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Monthly Seminar taking place at the
Laboratoire JacquesLouis Lions
Sorbonne Université, Paris
Organizers
Philippe G. LeFloch (Sorbonne, Paris)
Jacques Smulevici (Sorbonne, Paris)
Jérémie Szeftel (Sorbonne, Paris)
Lectures given during the Academic year 2021–2022
Wednesday December 8, 2021
lecture room 1516–309
14h Renato Velozo Ruiz (Cambridge, UK)
Stability of Schwarzschild for the spherically symmetric Einsteinmassless 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 highfrequency spacetimes
Abstract. I will present a recent work on highfrequency 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 highfrequency gravitational waves, which enjoy some polarizationrelated 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 wellposedness in harmonic gauge for highfrequency solutions.
Wednesday November 10, 2021
lecture room 1525104
14h José Luis Jaramillo (Université de Bourgogne)
On the stability of black hole quasinormal modes: a pseudospectrum approach
Abstract. Black hole (BH) quasinormal 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 pseudospectrum 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 nonselfadjoint operator by adopting a hyperboloidal formulation of spacetime. Nonselfadjoint (more generally nonnormal) operators suffer potentially of spectral instabilities, the notion of pseudospectrum providing a tool suitable for their study. We explore this problem in a numerical methodology based on pseudospectral methods. As a result, we find evidence that perturbed Nollert & Price BH QNM branches track the pseudospectrum 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 Kerrde Sitter
Abstract. The nonlinear stability of the slowlyrotating Kerrde Sitter family was first proven by HIntzVasy in 2018 using microlocal techniques. I will present a novel proof of the nonlinear stability of Kerrde Sitter that avoids frequencyspace techniques outside of a neighborhood of the trapped set. Similar to the original work of HintzVasy, the critical step is to prove exponential decay for solutions of the linearized problem, which is done by using a highfrequency ILED estimate, and a mode stability result.
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International Conference
MATHEMATICAL GENERAL RELATIVITY
June 2nd to 5th, 2020
Unfortunately, we have to postpone this event,
and we will reschedule it in a few months.
Institut Henri Poincaré
11 rue Pierre et Marie Curie, Paris
Invited Speakers

Xinliang An (Singapore)

Mihalis Dafermos (Cambridge/Princeton)

David Fajman (Vienna)

Grigorios Fournodavlos (Sorbonne)

Elena Giorgi (Princeton)

Mahir Hadzic (London)

Gustav Holzegel (London)

Cécile Huneau (Palaiseau)

Sergiu Klainerman (Princeton)

Philippe G. LeFloch (Sorbonne)

Jonathan Luk (Stanford)

Siyuan Ma (Sorbonne)

Yue Ma (Xi’an)

Maxime Van De Moortel (Princeton)

Georgios Moschidis (Princeton)

Hans Ringström (Stockholm)

Jared Speck (Cambridge, USA)

Shiwu Yang (Beijing)

