# Research

My research focuses on partial differential equations arising in general relativity. Much of my work is related to stability and gluing problems for solutions of the Einstein field equations and with the precise control of regularity and 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.

## Preprints

- Gluing small black holes along timelike geodesics III: construction of true solutions and extreme mass ratio mergers, 114 pages, 2024. arXiv version.
- Gluing small black holes along timelike geodesics II: uniform analysis on glued spacetimes, 141 pages, 2024. arXiv version.
- Pseudodifferential operators on manifolds with scaled bounded geometry, 79 pages, 2024. arXiv version.
- Local theory of wave equations with timelike curves of conic singularities, 71 pages, 2024. arXiv version.
- Gluing small black holes along timelike geodesics I: formal solution, 165 pages, 2023. arXiv version.
- Linear waves on asymptotically flat spacetimes. I, 163 pages, 2023. arXiv version.
- Microlocal analysis near null infinity in asymptotically flat spacetimes, with András Vasy, 98 pages, 2023. arXiv version.
- Microlocal analysis of operators with asymptotic translation- and dilation-invariances, 147 pages, 2023. arXiv version.
- Underdetermined elliptic PDE on asymptotically Euclidean manifolds, and generalizations, 28 pages, 2023. arXiv version.

## Publications

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

- The linearized Einstein equations with sources. Letters in Mathematical Physics, 114(4):94, 2024.
- Gluing small black holes into initial data sets. Communications in Mathematical Physics, 405:114, 2024.
- Universality of the quantum energy flux at the inner horizon of asymptotically de Sitter black holes, with Christiane Klein. Classical and Quantum Gravity
**41**075006. - Asymptotically de Sitter metrics from scattering data in all dimensions. Philosophical Transactions of the Royal Society A, 382(2267), 2024.
- Mode stability and shallow quasinormal modes of Kerr–de Sitter black holes away from extremality, 2021. Accepted for publication in the
*Journal of the European Mathematical Society*. - Semiclassical propagation through cone points, 2021. Accepted for publication in
*Analysis&PDE*. - Exterior stability of Minkowski space in generalized harmonic gauge. Archive for Rational Mechanics and Analysis, 247:99, 2023.
- Elliptic parametrices in the 0-calculus of Mazzeo and Melrose. Pure and Applied Analysis, 5(3), 729–766, 2023.
- 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, 47(12):2363–2400, 2022.
- 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.

## Collaborators

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