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Monthly Seminar

Laboratoire Jacques-Louis Lions

Sorbonne Université, Paris

Organizers

 Cécile Huneau (Ecole Polytechnique, Palaiseau)

 Philippe G. LeFloch (Sorbonne Université, Paris)

 Jacques Smulevici (Sorbonne Université, Paris)

Jérémie Szeftel (Sorbonne Université, Paris)


Lectures given during the Academic year 2022–2023


Wednesday October 19, 2022

lecture room 15-16-309 (Jussieu)

14h Annalaura STINGO (Ecole Polytechnique, Palaiseau)

Global stability of Kaluza-Klein theories: a toy model

Abstract. The Kaluza-Klein theories represent the classical mathematical approach to the unification of general relativity with electromagnetism and more generally with gauge fields. In these theories, general relativity is considered in 1+3+d dimensions and in the simplest case d=1 dimensional gravity is compactified on a circle to obtain at low energies a (3+1)-dimensional Einstein-Maxwell-Scalar systems. In this talk I will discuss the problem of the classical global stability of Kaluza-Klein theories when d=1 and present a toy model we studied in collaboration with C. Huneau.

15h30 Dawei SHEN (Sorbonne University, Paris) ——-> Lecture–Shen-Dawei–Sorbonne-October-2022

General covariant modulated (GCM) procedure

Abstract. I will start by introducing the main idea of the proof of the “Kerr stability for small angular momentum” by Klainerman and Szeftel, as our motivation to introduce the General Covariant Modulated (GCM) procedure. Then, I will present the paper “Constructions of GCM spheres in perturbations of Kerr” of Klainerman and Szeftel concerning the construction of GCM spheres. Finally, by applying the GCM spheres I will explain the main result concerning the “Constructions of GCM hypersurfaces in perturbations of Kerr”.


Thursday November 17, 2022

lecture room 15-16-201 (Jussieu)

14h Dietrich HÄFNER (Université Grenoble Alpes)

On the linear stability of Kerr black holes

Abstract: I will explain a result obtained in collaboration with P. Hintz and A. Vasy on the linear stability of slowly rotating Kerr black holes as solutions of the Einstein vacuum equations: linearized perturbations of a Kerr metric decay at an inverse polynomial rate to a linearized Kerr metric plus a pure gauge term. We work in a natural generalized wave map/DeTurck gauge and show that the pure gauge term can be taken to lie in a fixed finite dimensional space with a simple geometric interpretation. Our proof rests on a robust general framework, based on recent advances in micro-local analysis and non-elliptic Fredholm theory. The restriction to small angular momentum mainly comes from the analysis of mode solutions and I will explain at the end of the talk how this analysis can be carried out also in the case of large angular momentum of the black hole. (This last part is based on joint work with L. Andersson and B. Whiting.)

15h30 Nicolas MARQUE (Université de Lorraine)

Energie pour la gravité du quatrième ordre

Abstract. J‘aborderai un travail mené en collaboration avec R. Avalos, P. Laurain et J. Lira. En considérant l’espace-temps comme point critique de courbures élastiques quadratiques (type Lovelock-Bach) généralisant l’énergie d’Einstein-Hilbert, nous obtenons des équations de courbure d’ordre 4 dont les espaces-temps d’Einstein sont des solutions naturelles. L’objectif de ce travail est d’étudier ces métriques de Lorentz d’ordre quatre via une analyse de quantités conservées inspirées de la masse ADM.  Nous nous appuierons sur ces quantités conservées et leurs liens avec la Q-courbure pour établir des théorèmes de rigidité pour des feuilles Riemanniennes de tels espaces-temps.