Pin Yu (Beijing)
Schedule of the conference: TBA
Schedule for TUESDAY
Schedule for WEDNESDAY
Schedule for THURSDAY
Schedule for FRIDAY
Titles and abstracts of the lectures: TBA
Organizers
Philippe G. LeFloch (Sorbonne), Jacques Smulevici (Sorbonne), Jérémie Szeftel (Sorbonne)
Funding
 GEOWAKI
“The analysis of geometric nonlinear wave and kinetic equations”
Principal investigator: Jacques Smulevici
ERC Starting Grant 2016
 EPGR
“The Evolution Problem in General Relativity”
Principal investigator: Jérémie Szeftel
ERC Consolidator Grant 2016
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Seminar at the
Laboratoire JacquesLouis Lions
Sorbonne Université, Paris
Organizers
Philippe G. LeFloch (Sorbonne, Paris)
Jacques Smulevici (Sorbonne, Paris)
Jérémie Szeftel (Sorbonne, Paris)
Seminar organized during the WinterSpring 2019
Tuesday February 19, 2019
lecture room 15/16309
14h João Costa (Lisbon)
Strong cosmic censorship, linear waves, and quasinormal modes
Abstract. I will present some recent results concerning the Strong Cosmic Censorship Conjecture (SCCC) in the presence of a positive cosmological constant. I will start by reviewing some of the progress made in the context of the EinsteinMaxwellscalar field system in spherical symmetry and the linear wave equation in the black hole interior of ReissnerNordström de Sitter. These results show that the validity of the SCCC hinges on the precise decay rates of perturbations along the event horizon, which are known to be determined by the black hole’s quasinormal spectrum. I will also discuss recent numerical computations of quasinormal modes that suggest the failure of the SCCC in a near extremal regime of charged de Sitter black holes.
15h30 Shijie Dong (Paris)
Evolution of the U(1) Higgs Boson: global nonlinear stability with energy bounds
Abstract. Relying on the hyperboloidal foliation method, we establish the nonlinear stability of the ground state for the socalled U(1) standard model of electroweak interactions. This amounts to establishing a globalintime theory for the initial value problem for a nonlinear waveKleinGordon system coupling massive (Dirac, scalar, gauge) equations together. In particular, we need to investigate here for the Dirac operator and the properties of energy functionals associated with the hyperboloidal foliation of Minkowski spacetime. We also provide a unified decay result for the Dirac equation when the mass coefficient can be arbitrarily small. Our energy bounds are uniform with respect to our (hyperboloidal) time variable, except for a mild loggrowth. This is a joint work with P.G. LeFloch (Paris) and Z. Wyatt (Edinburgh).
Monday March 18, 2019
lecture room 15/16101
14h MariaCaterina Valcu (Lyon)
Des équations de contrainte en relativité générale
Abstract. On s’intéresse à la caractérisation des données initiales en relativité générale, c’est à dire aux solutions des équations de contrainte. On s’appuie sur une version modifiée de la méthode conforme, introduite cette fois par David Maxwell, qui semble mieux adaptée à l’étude du système dans le cas où la courbure moyenne n’est pas constante. Par contre, le système devient bien plus compliqué du point de vue analytique. On travaille sous des conditions de petitesse sur nos paramètres, en petite dimension (n=3,4,5) et en présence d’un champ scalaire avec potentiel positif, ce qui mène à un terme nonlinéaire dominant focalisant. L’analyse est assez fine et implique une série d’outils différents, dont des résultats de compacité et un théorème du point fixe.
15h30 Léo Bigorgne (Orsay)
Sharp asymptotic behavior of solutions of the 3d VlasovMaxwell system with small data
Abstract. The VlasovMaxwell system is a classical model in plasma physics. Glassey and Strauss proved global existence for the small data solutions of this system under a compact support assumption on the initial data. They used in particular that under these hypotheses, the velocity support of the particle density remains compact. This allows a clean separation between the characteristics of the wave equations (which are null geodesics) and those of the transport equation (which are timelike). We will see how vector field methods can be applied to revisit this problem. In particular, it allows to remove all compact support assumptions on the initial data and obtain sharp asymptotics on the solutions and their derivatives. We will also study the null structure of the system, which constitutes a crucial element of the proof and allows us to deal with high velocities.
