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**_________________________________________**

#### Seminar at the

#### Laboratoire Jacques-Louis Lions

#### Université Pierre et Marie Curie, Paris

*Organizers*

#### Philippe G. LeFloch (Paris)

#### Jacques Smulevici (Orsay)

#### Jérémie Szeftel (Paris)

This Fall: **October 10, November 21, and December 12, 2016**

#### Monday October 10, 2016

*room 15/25-104*

#### 14h Peter Hintz (Berkeley)

**Nonlinear stability of Kerr-de Sitter black holes**

**Abstract. **In joint work with András Vasy, we recently established the stability of the Kerr-de Sitter family of black holes as solutions of the initial value problem for the Einstein vacuum equations with positive cosmological constant, for small angular momenta but without any symmetry assumptions on the initial data. I will explain the general framework which enables us to deal systematically with the diffeomorphism invariance of Einstein’s equations, and thus how our solution scheme finds a suitable (wave map type) gauge within a carefully chosen finite-dimensional family of gauges; I will also address the issue of finding the mass and the angular momentum of the final black hole.

#### 15h30 Stefan Czimek (Paris)

**An extension procedure for the constraint equations**

**Abstract. **In this talk we present a new extension procedure for the maximal constraint equations of general relativity, motivated by applications to the Cauchy problem. Given a small solution on the unit ball, we can extend it to an asymptotically flat global solution. The main features are that our extension procedure does not need a gluing region, preserves regularity and works in weak regularity. For the proof, we use new methods to solve the prescribed divergence equation for the second fundamental form and the prescribed scalar curvature equation for the metric. We use the under-determinedness of the constraint equations to conserve regularity.

#### Monday November 21, 2016

*room 15/16-413*

#### 14h The-Cang Nguyen (Paris)

**Progress and recent results for the conformal equations**

**Abstract. **The presentation will be divided into two parts. First, I will introduce the conformal equations and present recent results for these equations as well as questions arising naturally. In a second part, I will talk about the “half-continuity method” and explain how to use this method for giving answers to the questions posed in the first part.

#### 15h30 Volker Schlue (Paris)

**On the nonlinear stability of expanding black hole cosmologies**

#### Monday December 12, 2016

*room 15/25-102*

#### 14h Michał Wrochna (Grenoble)

**The quantum stress-energy tensor and its intricate relationship with spacetime geometry**

**Abstract. **It is widely believed that at low energies, quantum gravity should yield an effective theory described by Einstein equations with a stress-energy tensor made of averaged fluctuations of quantum fields. The construction of that stress-energy tensor is however very problematic and its intricate dependence on spacetime geometry results in highly non-linear equations that possess no qualitative theory to date. In this talk I will review this problem as a motivation for improving the construction of linear Klein-Gordon quantum fields, and discuss recent progress that allows for a better control of the dependence on the spacetime metric (partly based on joint work with Christian Gérard).

#### 15h30 Guillaume Idelon-Riton (Regensburg)

**Some results about the scattering theory for the massive Dirac fields in the Schwarzschild-Anti-de Sitter space-time**

**Abstract. **I will first give a brief presentation of the Schwarzschild-Anti-de Sitter spacetime and of some of its geometrical properties that will concern us. Then I will present the massive Dirac equation in this space-time and first study the Cauchy problem which is not completely obvious since our spacetime is not globally hyperbolic. I will then give a result concerning the asymptotic completeness for these fields. By means of a Mourre estimate, it is possible to obtain that the minimal velocity for these fields is 1. I will then show that our dynamics behaves in asymptotic regions like a transport at unit speed in the direction of the black hole. In a third part, I will study the local energy decay for these fields. First, using the existence of exponentially accurate quasi-modes, I will show a logarithmic lower bound on the local energy decay which is in accordance with the results of G. Holzegel and J. Smulevici in the Kerr-Anti-de Sitter spacetime for the Klein-Gordon fields. In order to obtain an upper bound, I will prove the existence of resonances and give some tools in order to localize them.

**September 14, 2015 to December 18, 2015**

#### Trimester Program at the

#### Centre Emile Borel

#### Financial support provided by Institut Henri Poincaré

#### and ANR Project *“Mathematical General Relativity”*

*Organizers*

*Lars Andersson (Potsdam)*

*Sergiu Klainerman (Princeton) *

*Philippe G. LeFloch (Paris) *

**MAIN THEMES OF THE PROGRAM**

Einstein’s field equation of general relativity is one of the most important geometric partial differential equations. Over the past decade, the mathematical research on Einstein equation has made spectacular progress on many fronts (Cauchy problem, cosmic censorship, asymptotic behavior). These developments have brought into focus the deep connections between the Einstein equation and other important geometric PDE’s, including the wave map equation, Yang-Mills equation, Yamabe problem, as well as Hamilton’s Ricci flow. The field is of growing interest for mathematicians and of intense current activity, as is illustrated by major recent breakthrough, concerning the uniqueness and stability of the Kerr black hole model, the formation of trapped surfaces, and the bounded L2 curvature problem. Specifically, the themes of mathematical interest that will be developed in the present Program and are currently most active include:

