This one-week conference is a main event of the Trimester Program taking place at the Institut Henri Poincaré in order to celebrate the 100th Anniversary of General Relativity. This* *Conference will provide an update on current research in mathematical general relativity.

*Emile Borel Centre of Henri Poincaré Institute, Paris*

*November 16 to 20, 2015*

**Monday Nov. 16**

**Robert Wald**(Chicago) [chairman P.G. LeFloch] VIDEO

*(coffee break)*

**Jean-Pierre Bourguignon**(Bures-sur-Yvette) VIDEO

*(lunch break)*

**Roger Penrose**(Oxford) VIDEO [chairman S. Klainerman]

*(coffee break)*

**Tuesday Nov. 17**

**Mihalis Dafermos**(Princeton) VIDEO [chairman M. Herzlich]

*(coffee break)*

**Jonathan Luk**(Cambridge, UK)

*(blackboard talk) VIDEO*

*(lunch break)*

**Jérémie Szeftel**(Paris) [chairman V. Moncrief] VIDEO

**Wednesday Nov. 18 **

*Movie on the History of General Relativity:**Einstein et la relativité générale, une histoire singulière*(with English subtitles)

*(Amphitheater Hermite)*

*Ceremony in honor of Y. Choquet-Bruhat (hold at IHES, Bures-sur-Yvette)*

**Thursday Nov. 19**

**Thibault Damour**(Bures-sur-Yvette) VIDEO [chairman R. Wald]

*(coffee break)*

**Demetrios Christodoulou**(Zürich & Athens)

*(blackboard talk) VIDEO*

*Buffet Dinner Reception (ground floor of IHP)*

**Friday Nov. 20**

**Georges Ellis**(Cape Town) VIDEO [chairman L. Andersson]

*(coffee break)*

**Richard Schoen**(Stanford & Irvine) VIDEO

*end of the Conference)*

**LIST OF SPEAKERS,TITLES and ABSTRACTS**

Jean-Pierre Bourguignon (Bures-sur-Yvette) **General Relativity and Geometry: Interactions and Missed Opportunities**

**Abstract. **Physics and Geometry have a long history in common, but the Theory of General Relativity, and theories it triggered, have been a great source of challenges and inspiration for geometers. It started even before its birth with the motivation coming from the theory of ether in the works of B. Riemann and W.K. Clifford, and the advent of Special Relativity by A. Einstein. The central role Ricci curvature plays in the Theory of General Relativity changed completely the vision geometers have of the importance of this geometric object. Several other instances of geometric notions which were triggered through this interaction will be presented from Conformal Geometry, Riemannian submersions, the space of Riemannian metrics, Killing spinors and topological Lagrangians. These positive interactions culminate of course in the spectacularly deep understanding of the Einstein field equations as a non-linear system of partial differential equation, which will be reported upon at this conference by a number of speakers.

Demetrios Christodoulou (Zürich & Athens) **The Formulation of the Two-Body Problem in General Relativity** *(blackboard talk)*

**Abstract. **In my talk I shall discuss the formulation of the two-body problem in general relativity. The setup is to represent two stars coming in from infinity with asymptotically uniform velocities and with no incoming radiation. In electromagnetic theory, where the field equations are linear, the no incoming radiation condition is fulfilled by selecting the retarded Green’s function. In contrast, in general relativity by reason of the nonlinearity of the Einstein field equations, imposing the no incoming radiation condition requires us to consider a sequence of initial value problems with the initial hypersurfaces tending to the infinite past and to choose appropriately the associated sequence of initial dat. I shall explain how this is to be accomplished. I shall discuss the choice of fluid model which leads to the simplest problem. I shall also discuss the Newtonian limit and what is presently known in this case. Finally, in regard to the relativistic problem, I shall point out what seem to me to be the most fruitful directions of research.

Mihalis Dafermos (Princeton) **The Stability Problem for Black Holes and the Cosmic Censorship Conjectures**

Thibault Damour (Bures-sur-Yvette)** ****Through a Glass Darkly: the Structure of Cosmological Singularities**

**Abstract. **Belinskii, Khalatnikov and Lifshitz (BKL) introduced the idea that the geometry of spacetime near a generic, inhomogeneous cosmological (i.e. spacelike) singularity has (for the four-dimensional vacuum Einstein equations) a chaotic structure. After reviewing the status of BKL ideas as well as their extension to the higher-dimensional vacuum Einstein equations, and to the Einstein-matter systems of most relevance in the context of current theoretical physics, we shall present a precise formulation of the BKL conjecture (in the chaotic case) in terms of an Iwasawa-type parametrization of the spatial metric. We shall also present recent developments suggesting that the chaotic BKL behavior is the shadow of a hidden symmetry (possibly related to M-theory dualities) involving infinite-dimensional hyperbolic Kac-Moody algebras.