Monday May 6, 2019
lecture room 15/16309
14h Erwann Delay (Avignon)
Le théorème d’énergie positive hyperbolique
Abstract. Le théorème d’énergie positive hyperbolique affirme que toute variété riemannienne complète, asymptotique à l’espace hyperbolique réel, et dont la courbure scalaire est minorée par celle du modèle, possède un vecteur énergieimpulsion de genre temps dirigé vers le futur, ce vecteur étant nul seulement pour le modèle. Nous verrons une preuve de ce résultat en toutes dimensions et sans condition spin. Il s’agit d’un travail en collaboration avec Piotr Chrusciel.
15h30 Olivier Graf (Sorbonne)
The spacelikecharacteristic Cauchy problem with L2 bounded curvature
Abstract. The bounded L2 curvature theorem by Klainerman, Rodnianski, and Szeftel states that the time of existence of a solution to Einstein’s vacuum equations is controled by the L2 norm of its curvature on spacelike Cauchy hypersurfaces. I will present a version of this result where the curvature is bounded in L2 on null hypersurfaces. This provides a first breakdown criterion on characteristic hypersurfaces at this level of regularity. The proof relies on an extension procedure, as well as on the existence and control at low regularity of a new parabolic foliation of null hypersurfaces. This is a joint work with Stefan Czimek (Toronto).
Monday June 24, 2018
lecture room 15/16309
14h Oscar J. C. CamposDias (Southampton)
Strong cosmic censorship (in de Sitter backgrounds)
Abstract. Generically, strong cosmic censorship (SCC) is the statement that physics within general relativity should be predicted from initial data prescribed on a Cauchy hypersurface. In this talk I will review how finetuned versions of SCC have been formulated and evolved along the last decades up to the point where we believe that Christodoulou’s version is true in asymptotically flat spacetimes. However, I will also describe that in the last 2 years it was found that this is no longer necessarily true for some other backgrounds, namely in de Sitter (with a positive cosmological) spacetimes.
15h30 Shiyuan Ma (Sorbonne)
Linear stability for the Kerr spacetime
Abstract. The Teukolsky Master Equation governs the dynamics of linearized gravity on the Kerr rotating black hole spacetime. In this talk, based on recent works on basic energy and Morawetz estimates for solutions of the Teukolsky equation, I shall show how to derive improved decay estimates for the Teukolsky equation and explain how such results can be used to prove linear stability for the Kerr spacetime. The proof relies on using a radiation gauge. This is joint work with Lars Andersson, Thomas Bäckdahl, and Pieter Blue.
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Seminar at the
Laboratoire JacquesLouis Lions
Sorbonne Université, Paris
Organizers
Philippe G. LeFloch (Sorbonne, Paris)
Jacques Smulevici (Sorbonne, Paris)
Jérémie Szeftel (Sorbonne, Paris)
Dates of the Seminar this Fall 2018
September 24, October 29, November 19
Monday September 24, 2018
room 15/16309
14h Dietrich Häfner (Grenoble)
Scattering for Dirac and KleinGordon fields on the (De Sitter) Kerr metric and the Hawking effect
Abstract. We consider Dirac and KleinGordon fields on the (De Sitter) Kerr metric which describes rotating black holes. Whereas there exists a conserved L2 norm for the Dirac field, no positive conserved quantity exists for the KleinGordon field, which makes the analysis more difficult for the latter. We obtain asymptotic completeness results for the Dirac field on the Kerr and for the KleinGordon field on the De Sitter Kerr metric. We then present a rigorous result about the Hawking effect for fermions in the setting of a collapse of a rotating charged star. This effect predicts the creation of particles by black holes.
15h30 JeanPhilippe Nicolas (Brest)
Peeling for scalar fields on the Kerr metric
Abstract. The peeling is an asymptotic behavior of massless fields along outgoing null geodesics in asymptotically flat spacetimes, initially observed by Sachs at the beginning of the 1960’s, then reformulated in very simple terms by Penrose in 1965 using conformal geometry. The question of its genericity, especially when talking about the peeling of the Weyl tensor of an Einstein spacetime, was controversial for several decades after Penrose’s paper. For Einstein’s equations, the question is now essentially settled, but given an Einstein spacetime, it is not clear whether there is a large class of Cauchy data giving rise to solutions with a good peeling. Lionel Mason and the speaker answered the question for fields of spin 0, 1/2 and 1 on Schwarzschild’s spacetime in 2009 and 2012. We extended recently the results to linear and non linear scalar fields on the Kerr geometry in a joint work with Pham Truong Xuan. We shall recall the history of the subject, describe the principles of the approach developed with Lionel Mason and talk about the specific features of our work for Kerr metrics.
