In a broad sense, my research lies in the areas of Analysis of PDEs and Inverse Problems in Mathematical Physics.
Specific fields and topics include
Microlocal Analysis: differential analysis on manifolds with ends or singularities, semiclassical analysis, propagation of singularities;
Inverse Problems: time-dependent hyperbolic PDEs (uniqueness and stability of inversion, Bayesian inversion);
Spectral Theory: spectral measure and eigenvalues of Laplace and Schrödinger operators on manifolds (with ends or singularities);
Harmonic Analysis: spectral multipliers on manifolds (with singularities or of exponential volume growth).
Microlocal analysis originated in the 1950s from the use of Fourier transform techniques in the study of variable-coefficient PDEs; its intellectual roots lie in geometric optics and the WKB approximation. The field took on a coherent identity starting in the 1960s with the development of pseudodifferential and, later, Fourier integral operators as fundamental tools. The microlocal machinery provides precise descriptions of the geometric and analytic structures of the fundamental solutions to variable-coefficient PDEs on the cotangent bundle of physical spaces.
My research is focused on differential analysis on manifolds with singularities and ends, semiclassical analysis, and propagation of singularities/waves. The intellectual roots of these topics lie in classical physics, general relativity, and quantum field theory. Specifically, I contributed to microlocal descriptions of geodesic flows and the semiclassical resolvent / spectral measure on asymptotically hyperbolic manifolds [1, 2]; microlocal descriptions of diffractions of singularities and the semiclassical resolvent / spectral measure on conic manifolds [9].
Inverse problems concern the reconstruction of background information of physical models from measurable local data. It is the research that makes invisible objects observable from conveniently measurable information. Inverse problems formulate the theoretical basis of CT, PET and ultrasound techniques in medical imaging, and establish the framework of determining the inner structure of Earth in geophysics as well as the geometric structure of black holes in cosmology.
I investigates inverse problems of PDEs arising in gauge theory. In the Standard Model of Particle Physics, the dynamics of the force fields of elementary particles (bosons and fermions) are modeled by the Euler–Lagrange equations of the Yang–Mills–Higgs–Dirac–Yukawa Lagrangians, which are time-dependent nonlinear hyperbolic PDEs. I contributed to the method of broken light ray transforms for the recovery of lower order coefficients of time-dependent nonlinear hyperbolic PDEs [7] and the inverse problems of such PDEs from the Standard Model [7, 8, 10].
I apply microlocal approaches and results to the related problems in spectral theory and harmonic analysis. For example, I contributed to restriction estimates for the spectral measure [2, 4]; boundedness of spectral multipliers and Riesz transforms [2, 5]; Strichartz estimates for Schrödinger equations [3, 9]; heat kernel bounds [5]; uniform Sobolev estimates [6]; eigenvalue estimates for Schrödinger operators [6].