Technically, my research mainly focuses on microlocal analysis. In terms of topics, I am interested in wave and dispersive equations, and mathematical physics.
Research papers (newest first):
Microlocal approach to PDEs:
The scattering map for the Schrödinger operator on curved spaces (with Andrew Hassell), preprint.
Microlocal analysis of the non-relativistic limit of the Klein--Gordon equation: Asymptotics (with Andrew Hassell, Ethan Sussman, András Vasy), preprint.
Microlocal analysis of the non-relativistic limit of the Klein--Gordon equation: Estimates (with Andrew Hassell, Ethan Sussman, András Vasy), preprint.
The essential self-adjointness of the wave operator on radiative spacetimes (with Mikhail Molodyk, Ethan Sussman), published in Communications in Mathematical Physics.
Propagation of Singularities with Normally Hyperbolic Trapping , published in Annales Henri Poincaré.
Dynamical systems:
The topological pressure of trapped sets in Kerr-(de Sitter) spacetimes ,preprint.
Dispersive PDEs and Harmonic analysis:
Decay and Strichartz estimates for critical electromagnetic wave equations on conic manifolds (with Junyong Zhang), preprint.
Strichartz estimates for the Schrödinger equation in high dimensional critical electromagnetic fields (with Junyong Zhang), preprint.
The final state problem for the nonlinear Schrodinger equation in dimensions 1, 2 and 3 (with Andrew Hassell), published in Pure and Applied Analysis.
Pointwise dispersive estimates for Schrodinger and wave equations on conical singular spaces (with Junyong Zhang), preprint.
Bochner-Riesz means on a conical singular manifold (with Junyong Zhang, Jiqiang Zheng), preprint.
Geometric inverse problems:
Determining metrics from the scattering map of the time-dependent Schrodinger equation , preprint.
The tensorial X-ray transform on asymptotically conic spaces (with András Vasy), published in Inverse Problems and Imaging.
2D Inverse Problems with a Foliation Condition , preprint.
Algebraic geometry/Mathematical physics:
Interior analysis, stretched technique and bubbling geometries (with Hai Lin), published in Annals of Physics.
Calabi–Yau generalized complete intersections and aspects of cohomology of sheaves (with Hai Lin), published in Journal of Mathematical Physics.
Deep learning:
Dimensional Reweighting Graph Convolutional Networks (with Xu Zou et al.), preprint.
Notes/Expository papers:
Propagation of Singularities on Kerr(-de Sitter) Spacetimes , published in the 2023 volume of the book series MATRIX Annals. This is an expository version of the Propagation of Singularity with Normally Hyperbolic Trapping above.
Lecture notes on non-elliptic Fredholm theory , arXiv: 2604.18956.
Automorphisms of the algebra of smooth functions on compact manifolds (I found later that this is already contained in [Lemma2, Duistermaat-Singer 1976]. )