Monday October 29, 2018
room 15/16101
14h Joe Keir (Cambridge)
The weak null condition and the pweighted energy method
Abstract. The Einstein equations in wave coordinates are an example of a system which does not obey Klainerman’s “null condition”. Their failure to satisfy this condition leads to many difficulties, both in LindbladRodnianski’s proof of global existence and in any attempt to apply other techniques to these equations. One such technique is the “pweighted energy method” of Dafermos Rodnianski, which is a very powerful and robust method that can easily be adapted to understand the behavior of waves in many interesting situations, including black holes. In this talk I will explain how to modify this method to systems which only obey the “weak null condition”, including the Einstein equations. This involves adapting the pweighted energy method, and combining it with the many of the geometric methods used by Christodoulou and Klainerman. Among other things, this allows us to enlarge the class of wave equations which are known to admit smalldata global solutions, and it also yields a detailed description of null infinity. In particular, in some situations we can understand the geometric origin of the slow decay towards null infinity exhibited by these systems: it is due to the formation of “shocks at infinity”.
15h30 Volker Schlue (Paris)
Scattering from infinity for semilinear wave equations
Abstract. I will discuss the construction of global solutions from scattering data (at null infinity) for various semilinear wave equations on Minkowski space satisfying the (weak) null condition. I will elaborate on the proof which relies, i) on a fractional Morawetz estimate, and (ii) on the construction of suitable approximate solutions from the scattering data. Finally I will outline the application of these results to Einstein’s equations in harmonic coordinates. This is joint work with Hans Lindblad.
Monday November 19, 2018
room 15/16309
14h Adam Layne (Stockholm)
Stability within T2symmetric expanding spacetimes
Abstract. We present a recently completed, nonpolarized analogue of the asymptotic characterization of T2symmetric Einstein flow solutions by P. LeFloch and J. Smulevici. We impose a far weaker condition, but obtain similar rates of decay for the normalized energy and associated quantities. Critical to this work have been novel numerical simulations which indicate that there is locally attractive behavior for those T2symmetric solutions not subject to this weakened condition. This local attractor is distinct from the local attractor in our main theorem, thereby indicating that the polarized asymptotics are on one hand stable within a larger class than merely polarized solutions, but unstable within all T2symmetric solutions.
15h30 Grigorios Fournodavlos (Sorbonne)
Dynamics of the Einstein vacuum equations about the Schwarzschild black hole interior
Abstract. We will talk about the dynamical behavior of the Schwarzschild black hole singularity, in the context of the Einstein equations in vacuum, from the point of view of the Cauchy problem in general relativity. As it is well known, the Schwarzschild singularity is highly unstable under arbitrarily small perturbations, which makes the study of its dynamics in full generality a difficult problem. We will begin by giving an overview of the current status of the nearSchwarzschildblack hole interior problem and we will compare it to the dynamics observed near other singularity models, in black hole interiors or Big Bangs. Then we will discuss linear and nonlinear partial results in the nearSchwarzschild case, both backwards and forwards in time, with and without symmetries.
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International Conference
MATHEMATICAL GENERAL RELATIVITY
Monday May 28 to Friday June 1rst 2018
Institut Henri Poincaré
11 rue Pierre et Marie Curie, Paris
Invited Speakers
 Spyros Alexakis (Univ. of Toronto)
 Xinliang An (Univ. Toronto)
 Lars Andersson (Einstein Inst., Potsdam)
 Stefanos Aretakis (Princeton)
 Grigorios Fournodavlos (Univ. of Cambridge)