- The initial value problem for Einstein equation and the causal geometry of spacetimes with low regularity, formation of trapped surfaces

- Techniques of Lorentzian geometry: injectivity radius estimates, geometry of null cones; construction of parametrix

- Geometry of black hole spacetimes: uniqueness theorems, censorship principles

- Coupling of Einstein equation for self-gravitating matter models, weakly regular spacetimes, nonlinear stability of Minkowski space with matter

General schedule for the Trimester

**SCIENTIFIC ACTIVITIES during the Trimester**

**Workshops and Conferences**(see below)**On the Mathematical Theory of Black Holes,**Course- Begins on October 13 from 2pm to 4:30pm, and the following Tuesdays.
- Video of the lectures

**An Introduction to Self-Gravitating Matter**__,__Course by P.G. LeFloch.- Begins on October 9 from 2pm to 4:30pm, and the following Fridays.
- LECTURE NOTES for this course
- Video 1: An introduction to self-gravitating matter
- Video 2: Modified gravity and weakly regular spacetimes
- Video 3: Weak solutions to the Einstein equations
- Video 4: Weakly regular Cauchy developments
- Video 5: Self-gravitating fluids with bounded variation
- Video 6: The geometry of weakly regular spacetimes
- Video 7: Nonlinear stability of Minkowski space for massive fields

**Weekly Seminar on Mathematical General Relativity**- organized by L. Andersson, jointly with S. Klainerman, P.G. LeFloch, J. Szeftel (Paris), and A. Zeghib (Lyon).
- Begins on October 7 from 2pm to 4:30pm, and the following Wednesdays.
- Speakers in the Seminar :
- Bruno Premoselli (Cergy-Pontoise), Martin Taylor (Cambridge, UK)
- Steffen Aksteiner (Potsdam), Lars Andersson (Potsdam)
- Jesus Oliver (San Diego)
- Anne Franzen (Utrecht), Jared Wunsch (Evanston)
- Sung-Jin Oh (Berkeley), Anna Sakovich (Vienna)
- Hans Lindblad (Baltimore), Florian Beyer (Dunedin)

- VIDEOS available (courses, main conference)
**Tea break**every day 3pm-3:30pm

**WORKSHOPS AND CONFERENCES**

**Sept. 14 to 18, 2015** **Summer School – INTRODUCTION TO MATHEMATICAL GENERAL RELATIVITY**

**List of speakers**

Greg Galloway (Miami)

Gerhard Huisken (Tuebingen)

Hans Ringstrom (Stockholm)

**Sept. 23 to 25, 2015 ** **Workshop – RECENT ADVANCES IN MATHEMATICAL GENERAL RELATIVITY**

**List of speakers**

Spyros Alexakis (Toronto)

Piotr Chrusciel (Vienna)

Joao Costa (Lisbon)

Semyon Dyatlov (Cambridge, USA)

Stefan Hollands (Cardiff)

Alexandru Ionescu (Princeton)

Lionel Mason (Oxford)

Vincent Moncrief (Yale)

Jean-Philippe Nicolas (Brest)

Harvey Reall (Cambridge, UK)

Hans Ringstrom (Stockholm)

Mu-Tao Wang (New York)

**Sept. 28 to Oct. 1, 2015 ** ** Workshop – GEOMETRIC ASPECTS OF MATHEMATICAL RELATIVITY** (Hold in Montpellier and organized by Marc Herzlich and Erwann Delay)

**List of speakers**

Piotr Chrusciel (Vienna)

Michael Eichmair (Zürich)

Mu-Tao Wang (New York)

**Oct. 26 to 29, 2015 Workshop – DYNAMICS OF SELF-GRAVITATING MATTER**

**List of speakers**

Hakan Andreasson (Gothenburg)

Thierry Barbot (Avignon)

Robert Beig (Vienna)

David Fajman (Vienna)

Marc Mars (Salamanca)

David Maxwell (Fairbanks)

Todd Oliynyk (Monash)

Volker Schlue (Toronto)

Bernd Schmidt (Potsdam)

Jared Speck (Cambridge, USA)

Shadi Tahvildar-Zadeh (Rutgers)

Eric Woolgar (Alberta)

