Research

Publications and Preprints:

Differential Geometry, Microlocal Analysis:

1. A gluing construction of $D_{k}$ ALF gravitational instantons and existence of non-holomorphic minimal spheres.
   arXiv:2407.20149
2. Degenerating hyperbolic surfaces and spectral gaps for large genus. Joint with Yunhui Wu and Haohao Zhang.
   Analysis and PDE, Vol. 17 (2024), No. 4, 1377–1395. Journal version
3. Spherical conical metrics and harmonic maps to spheres. Joint with Mikhail Karphukhin.
   Transactions of the AMS, Volume 375, Number 5, May 2022, Pages 3325–3350. Journal version
4. Fundamental gaps of spherical triangles. Joint with Shoo Seto and Guofang Wei.
   Annals of Global Analysis and Geometry 61, pages1–19 (2022). Journal version
5. Spectral properties of reducible conical metrics. Joint with Bin Xu.
   Illinois Journal of Mathematics, Vol. 65, No. 2 (2021), 313–337. Journal version
6. Conical metrics on Riemann surfaces, II: spherical metrics. Joint with Rafe Mazzeo.
   International Mathematics Research Notices, Volume 2022, Issue 12, June 2022, Pages 9044–9113. Journal version
7. Rigidity of a family of spherical conical metrics.
   New York Journal of Mathematics 26 (2020) 272–284. Journal version
8. Spherical conic metrics and realizability of branched covers.
   Proceedings of the AMS, Volume 147, Number 4, April 2019, Pages 1805–1815. Journal version
9. Conical metrics on Riemann surfaces, I: the compactified configuration space and regularity. Joint with Rafe Mazzeo.
   Geometry and Topology 24 (2020) 309–372. Journal version
10. Boundary behaviour of Weil-Petersson and fiber metrics for Riemann moduli spaces. Joint with Richard Melrose.
    International Mathematics Research Notices, Volume 2019, Issue 16, Pages 5012–5065. Journal version
11. Eleven dimension supergravity equations on edge manifolds.
    Annales Henri Poincaré, Volume 19, Issue 8 (2018), 2347–2400. Journal version
12. Eigenvalue resolution of self-adjoint matrices.
    Pacific Journal of Mathematics, Vol. 288 (2017), No. 1, 241–255. Journal version
13. Resolution of the canonical fiber metrics for a Lefschetz fibration. Joint with Richard Melrose.
    Journal of Differential Geometry, Volume 108, Number 2 (2018), 295-317. Journal version
14. The eleven dimensional supergravity equations, resolutions and Lefschetz fiber metrics. PhD thesis

Interdisciplinary Research:

1. Closed-Form Adaptive Tracking Control of Heat Equations Aided by Fourier Regularization and Bi-orthogonal Series.
   Joint with Tong Ma. Submitted.