 Dejan Gajic (Cambridge University)
 Peter Hintz (Univ. of California, Berkeley)
 Gustav Holzegel (Imperial College, London)
 Cécile Huneau (Ecole Polytechnique, Palaiseau)
 Jérémie Joudioux (Univ. of Vienna)
 Jonathan Luk (Stanford Univ.)
 Sunjin Oh (Korea Inst. Advanced Study)
 Andrzej Rostworowski (Univ. Krakow)
 Jan Sbierski (Oxford Univ.)
 Yakov ShlapentokhRothman (Princeton)
 Volker Schlue (Sorbonne Univ.)
 Jérémie Szeftel (Sorbonne Univ.)
Schedule of the conference
Schedule for MONDAY
 9:00am coffee
 9:30am L. Andersson
 10:30am coffee break.
 11:00am S. Alexakis
 2:30pm P. Hintz
 3:30pm coffee break
 4:00pm V. Schlue
Schedule for TUESDAY
 9:00am: coffee
 9:30am J Joudioux
 10:30am coffee break
 11:00am G. Fournodavlos
 2:30pm Sunjin Oh
 3:30pm coffee break
 4:00pm D Gajic
Schedule for WEDNESDAY
 9:00am coffee
 9:30am G Holzegel
 10:30am coffee break
 11:00am Y. ShlapentokhRothman
 noon: C. Huneau
 7:00pm Reception at Sorbonne Univ.
Central Tower, Jussieu campus, 4 Place Jussieu
Schedule for THURSDAY
 9:00am coffee
 9:30am S. Aretakis
 10:30am coffee break
 11:00am J. Sbierski
 2:30pm A. Rostworowski
 3:30pm coffee break
 4:00pm X. An
Schedule for FRIDAY
 9:00am: coffee
 9:30am J. Szeftel
 10:30am coffee break
 11:00am J Luk
Titles and attracts of the lectures
 Spyros Alexakis (Univ. of Toronto): Singularity formation in black hole interiors: Polarized perturbations of SchwarzschildAbstract: We consider the stability of the Schwarzschild singularity in vacuum under polarized and axially symmetric perturbations. We find that the spacelike singularities persist under such perturbations, but their dynamics exhibit a great richness, consistent with the asymptotically velocity term dominated behavior. The result relies crucially on a new approach for the Einstein equations in axial symmetry. Joint work with G. Fournodavlos.
 Xinliang An (Univ. Toronto): On apparent horizon formation.
Abstract:Combining both hyperbolic and elliptic techniques, we study the formation of a marginally outer trapped tube (apparent horizon) in gravitational collapse. Analytic and geometric properties of this apparent horizon will also be discussed.
 Lars Andersson (Einstein Inst., Potsdam): Linear stability for the Kerr spacetime.
Abstract: The Teukolsky Master Equation governs the dynamics of linearized gravity on the Kerr rotating black hole spacetime. Recent work of Ma, and of Dafermos, Holzegel and Rodnianski provides energy, Morawetz, and pointwise decay estimates for solutions of the Teukolsky equation. In this talk I shall show how to derive improved decay estimates for the Teukolsky equation and explain how such results can be used to prove linearized stability for the Kerr spacetime, including energy, Morawetz, and pointwise estimates for the linearized metric. The proof relies on using a radiation gauge. This is based on ongoing joint work with Thomas Bäckdahl, Pieter Blue, and Siyuan Ma.
 Stefanos Aretakis (Princeton): Asymptotics for the wave equation on black hole backgrounds
Abstract: We will present asymptotic results for solutions to the wave equation for the full ReissnerNostrom family of black holes. These spacetimes are spherically symmetric asymptotically flat solutions to the EinsteinMaxwell system. We will consider both the subextremal and the extremal cases. We will see that conservation laws on null hypersurfaces play an important role in the precise latetime asymptotics for solutions to the wave equation. For the extremal ReissnerNordstrom the situation is more subtle given that there are two independent conservation laws (in contrast to the sub extremal where this is only one such conservation law). We will also present a scattering theory in the extremal case which in particular allows us to construct exponentially decaying smooth solutions. This work is joint with Gajic (Cambridge) and Angelopoulos (UCLA).
 Grigorios Fournodavlos (Univ. of Cambridge): On ‘hard stars’ in general relativity
Abstract: After a brief review of the classical results on gravitational collapse in spherical symmetry, from the OppenheimerSnyder model (1939) to Christodoulou’s twophase model (1995), I will discuss one possible end state in the latter model: hard stars. These are idealized models of neutron stars. I will present a variational characterization and discuss its relevance to the orbital stability problem in spherical symmetry. Various obstacles to a global in time result are outlined, in particular the absence of a dispersion mechanism, the trapped surface formation scenario due to reflecting boundary conditions (cf. AdSscalar field) and the possibility of phase transitions within the two phase model to avoid RayleighTaylor instabilities. This is a joint work with Volker Schlue.