**Nov. 16 to 20, 2015 ** **International Conference – GENERAL RELATIVITY – A Celebration of the 100th Anniversary**

**List of speakers Schedule and title**

Jean-Pierre Bourguignon (Bures-sur-Yvette)

Demetrios Christodoulou (Zürich & Athens)

Mihalis Dafermos (Princeton)

Thibault Damour (Bures-sur-Yvette)

Georges Ellis (Cape Town)

Richard Hamilton (New York)

Gustav Holzegel (London)

Jonathan Luk (Cambridge, UK)

Roger Penrose (Oxford)

Richard Schoen (Stanford & Irvine)

Jacques Smulevici (Orsay)* *

Jérémie Szeftel (Paris)

Robert Wald (Chicago)

Qian Wang (Oxford)

**Dec. 14 to 16, 2015 ** ** International Conference- RELATIVITY AND GEOMETRY – IN MEMORY OF A. LICHNEROWICZ ** (Organized by Giuseppe Dito, Jean-Pierre Francoise, Paul Gauduchon, Richard Kerner, Yvette Kosmann-Schwarzbach et Daniel Sternheimer)

**List of speakers**

Olivier Biquard (Paris 6)

Robert Bryant (Durham)

Pierre Cartier (Gif-Sur-Yvette)

Thibault Damour (Gif-Sur-Yvette)

Nathalie Deruelle (Paris 7)

Simon Donaldson (Stony Brook & London)

Michel Dubois-Violette (Paris 11)

Charles Francès (Strasbourg)

Edward Frenkel (Berkeley)

Christian Fronsdal (Los Angeles)

Simone Gutt (Bruxelles)

James Isenberg (Eugene)

Sergiu Klainerman (Princeton)

Maxim Kontsevich (Gif-Sur-Yvette)

Alan Weinstein (Berkeley)

Program coordinated by the Centre Emile Borel at IHP. Financial support provided by the Institut Henri Poincaré and the ANR Project *“Mathematical General Relativity. Analysis and geometry of spacetimes with low regularity”.*

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*Organizers*

#### Michael Anderson (Stony Brook)

#### Sergiu Klainerman (Princeton)

#### Philippe G. LeFloch (Paris)

#### Jared Speck (Cambridge, USA)

#### Location: Simons Center for Geometry and Physics

#### Date: One-month concentration period in January 2015

#### Workshop from January 20 to 23, 2015

Einstein’s field equation of general relativity is one of the most important geometric partial differential equations. Over the past decade, the mathematical research on Einstein equation has made spectacular progress on many fronts, including the Cauchy problem, cosmic censorship, and asymptotic behavior. These developments have brought into focus the deep connections between the Einstein equation and other important geometric partial differential equations, including the wave map equation, Yang-Mills equation, Yamabe equation, as well as Hamilton’s Ricci flow. The field is of growing interest for mathematicians and of intense current activity, as is illustrated by major recent breakthroughs concerning the uniqueness and stability of black hole models, the formation of trapped surfaces, and the bounded L2 curvature problem. The themes of mathematical interest that will be particularly developed in the present Program include the formation of trapped surfaces and the nonlinear interaction of gravitational waves. The new results are based on a vast extension of the earlier technique by Christodoulou and Klainerman establishing the nonlinear stability of the Minkowski space. This Program will be an excellent place in order to present the recent breakthrough on the bounded L2 curvature problem for the Einstein equation, which currently provides the lower regularity theory for the initial value problem, as well as the recently developed theory of weakly regular Einstein spacetimes with distributional curvature.

**Long-term participants**

Michael Anderson (Stony Brook)

Piotr Chrusciel (Vienna)

Mihalis Dafermos (Princeton)

Cécile Huneau (Paris)

Alexandru D. Ionescu (Princeton)

James Isenberg (Eugene)

Sergiu Klainerman (Princeton)

Philippe G. LeFloch (Paris)

Jared Speck (Cambridge, USA)

Jinhua Wang (Hangzhou)

Mu-Tao Wang (New York)

Qian Wang (Oxford)

Willie Wong (Lausanne)

**Speakers during the Workshop**

- Tuesday January 20
- Sung-Ji Oh (Berkeley) Linear instability of the Cauchy horizon in subextremal Reissner-Nordström spacetime under scalar perturbations
- Volker Schlue (Toronto) Stationarity of time-periodic vacuum spacetimes
- Alexandru D. Ionescu (Princeton) The Euler–Maxwell system for electrons: global solutions in 2D
- Joachim Krieger (Lausanne) Concentration-compactness for the critical Maxwell-Klein-Gordon equation

- Wednesday January 21
- Xianliang An (Piscataway) Two results on formation of trapped surfaces
- Tahvildar-Zadeh (Piscataway) The Dirac electron and the Kerr-Newman spacetime
- Mihalis Dafermos (Princeton)
- Jim Isenberg (Eugene) Asymptotically hyperbolic shear-free solutions of the Einstein constraint equations

