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#### Monthly Seminar taking place at the

*Organizers*

**Lectures given during the Academic year 2021–2022**

#### Wednesday December 8, 2021

*lecture room 15-16–309*

#### 14h Renato Velozo Ruiz (Cambridge, UK)

*Stability of Schwarzschild for the spherically symmetric Einstein-massless Vlasov system*

##### Abstract. The Einstein–massless Vlasov system is a relevant model in the study of collisionless many particle systems in general relativity. In this talk, I will present a stability result for the exterior of Schwarzschild as a solution of this system assuming spherical symmetry. We exploit the hyperbolicity of the geodesic flow around the black hole to obtain decay of the energy momentum tensor, despite the presence of trapped null geodesics. The main result requires a precise understanding of radial derivatives of the energy momentum tensor, which we estimate using Jacobi fields on the tangent bundle in terms of the Sasaki metric.

#### 15h30 Arthur Touati (Ecole Polytechnique, Palaiseau)

#### Construction of high-frequency spacetimes

##### Abstract. I will present a recent work on high-frequency solutions of Einstein’s vacuum equations. The motivation behind the study of such solutions comes from physical and mathematical questions. These solutions model the propagation of high-frequency gravitational waves, which enjoy some polarization-related properties. From a mathematical point of view, they partially answer Burnett’s conjecture in general relativity, which concerns the lack of compactness of a family of solutions to Einstein’s vacuum equations. I will start by reviewing the existing literature, and then discuss my results for a toy model. I will then sketch the proof of the local well-posedness in harmonic gauge for high-frequency solutions.

#### Wednesday November 10, 2021

*lecture room 15-25-104*

#### On the stability of black hole quasi-normal modes: a pseudo-spectrum approach

##### Abstract. Black hole (BH) quasi-normal modes (QNM) encode the resonant response of black holes under linear perturbations, their associated complex frequencies providing an invariant probe into the background spacetime geometry. In the late nineties, Nollert and Price found evidence of a BH QNM instability phenomenon, according to which perturbed QNMs of Schwarzschild spacetime migrate to new perturbed QNM branches of different qualitative behavior and asymptotics. Here we revisit this BH QNM instability issue by adopting a pseudo-spectrum approach. Specifically, rather than starting from the formulation of QNMs in scattering resonance theory, we cast the QNM problem as an eigenvalue problem for a non-self-adjoint operator by adopting a hyperboloidal formulation of spacetime. Non-selfadjoint (more generally non-normal) operators suffer potentially of spectral instabilities, the notion of pseudo-spectrum providing a tool suitable for their study. We explore this problem in a numerical methodology based on pseudo-spectral methods. As a result, we find evidence that perturbed Nollert & Price BH QNM branches track the pseudo-spectrum contour lines, therefore probing the analytic structure of the resolvent. Specifically, we find strong support to claim: i) the stability of the slowest decaying (fundamental) mode, and ii) the instability of all QNM ‘overtones’. But numerical evidence is not a proof. Or goal in this talk is to boost the interaction between physicists and analysts to fully assess this BH QNM instability problem.

#### 15h30 Allen Fang (Paris)

#### Nonlinear stability of Kerr-de Sitter

##### Abstract. The nonlinear stability of the slowly-rotating Kerr-de Sitter family was first proven by HIntz-Vasy in 2018 using microlocal techniques. I will present a novel proof of the nonlinear stability of Kerr-de Sitter that avoids frequency-space techniques outside of a neighborhood of the trapped set. Similar to the original work of Hintz-Vasy, the critical step is to prove exponential decay for solutions of the linearized problem, which is done by using a high-frequency ILED estimate, and a mode stability result.