
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:
- The semiclassical resolvent on conic manifolds and application to Schrödinger equations, arXiv: 2009.12895, submitted.
- (With M. Lassas, L. Oksanen, G. Paternain) Inverse problem for the Yang-Mills equations, arXiv:2005.12578, Comm. Math. Phys., to appear.
- (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.
- The Sobolev inequalities on real hyperbolic spaces and eigenvalue bounds for Schrödinger operators with complex potentials, arXiv:1811.08874, submitted.
- (With A. Hassell) The heat kernel on asymptotically hyperbolic manifolds, Comm. Partial Differential Equations 45 (2020), no. 9, 1031-1071.
- Stein-Tomas restriction theorem via spectral measure on metric measure spaces, Math. Z. 289 (2018), no. 3-4, 829-835.
- 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.
- (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.
- (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)
Publications
The heat kernel on asymptotically hyperbolic manifolds
– Communications in Partial Differential Equations
(2020)
45,
1031
Resolvent and spectral measure on non-trapping asymptotically hyperbolic manifolds III: Global-in-time Strichartz estimates without loss
– Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire
(2018)
35,
803
(DOI: 10.1016/j.anihpc.2017.08.003)
Resolvent and Spectral Measure on Non-Trapping Asymptotically Hyperbolic Manifolds II: Spectral Measure, Restriction Theorem, Spectral Multipliers
– Annales de l’institut Fourier
(2018)
68,
1011
(DOI: 10.5802/aif.3183)
Stein–Tomas restriction theorem via spectral measure on metric measure spaces
– Mathematische Zeitschrift
(2017)
289,
829
(DOI: 10.1007/s00209-017-1976-y)
Resolvent and spectral measure on non-trapping asymptotically hyperbolic manifolds I: Resolvent construction at high energy
– Communications in Partial Differential Equations
(2016)
41,
515
On multilinear Littlewood-Paley operators
– Nonlinear Analysis: Theory, Methods & Applications
(2015)
115,
25
(DOI: 10.1016/j.na.2014.12.001)
Weighted estimates for a class of multilinear fractional type operators
– Journal of Mathematical Analysis and Applications
(2010)
362,
355
(DOI: 10.1016/j.jmaa.2009.08.022)
Weighted Estimates for the Maximal Operator of a Multilinear Singular Integral
– Bulletin of the Polish Academy of Sciences Mathematics
(2010)
58,
129
(DOI: 10.4064/ba58-2-4)
The Sobolev Inequalities on Real Hyperbolic Spaces and Eigenvalue Bounds
for Schrödinger Operators with Complex Potentials
Inverse problem for the Yang-Mills equations
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