- Thursday January 22
- Cécile Huneau (Paris) Stability in exponential time of Minkowski
- Jacques Smulevici (Orsay) Vector field methods for transport equations with applications to the Vlasov-Poisson system
- Mu-Tao Wang (New York) Quasi-local angular momentum and the limit at infinity
- Spyros Alexakis (Toronto) The Penrose inequality for perturbations of the Schwarzschild exterior

- Friday January 23
- Mihai Tohaneanu (Statesboro) Pointwise decay for the Maxwell system on black holes
- Qian Wang (Oxford)
- Peter Blue (Edinburgh) Revisiting decay of fields outside a Schwarzschild black hole
- Philippe G. LeFloch (Paris) Weak solutions to the Einstein equations in spherical or T2 symmetry

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**_________________________________________**

*Organizers*

#### Philippe G. LeFloch (Paris)

#### Jérémie Szeftel (Paris)

#### Ghani Zeghib (Lyon)

**ANR Project**

#### “Mathematical General Relativity. Analysis and geometry of spacetimes with low regularity”

**Wednesday June 17, 2015**

### Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris

#### Lecture room 15/25-326

**11h Emmanuel Hebey (Cergy-Pontoise) **Systèmes de Kirchhoff critiques stationnaires sur des variétés compactes

**14h Lydia Bieri (Ann Arbor) **Gravitational radiation and two types of memory

**Abstract. **We are believed to live on the verge of detection of gravitational waves, which are predicted by General Relativity. In order to understand gravitational radiation, we have to investigate analytic and geometric properties of corresponding solutions to the Einstein equations. Gravitational waves leave a footprint in the spacetime regions they pass, changing the manifold – and therefore displacing test masses – permanently. This is known as the memory effect. It has been believed that for the Einstein equations, being nonlinear, there exists one such effect with a small `linear’ and a large `nonlinear’ part. In this talk, I present some of my joint work with D. Garfinkle showing that these are in fact two different effects.

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**_________________________________________**

*Organizers*

#### Philippe G. LeFloch (Paris)

#### Jérémie Szeftel (Paris)

#### Ghani Zeghib (Lyon)

**ANR Project**

#### “Mathematical General Relativity. Analysis and geometry of spacetimes with low regularity”

**Wednesday May 27, 2015**

### Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris

#### Lecture room 15-25–326

**14h Thierry Barbot (Avignon) **Surfaces polygonales fuchsiennes et espace de Teichmüller décoré

**Abstract. **Dans l’article “Fuchsian polyhedra in Lorentzian space-forms, Mathematische Annalen 350, 2, pp. 417-453, 2011″, F. Fillastre a montré que toute métrique euclidienne avec singularités coniques d’angles > 2 pi sur une surface compacte se réalise de manière unique comme une surface de Cauchy polygonale dans un espace-temps globalement hyperbolique localement plat radial (i.e. dont le groupe d’holonomie fixe un point de l’espace de Minkowski). Dans cet exposé, j’évoquerai le travail de L. Brunswic dans son travail de thèse sous ma direction, qui vise à reprouver ce résultat et à l’étendre au cas des surfaces polygonales dans un espace-temps localement plat mais admettant des particules massives. Le but est de montrer qu’il y a encore existence et unicité une fois prescrit la masse des particules massives (le cas régulier montré par Fillastre correspondant au cas où l’angle singulier des particules massives est 2pi). Je montrerai aussi que la situation étudiée par R. Penner dans l’article “The Decorated Teichmϋller Space of Punctured Surfaces, Commun. Math. Phys. 113, 299-339 (1987)” est un cas limite de la situation étudiée par Brunswic, et correspond au cas où les particules sont d’angle conique nul. Je montrerai aussi comment répondre positivement à la question dans le cas où il n’y a qu’une singularité.

**15h30 Andrea Seppi (Pavia) **Convex surfaces in (2+1)-dimensional Minkowski space

**Abstract. **It is known that the hyperbolic plane admits an isometric embedding into Minkowski space; in 1983 Hanu and Nomizu first observed the existence of non-equivalent isometric embeddings, thus showing a relevant difference with the Euclidean case. In this talk, I will introduce some natural properties of a convex surface in Minkowski space, concerning causality and asymptotic behavior. I will then explain some new results (jointly with Francesco Bonsante) on the classification of constant curvature surfaces with bounded principal curvatures and on the solvability of Minkowski problem in (2+1)-dimensional Minkowski space. If time permits, I will give the main ideas of the proof and especially the relation to some type of Monge-Ampere equations.