This workshop **RECENT ADVANCES IN MATHEMATICAL GENERAL RELATIVITY **is an event of the Trimester Program taking place at the Institut Henri Poincaré in order to celebrate the 100th Anniversary of General Relativity. It will present recent progress in several areas of mathematical general relativity.

*Emile Borel Centre of Henri Poincaré Institute, Paris*

*September 23 to 25, 2015*

**SCHEDULE**

**Wednesday Sept. 23** (chairman P. LeFloch)

9am—9:30am : Registration

9:30am-—10:30am : **Vincent Moncrief **

10:30am—11am (coffee break)

11am—12am : **Harvey Reall**

12am—2pm : (lunch break)

2pm—3pm : **Joao Costa**

3pm—3:30pm : (coffee break)

3:30pm—4:30pm :** Mu-Tao Wang**

**Thursday Sept. 24** (chairman L. Andersson)

9:30am-—10:30am : **Piotr Chrusciel**

10:30am—11am (coffee break)

11am—12am : **Lionel Mason**

12am—2pm : (lunch break)

2pm—3pm : **Hans Ringstrom**

3pm—3:30pm : (coffee break)

3:30pm—4:30pm : **Jean-Philippe Nicolas**

6pm : *Buffet Dinner Reception*

**Friday Sept. 25** (chairman S. Klainerman)

9:30am-—10:30am : **Stefan Hollands **

10:30am—11am (coffee break)

11am—12am : **Alexandru Ionescu**

12am—2pm : (lunch break)

2pm—3pm : **Semyon Dyatlov**

3pm—3:30pm : (coffee break)

3:30pm—4:30pm : **Spyros Alexakis**

**LIST OF SPEAKERS **

**Spyros Alexakis** (Toronto) The Penrose inequality on perturbations of the Schwarzschild exterior

**Abstract. **We prove the Penrose inequality on single-black hole space-times which are close to the Schwarzschild exterior in a specific slab containing a marginally outer trapped surface that extends to past null infinity. We will discuss this work in the context of the final state of dynamical black hole evolution.

**Piotr Chrusciel** (Vienna) The vacuum relativistic constraint equations with a positive cosmological constant

**Abstract.** I will review various known results about the general relativistic constraint equations in the presence of a positive cosmological constant, including recent progress by B. Premoselli, and some joint work with R. Gicquaud.

**Joao Costa** (Lisbon) Global uniqueness for the Einstein-Maxwell-scalar field system with a cosmological constant

**Abstract.** Motivated by the Strong Cosmic Censorship Conjecture (SCCC) we consider the problem of global uniqueness for the Einstein-Maxwell-scalar field system with a cosmological constant, for spherically symmetry characteristic initial data. First we consider the situation where the outgoing data is stationary (i.e., prescribed by a sub-extremal Reissner Nordström black hole event horizon) and the remaining data is otherwise free. For large classes of sufficiently fast decaying free data the Hawking mass remains bounded and it is then possible to construct regular extensions of the maximal (globally hyperbolic) development. This suggests a potential failure of the SCCC in the case of a positive cosmological constant. To try to clarify the previous situation we then consider the case where the outgoing data, instead of stationary, satisfies Price’s law. Once again we identify open sets of data for which the mass remains bounded. We finish by discussing the potential consequences of this fact for the SCCC. This is joint work with: P. Girão, J. Natário and J. D. Silva.

**Semyon Dyatlov** (Cambridge, USA) Quasi-normal modes: the spectrum of Kerr-de Sitter black holes

**Abstract.** Consider linear waves on the Kerr-de Sitter spacetime, which models a rotating black hole with a positive cosmological constant. In contrast with the Kerr solution, solutions to the wave equation decay exponentially up to a finite dimensional subspace. This makes it possible to expand waves asymptotically in terms of quasi-normal modes, which are the complex characteristic frequencies associated to the spacetime. I present several recent results, giving a rigorous definition of quasi-normal modes and describing their asymptotic behavior in the high frequency limit. The high frequency picture relies on the normally hyperbolic structure of the set of trapped light rays.

**Stefan Hollands** (Cardiff) (In-)stabilities of black holes and black hole thermodynamics

**Abstract. **Stationary black hole solutions are well-known to obey mathematical identities that are strikingly similar to the ordinary laws of phenomenological thermodynamics. These similarities suggest, among other things, that one might be able to make predictions about the stability of black holes based on stability criteria analogous to phenomenological thermodynamics (“negative heat capacity”). In this talk, I explain how identities related to the “canonical energy” of gravitational perturbations can be used to make such arguments rigorous, thereby — in essence — establishing a further law of black hole thermodynamics. I also explain how these ideas can be used to prove previous conjectures about black hole instabilities in higher dimensions, such as the Gutser-Mitra and Durkee-Reall conjectures for higher dimensional stationary black holes, as well as a conjecture about so-called “super radiant” instabilities of AdS-type black holes.

**Alexandru Ionescu** (Princeton) On the stability of the wave-map equation in Kerr spaces

**Abstract.** I will discuss some recent work on the stability of the wave-map equation in Kerr spaces with small angular momentum. This problem should be viewed in the context of the larger stability/rigidity problem for the family of Kerr spaces. The talk is based on joint work with S. Klainerman and S. Alexakis.