Georges Ellis (Cape Town) *The Steps along the Way Whereby Einstein’s General Theory Enabled the Rise of Present Day Cosmology *

Gustav Holzegel (London) *The Linear Stability of the Schwarzschild Solution under Gravitational Perturbations*

Jonathan Luk (Cambridge, UK)** ****Interior of Dynamical Vacuum Black Holes** *(blackboard talk)*

**Abstract. **I will discuss recent work on the structure of black hole interiors for dynamical vacuum spacetimes (without any symmetry) and what this means for the question of the nature of generic singularities in general relativity and the celebrated strong cosmic censorship of Penrose. This is joint work with Mihalis Dafermos.

Roger Penrose (Oxford) **Palatial Twistor Theory: a Quantum Approach to Classical (and Quantum?) Space-Time **

**Abstract.** Up until recently, the applications of twistor theory to general relativity have been rather limited, applicable mainly to special solutions of the Einstein equations and to complex solutions which are anti-self-dual, these describing left-handed “non-linear gravitons”. Recently, however, a new approach—palatial twistor theory—has emerged which, though related to the earlier ambitwistor approach, is essentially different, bringing the quantum ideas of non-commutativity into the geometry. Though the (Einstein) space-times treated this way are still classical, the approach suggests a possible route to a quantum gravity theory.

Richard Schoen (Stanford & Irvine) **Localizing Solutions of the Einstein Equations**

**Abstract. **In general it is not possible to localize solutions of the Einstein equations since there are asymptotic conserved quantities such as the total mass which are nonzero for every nontrivial space-time. In this lecture we will describe work with A. Carlotto which achieves a localization in cones which is in a certain sense optimal. The construction leads to a superposition theorem for solutions of the Einstein vacuum equations in such a way that there is no interaction between the solutions for a fixed time. It also leads to some surprising new phenomena for the geometry of initial data sets.

Jacques Smulevici (Orsay) **Dynamics in Asymptotically AdS Spacetimes**

**Abstract. **I will review several recent results concerning the study of wave equations in spacetimes which behaves near spatial infinity as the Anti-de-Sitter (AdS) space. In particular, my talk will cover the behavior of linear waves in asymptotically AdS black hole spacetimes and the conjectured instability of AdS for Dirichlet type boundary conditions as well as the decay properties of linear fields in AdS, including electromagnetic and Weyl fields, for absorbing boundary conditions obtained in joint work with Holzegel, Luk and Warnick.

Jérémie Szeftel (Paris)* **Remarks on the Nonlinear Stability of Schwarzschild*

Robert Wald (Chicago) *Dynamic and Thermodynamic Stability of Black Holes and Black Branes*

**Abstract. **I describe work with Stefan Hollands that establishes a general criterion for the dynamical stability of black holes and black branes in arbitrary spacetime dimensions with respect to axisymmetric perturbations. We show that the positivity of the canonical energy on a subspace of linearized solutions that have vanishing linearized ADM mass and angular momentum implies mode stability. Conversely, failure of positivity of canonical energy on this subspace implies instability in the sense that there exist perturbations that cannot asymptotically approach a stationary perturbation; furthermore, failure of positivity on a solution that can be written as the time derivative of another solution implies exponential growth. We further show that positivity of canonical energy is necessary and sufficient for thermodynamic stability (maximum of area at fixed mass and angular momentum) and is also equivalent to the satisfaction of a local Penrose inequality. For black branes, we show that a sufficient condition for instability is the failure of the Hessian of area with respect to mass and angular momentum to be negative, thus proving a conjecture of Gubser and Mitra. Our methods can be applied quite generally to diffeomorphism covariant theories derived from a Lagrangian.

Qian Wang (Oxford) *Global Existence for the Einstein Equations with Massive Scalar Fields*

**Abstract. **I will present the small data global existence result for the (3+1) Einstein equations with massive scalar fields. The proof is based on the intrinsic construction and control of Lorentz boost in the dynamic spacetime. This is a joint work with Dr. Jinhua Wang.

**ORGANIZERS**

*Lars Andersson (Potsdam)*

*Sergiu Klainerman (Princeton) *

Philippe G. LeFloch *(Paris)*

This conference is part of the **Three-Month Program on MATHEMATICAL GENERAL RELATIVITY** — Institut Henri Poincaré, Paris