Periodic Reporting for period 3 - BHSandAADS (The Black Hole Stability Problem and the Analysis of asymptotically anti-de Sitter spacetimes) Reporting period: 2020-11-01 to 2022-04-30 Summary of the context and overall objectives of the project Black holes are one of the most fascinating predictions of Einstein's theory of general relativity developed in 1915. The recent detection of gravitational waves produced by the merger of two black holes (leading to the Nobel Prize 2017), the first "photo" of a black hole by the Event Horizon Telescope in 2019, and the recent award of the Nobel Prize 2020 to Roger Penrose "for the discovery that black hole formation is a robust prediction of the general theory of relativity" are jut three very recent examples of the fundamental role that the analysis of black holes plays in modern science. The study of black holes, or more generally, general relativity, unites the mathematical areas of geometry and non-linear partial differential equation with both theoretical and experimental physics. Mathematically, black holes are solutions of the Einstein equations, which themselves form a complicated non-linear system of partial differential equations -- so complicated that Einstein was not sure whether (non-trivial) solutions of his equations could ever be found. It came as a big surprise when the astronomer Karl Schwarzschild published an exact solution only a year after Einstein had published his equations. This solution, the Schwarzschild solution, turns out to describe the simplest (spherically symmetric) black hole geometry, a fact that was however only understood decades later! Over the years, more exact solutions have been found. Most remarkable is the Kerr family of solutions, describing rotating black holes, discovered in 1963. It is believed that all stationary asymptotically flat black holes are described by the Kerr family.The main objectives of the project is to understand the stability of these stationary solutions. Roughly speaking: If the (Kerr)black hole is slightly perturbed, will the geometry remain close to the geometry one started with or will it "run away" and for instance form new black holes? The correct mathematical framework to address this question is the initial value problem (or Cauchy problem). Initial data are perturbed on a spacelike hypersurface and the development of the data is then analysed using modern techniques of partial differential equations and geometry. While the final goal is to understand the stability of the entire family of Kerr solutions, there are several smaller scale problems that have to be addressed on the way. One is to prove linear stability before non-iinear stability, another is to first prove stability for the simpler Schwarzschild family of black holes. As explained in the next paragraph, some of these steps have been successfully carried out during the first phase of the project.A second objective of the proposal is the analysis of so-called asymptotically anti-de Sitter spacetimes. These spacetimes are solutions of the Einstein equations with a negative cosmological constant, whose distinguished feature in the presence of a conformal boundary at infinity. Such spacetimes appear frequently in high energy physics as models for superconductors and also feature as models for the overarching idea of "holography" in theoretical physics. The project looks at understanding such spacetimes both from the dynamical point of view, by proving, for instance, the non-linear stability of the maximally symmetric solution, anti-de Sitter space if dissipative boundary conditions are imposed at the conformal boundary. Another class of objectives is to prove unique continuation theorems for the Einstein equations at the conformal boundary to provide rigorous formulations of the principle of holography.To conclude, understanding the Einstein equations in a mathematically rigorous fashion is essential for our understanding of the theory (e.g. the question whether black holes are "stable" predictions of theory) and complements experimental results from astrophysics. From an even broader perspective, a complete understanding of the theory is necessary to eventually move beyond the classical Einstein equations and understand settings in which quantum effects might play a role. Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far The main achievement of the project so far is the 500 page preprint posted on the arXiv in April 2020 providing a complete proof of "The Non-Linear Stability of the Schwarzschild family of black holes". Almost all research efforts during the first period went into this joint work with Mihalis Dafermos (Cambridge/ Princeton), Igor Rodnianski (Princeton) and Martin Taylor (Imperial, who is a former ERC postdoc). The work contains several innovations (to be described in the next section) and is a key step on the way to a full proof of the stability of the entire family of Kerr solutions.Finally, during the first period we also wrote a short paper with my ERC postdoc Chris Kauffman on "A note on the wave equation on black hole spacetimes with small non-decaying first order terms". That paper provides a physical space framework to prove decay estimates for the linear wave equation on the Schwarzschild black hole. We also made progress on program of unique continuation theorems on asymptotically anti-de Sitter spacetimes (with Arick Shao (Queen Mary, London)) developing a unifying framework to prove result for the Einstein equations. This work is currently being written up. Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far) The 500 page preprint mentioned contains several innovations that move the subject beyond the state of the art: Among others these are (1) The construction of (two) carefully chosen teleological double null foliations and the estimation of their dynamical interaction (2) Novel estimates for the Teukolsky equation (3) A modulation theory for the Einstein equations (required to prove the co-dimension 3 statement). Furthermore, we expect that our framework generalises to prove the stability of the Kerr family of solutions. In fact, the case of slowly rotating Kerr black holes should be doable within the remaining time of the project (ongoing work with Dafermos/ Rodnianski/ Taylor). Regarding the analysis of asymptotically anti-de Sitter spacetimes, we are currently writing up a proof (with Luk (Stanford), Smulevici (Paris) and Warnick (Cambridge)) of the stability of Anti-de Sitter space with respect to dissipative boundary conditions which should also be completed before the end of the project. This result will complete the stability results for de Sitter space and Minkowski space (the other maximally symmetric solutions of the Einstein equations, with positive and zero cosmological constant respectively) obtained more than 30 years ago.