**Lionel Mason** (Oxford) Perturbative approaches to gravitational scattering from null infinity

**Abstract.** Recently remarkably simple and compact formulae for gravitational scattering have been discovered. These formulae involve an auxiliary complex variable and have a very simple relationship with corresponding formulae for Yang-Mills embodying a gravity = Yang-Mills squared principle. I will explain how these arise from a formulation of gravity in terms of strings in ambitwistor space, the space of complexified null geodesics. This theory can be expressed in terms of null infinity and yields a remarkable relationship between the scattering of light rays from past to future null infinity and the corresponding perturbative scattering of the gravitational field via quantization. BMS symmetries then arise in the limit when gravitons have vanishing momentum. If there is time, I will explain how these ideas might be extended to give formulae for scattering on nontrivial backgrounds.

**Vincent Moncrief** (Yale) Euclidean-signature semi-classical methods for quantum cosmology

**Abstract.** We show how certain microlocal analysis methods, already well-developed for the study of conventional Schrödinger eigenvalue problems, can be extended to apply to the (mini-superspace) Wheeler-DeWitt equation for the quantized Bianchi IX (or ‘Mixmaster’) cosmological model. We use the methods to construct smooth, globally defined asymptotic expansions, for both ‘ground’ and ‘excited state’ wave functions, on the Mixmaster mini-superspace. A crucial step in this extension involves handling the fact that, for spatially closed universe models, all of the relevant eigenvalues to the Wheeler-DeWitt operator must vanish identically-̶̶̶̶ a sharp contrast to the situation normally arising for Schrödinger operators. We then briefly review an expansive, ongoing program to further extend the scope of such microlocal methods to encompass a class of interacting, bosonic quantum field theories and conclude with a discussion of the feasibility of applying this ‘Euclidean-signature semi-classical’ quantization program to the Einstein equations themselves ̶ in the general, non-symmetric case ̶ by exploiting certain established geometric results such as the positive action theorem.

**Jean-Philippe Nicolas** (Brest) Conformal scatttering

**Abstract. **The idea of conformal scattering is due to Roger Penrose in the 1960’s. The principle is use conformal methods to construct a scattering theory and not to merely reinterpret an existing scattering theory in conformal terms as it is sometimes believed. The first actual conformal scattering construction was due to Friedlander in 1980, making a link between the Lax-Phillips scattering theory and the notion of radiation fields. The method was then used in the 1980-s-1990’s by Baez and collaborators in static situations. It is only recently that conformal scattering has been applied to develop scattering theories in situations not easily accessible to the spectral analytic approach. This talk shall present the history of conformal scattering, its principles and recent results.

**Harvey Reall** (Cambridge, UK) Causality, hyperbolicity and shock formation in Lovelock theories of gravity

**Abstract. **In four spacetime dimensions, the LHS of the Einstein equation is uniquely determined by requiring that it is a symmetric, conserved tensor depending only on the metric and its first two derivatives. In higher dimensions, Lovelock classified tensors with these properties and showed that additional terms can arise. Hence the Einstein equation is not rigid in higher dimensions. The deformed theories are referred to as Lovelock theories of gravity. They have the interesting property that causality is not determined by the light cone: gravity can travel faster or slower than light. I will discuss characteristic surfaces and hyperbolicity of Lovelock theories. I will argue that Lovelock theories suffer from shock formation, unlike GR.

**Hans Ringstrom** (Stockholm) On the cosmic no-hair conjecture in the Einstein-Vlasov setting

**Abstract. **The standard starting point in cosmology is the assumption of spatial homogeneity and isotropy. However, it is preferable to prove that solutions generally isotropise and that the spatial variation (as seen by observers) becomes negligible. This is expected to happen in the presence of a positive cosmological constant; in fact, solutions are in that case expected to appear like the de Sitter spacetime to observers at late times. The latter expectation goes under the name of the cosmic no-hair conjecture. In the talk, we present a result (based on joint work with Håkan Andréasson) concerning a class of spacetimes (T^3-Gowdy, in the Einstein-Vlasov setting) whose members are neither spatially homogeneous nor isotropic, but which all satisfy the cosmic no-hair conjecture. Moreover, we demonstrate that the members of this class are future stable under general perturbations (without symmetries), and that the perturbed solutions satisfy the cosmic no-hair conjecture.

**Mu-Tao Wang** (New York) Quasi-local angular momentums and their limits at infinity

**Abstract.**The notion of angular momentum is of most fundamental importance in any branch of physics. However, there have been great difficulties in finding physically acceptable definition of this concept in general relativity except for a few cases. In this talk, I introduced new definitions of angular momentum at both the quasi-local and total level. The construction was based on previous work on quasilocal mass and optimal isometric embedding system, which anchors the reference system. The new definition of total angular momentum satisfies desirable properties such as invariance and conservation. At last, a new Bondi type mass loss formula and a new definition of total angular momentum were introduced in the asymptotically hyperbolic setting. The talk is based on joint work with Po-Ning Chen and Shing-Tung Yau.

**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