 Dejan Gajic (Cambridge University): Conservation laws and latetime tails of waves on Schwarzschild for all angular momenta
Abstract: In 1972, Price suggested that inverse polynomial tails should be present in the latetime behaviour of scalar fields on Schwarzschild black holes with fixed angular momentum and the decay rates should depend in a precise manner on the angular momentum. In the decades since, many features of these tails have been explored both numerically and heuristically. The presence of polynomial tails along event horizons has important implications for the nature of singularities inside dynamical black holes. In this talk I will discuss work done in collaboration with Y. Angelopoulos and S. Aretakis that establishes rigorously the existence of these polynomial latetime tails in Schwarzschild spacetimes. I will give a sketch of how the decay rates of Price can be derived using only physical space methods and how the coefficients in the latetime asymptotics of the scalar field are related to the existence of conservation laws.  Peter Hintz (Univ. of California, Berkeley): Global stability problems
Abstract: I will discuss the problem of proving the stability of (families of) exact spacetimes (M,g) such as Minkowski space or the family of Kerrde Sitter (KdS) black holes as solutions of Einstein’s vacuum equation, focussing on geometric aspects of this problem: how to compactify M for the purpose of analyzing the underlying nonlinear wave equation; how to choose a gauge to break the diffeomorphism invariance of Einstein’s equation; and the role of constraint damping.
 Gustav Holzegel (Imperial College, London): Boundedness and Decay for Solutions to the Teukolsky Equation on slowly rotating Kerr spacetimesAbstract.
Abstract: I will outline a proof (joint work with M. Dafermos and I. Rodnianski) of boundedness and polynomial decay statements for solutions of the spin ±2 Teukolsky equations on a Kerr exterior background with parameters satisfying a << M. The estimates are obtained through natural generalisations of the higher order quantities P and \underline{P} introduced in our previous work on the linear stability of the Schwarzschild metric.
 Cécile Huneau (Ecole Polytechnique, Palaiseau): High frequency limit for Einstein equations with U(1) symmetry.
Abstract: I will present the construction of a family of solutions to vacuum Einstein equations with U(1) symmetry which consist of an arbitrary number of high frequency waves travelling in different directions. In the high frequency limit, our family of solutions converges to a solution of Einstein equations coupled to null dusts. This construction is an illustration of the so called backreaction, studied by physicists (Isaacson, Burnet, Green, Wald…) : the small scale inhomogeneities have an effect on the large scale dynamics in the form of an energy impulsion tensor in the righthand side of Einstein equations. This is a joint work with Jonathan Luk (Stanford).
 Jérémie Joudioux (Univ. of Vienna): The vectorfield method for the transport equation with application to the EinsteinVlasov system.
Abstract: The vectorfield method, developed by Klainerman, was a key tool to understand the global existence of solutions to quasilinear wave equations. In a series of work in collaboration with D. Fajman (Vienna), and J. Smulevici (Orsay), the vector field method is extended to the relativistic transport equation where it is used to derive decay estimates for velocity averages for solutions to the relativistic Vlasov equation. An important application of this method is the proof of the stability of Minkowski space as a solution to the EinsteinVlasov system. I will present in this talk this commutator technique for the transport equation, and describe the decay estimates for velocity averages, and sketch the key steps of the stability proof.Joint work with D. Fajman (Vienna) and J. Smulevici (Orsay).
 Jonathan Luk (Stanford Univ.): The interior of extremal black holes
Abstract: I will contrast the interior regions of subextremal and extremal black holes and present a recent result regarding the interior of dynamical extremal black holes for the EinsteinMaxwellcharged scalar field system in spherical symmetry. This is a joint work with Dejan Gajic.
