian Yang-Mills-Higgs equation. (Joint with Huy Nguyen). Preprint. 

Teaching:

• Math 371       Ordinary differential equations                       2018 Fall, Binghamton University
• Math 590F     Topics in Analysis                                           2019 Spring, Binghamton University
• Math 371       Ordinary differential equations                       2019 Spring, Binghamton University
• LTCC             Introduction to mean curvature flow               2020 Fall, London (online)
• MA4C0          Differential Geometry                                     2021  Fall, University of Warwick
• MTH 6151      Partial Differential Equations                         2022 Fall, Queen Mary University of London
• MTH 6151      Partial Differential Equations                         2023 Fall, Queen Mary University of London
• MTH 6151      Partial Differential Equations                         2024 Fall, Queen Mary University of London
• MTH 5113      Introduction to Differential Geometry             2025 Spring, Queen Mary University of London
• MTH 6151      Partial Differential Equations                         2025 Fall, Queen Mary University of London
• SEF 041         Mathematics B                                              2026 Spring, Queen Mary University of London
• LTCC              Elliptic Partial Differential Equations             2026 Spring, London