# Research

My research focuses on partial differential equations arising in general relativity. Much of my work is concerned with stability problems for solutions of the Einstein field equations and with the global asymptotic control (regularity, decay) of solutions to related linear and nonlinear wave equations. Methods and ideas from microlocal analysis and spectral/scattering theory feature prominently in my research.

## Collaborators

Vitor Cardoso, João Costa, Kyriakos Destounis, Aron Jansen, Gunther Uhlmann, András Vasy, YuQing Xie, Jian Zhai, Maciej Zworski

## List of publications

Differences, if any, between published versions and local pdf files are due to the journals' copy editing.

- Resolvents and complex powers of semiclassical cone operators. Mathematische Nachrichten, 295(10), 2022.
- The Dirichlet-to-Neumann map for a semilinear wave equation on Lorentzian manifolds, with Gunther Uhlmann, Jian Zhai. Communications in Partial Differential Equations (
*not yet in print*). - Quasinormal modes of small Schwarzschild--de Sitter black holes, with YuQing Xie. Journal of Mathematical Physics, 63:011509, 2022.
- A sharp version of Price's law for wave decay on asymptotically flat spacetimes.. Communications in Mathematical Physics 389:491–542, 2022.
- Quasinormal modes and dual resonant states on de Sitter space, with YuQing Xie. Physical Review D, 104(6), 064037, 2021.
- An inverse boundary value problem for a semilinear wave equation on Lorentzian manifolds, with Gunther Uhlmann, Jian Zhai. International Mathematics Research Notices, rnab088, 2021.
- Black hole gluing in de Sitter space. Communications in Partial Differential Equations, 46(7):1280–1318, 2021.
- Linear stability of slowly rotating Kerr black holes, with Dietrich Häfner, András Vasy. Inventiones mathematicae, 223:1227–1406 (2021).
- Normally hyperbolic trapping on asymptotically stationary spacetimes. Probability and Mathematical Physics, 2(1):71–126, 2021.
- Strong cosmic censorship in charged black-hole spacetimes: still subtle, with Vitor Cardoso, João Costa, Kyriakos Destounis, Aron Jansen. Physical Review D, 98(10), 104007, 2018.
- Quasinormal modes and Strong Cosmic Censorship, with Vitor Cardoso, João Costa, Kyriakos Destounis, Aron Jansen. Physical Review Letters 120(3), 031103, 2018. See also this Viewpoint: A possible Failure of Determinism in General Relativity by Harvey Reall. Popular science accounts are given by Berkeley News, New Scientist (paywall), Live Science.
- Stability of Minkowski space and polyhomogeneity of the metric, with András Vasy. Annals of PDE, 6(2), 2020.
- Reconstruction of Lorentzian manifolds from boundary light observation sets, with Gunther Uhlmann. International Mathematics Research Notices, rnx320, 2018.
- Wave decay for star-shaped obstacles in R
^{3}: papers of Morawetz and Ralston revisited, with Maciej Zworski. Mathematical proceedings of the Royal Irish Academy, 117A(2):47–62, 2017. - Resonances for obstacles in hyperbolic space, with Maciej Zworski. Communications in Mathematical Physics, 359(2):699–731, 2018.
- Uniqueness of Kerr–Newman–de Sitter black holes with small angular momenta. Annales Henri Poincaré, 19(2):607–617, 2018.
- Non-linear stability of the Kerr–Newman–de Sitter family of charged black holes. Annals of PDE, 4(1):11, Apr 2018.
- The global non-linear stability of the Kerr–de Sitter family of black holes, with András Vasy. Acta mathematica, 220:1–206, 2018. A popular science account is given by Quanta magazine.
- Analysis of linear waves near the Cauchy horizon of cosmological black holes, with András Vasy. Journal of Mathematical Physics, 58(8):081509, 2017.
- Boundedness and decay of scalar waves at the Cauchy horizon of the Kerr spacetime. Commentarii Mathematici Helvetici, 92(4):801–837, 2017.
- Resonance expansions for tensor-valued waves on asymptotically Kerr–de Sitter spaces. Journal of Spectral Theory, 7:519–557, 2017.
- Asymptotics for the wave equation on differential forms on Kerr–de Sitter space, with András Vasy. Journal of Differential Geometry, 110(2):221–279, 2018.
- Quasilinear waves and trapping: Kerr–de Sitter space, with András Vasy. Journées équations aux dérivées partielles, 21(6):1–15, 2014.
- Global analysis of quasilinear wave equations on asymptotically Kerr–de Sitter spaces", with András Vasy. International Mathematics Research Notices, 2016(17):5355–5426, 2016.
- Global analysis of quasilinear wave equations on asymptotically de Sitter spaces. Annales de l'Institut Fourier, 66(4):1285–1408, 2016
- Non-trapping estimates at normally hyperbolic trapping, with András Vasy. Mathematical Research Letters, 21(6):1277–1304, 2014
- Semilinear wave equations on asymptotically de Sitter, Kerr–de Sitter and Minkowski spacetimes, with András Vasy. Analysis&PDE, 8(8):1807–1890, 2015.

## Preprints

- Linear waves on asymptotically flat spacetimes. I, 2023. arXiv version.
- Exterior stability of Minkowski space in generalized harmonic gauge, 2023. arXiv version.
- Microlocal analysis near null infinity in asymptotically flat spacetimes, with András Vasy, 2023. arXiv version.
- Microlocal analysis of operators with asymptotic translations- and dilation-invariances, 2023. arXiv version.
- Underdetermined elliptic PDE on asymptotically Euclidean manifolds, and generalizations, 2023. arXiv version.
- Gluing small black holes into initial data sets, 2022. arXiv version.
- Mode stability and shallow quasinormal modes of Kerr–de Sitter black holes away from extremality, 2021. arXiv version.
- Elliptic parametrices in the 0-calculus of Mazzeo and Melrose, 2021. arXiv version.
- Semiclassical propagation through cone points, 2021. arXiv version.