Research fields:

I have broad interests in analysis and PDEs arising in physics and the geometric microlocal structures underneath the explicit analysis results. Mathematically, I self-identify as a 'microlocal analyst'.

My specific research fields lie in microlocal analysis, inverse problems, and interactions with harmonic analysis and spectral theories. 

• Microlocal Analysis: differential analysis on manifolds with ends or singularities, semiclassical analysis, propagation of singularities;
• Inverse Problems: geometric inverse problems in mathematical physics;
• Spectral Theory: eigenvalues and spectral measure of Laplacian and Schrödinger operators;
• Harmonic Analysis: spectral multipliers and evolution equations on manifolds of bounded geometry.

Recent papers:

1. The semiclassical resolvent on conic manifolds and application to Schrödinger equations, arXiv: 2009.12895, submitted. 
2. (With M. Lassas, L. Oksanen, G. Paternain) Inverse problem for the Yang-Mills equations, arXiv:2005.12578, Comm. Math. Phys., to appear. 
3. (With M. Lassas, L. Oksanen, G. Paternain) Detection of Hermitian connections in wave equations with cubic non-linearity, arXiv:1902.05711,  J. Eur. Math. Soc. (JEMS), to appear.
4. The Sobolev inequalities on real hyperbolic spaces and eigenvalue bounds for Schrödinger operators with complex potentials, arXiv:1811.08874, submitted.
5. (With A. Hassell) The heat kernel on asymptotically hyperbolic manifolds, Comm. Partial Differential Equations 45 (2020), no. 9, 1031-1071.
6. Stein-Tomas restriction theorem via spectral measure on metric measure spaces, Math. Z. 289 (2018), no. 3-4, 829-835.
7. Resolvent and spectral measure on non-trapping asymptotically hyperbolic manifolds III:  Global-in-Time Strichartz Estimates without Loss, Ann. Inst. H. Poincaré Anal. Non Linéaire 35(2018),  803-829.
8. (With A. Hassell) Resolvent and spectral measure on non-trapping asymptotically hyperbolic manifolds II: Spectral Measure, Restriction Theorem, Spectral Multipliers, Ann. Inst. Fourier (Grenoble) 68(2018), 1011-1075.
9. (With A. Hassell) Resolvent and spectral measure on non-trapping asymptotically hyperbolic manifolds I: Resolvent construction at high energy, Comm. Partial Differential Equations 41(2016), 515-578.

Collaborators:

• Andrew Hassell, Australian National University;
• Matti Lassas, University of Helsinki;
• Lauri Oksanen, University College London;
• Gabriel Paternain, University of Cambridge.

Conferences:

• Nonlinear Geometric Inverse Problems; 19-22 April 2021; University of Cambridge, Cambridge, UK. (With L. Oksanen and G. Paternain)
• Microlocal Analysis and Applications; 14-21 June 2019; Fudan University, Shanghai, China. (With K. Du, C. Guillarmou, G. Huang, and J. Wunsch)