 Sunjin Oh (Korea Inst. Advanced Study): Strong cosmic censorship and generic mass inflation for charged black holes in spherical symmetry.
Abstract: I will first review a recent joint work with J. Luk, in which the C2formulation of the strong cosmic censorship is proved for the EinsteinMaxwell(real)Scalar Field system in spherical symmetry for twoended asymptotically flat data. More precisely, it was shown that a “generic” class of data for this model gives rise to maximal future developments which are future C2inextendible. In the second part of the talk, I will present a new, complementary theorem (also joint with J. Luk) that for a further “generic” subclass of such data, the Hawking mass blows up identically along the Cauchy horizon. This result confirms, rigorously and unconditionally, the mass inflation scenario of PoissonIsrael and Dafermos for the model at hand.
 Andrzej Rostworowski (Univ. Krakow): New insights into nonlinear perturbations of vacuum spacetimes.
Abstract: I will present a systematic and robust approach to nonlinear gravitational perturbations of maximally symmetric black holes. In particular, I will show that at each order of perturbation expansion, the system of perturbative Einstein equations can be reduced to two (for each gravitational mode in 3+1 dimensions on which the study will be focused) scalar wave equations, and the metric perturbations can be explicitly obtained, once the solutions to these scalar wave equations are known. That is, this approach extends the field of gravitational
perturbations of black holes, initiated in the seminal Regge & Wheeler 57′ paper beyond linear order. The talk will be partially based on a recent work Phys. Rev. D96, 124026 (2017).  Jan Sbierski (Oxford Univ.): On the unique evolution of solutions to wave equations
Abstract: The wellknown theorem of ChoquetBruhat and Geroch states that for given smooth initial data for the Einstein equations there exists a unique maximal globally hyperbolic development. In particular, the timeevolution of globally hyperbolic solutions is unique. This talk investigates whether the same results hold for quasilinear wave equations defined on a fixed background. We first present an example of a quasilinear wave equation for which unique evolution of smooth globally hyperbolic solutions in fact fails and contrast this case with the Einstein equations. We then proceed by presenting conditions which guarantee unique evolution. This talk is based on joint work with Felicity Eperon and Harvey Reall.
 Yakov ShlapentokhRothman (Princeton Univ.): The asymptotically selfsimilar regime for the Einstein vacuum equations.
Abstract: We will dynamically construct singular solutions to the Einstein vacuum equations which are asymptotically selfsimilar in that successive rescalings around the singularity converge to a selfsimilar solution. Connections both to Christodoulou’s bounded variation solutions of the spherically symmetric Einsteinscalar field system and to the ambient metric construction of Fefferman and Graham will be elaborated on. This is joint work with Igor Rodnianski.
 Volker Schlue (Sorbonne Univ.): Expanding black hole spacetimes: Towards the stability of the cosmological region
Abstract: The Schwarzschild de Sitter spacetime is the simplest model of a black hole in the expanding universe. I will discuss the challenges arising in the stability problem for this solution of the Einstein vacuum equations with positive cosmological constant, and focus in particular on the evolution in the expanding region beyond the cosmological horizon of the black hole. I will present a result for the decay of the conformal Weyl curvature, and discuss its relation to the existence of asymptotic degrees of freedom in this problem. Moreover, I will discuss the behavior of solutions to the Eikonal equation in de Sitter, and questions related to the definition of asymptotic quantities.
 Jérémie Szeftel (Sorbonne Univ.): The nonlinear stability of Schwarzschild
Abstract: I will discuss a joint work with Sergiu Klainerman on the stability of Schwarzschild as a solution to the Einstein vacuum equations with initial data subject to a certain symmetry class.
Organizers
Philippe G. LeFloch (Paris), Jacques Smulevici (Orsay), Jérémie Szeftel (Paris)
Funding
 GEOWAKI
“The analysis of geometric nonlinear wave and kinetic equations”
Principal investigator: Jacques Smulevici
ERC Starting Grant 2016
 EPGR
“The Evolution Problem in General Relativity”
Principal investigator: Jérémie Szeftel
ERC Consolidator Grant 2016
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