Abstract : We are looking for Morawetz type energy inequalities for wave equations associated to metrics close to Schwarzschild. For such metrics, there are a number of stationnary bicharacteristics, which are an obstacle to local energy decay. We discuss the impact of this picture on energy inequalities. We prove that, for Kerr metrics, there are non positive multipliers with coefficients independent of t.

  Lars ANDERSSON: The trapped region

Abstract : I will discuss recent results on existence and regularity of marginal surfaces, blowup of Jang's equation, and regularity of the trapped region. This is joint work with Jan Metzger.

  Pieter BLUE: Decay for Maxwell's equation on the Schwarzschild manifold

Abstract : We provide a rate of decay in time for solutions to Maxwell's equation in the exterior of a Schwarzschild black hole, r>2M. We use the conformal energy, a well-known modification of the energy with space-time weights, to control all the components of the electromagnetic field. In this method, the main obstacle to decay comes from the presence of a surface (the photonsphere at r=3M) on which there are null geodesics which orbit the black hole. This builds on previous results for the wave equation outside a Schwarzschild black hole, but does not require a spherical harmonic decomposition or for the initial data to vanish on thebifurcation sphere.

Yvonne CHOQUET-BRUHAT: Global solution of the Einstein equations in higher dimension

Abstract : We prove, in collaboration with Piotr Chrusciel and Julien Loizelet, the global existence for small data of solutions of the Einstein equations in even spacetime dimension greater or equal to 6. We use wave coordinates and a conformal mapping between Minkowski spaces; we suppose that the data are Schwarszchild outside of a compact set.

  Mihalis DAFERMOS: A proof of the uniform boundedness of solutions to the wave equation on axisymmetric stationary black hole backgrounds

Abstract : I will present a proof of the uniform boundedness (pointwise and in energy) of sufficiently regular solutions to the wave equation on stationary axisymmetric black hole exterior backgrounds sufficiently close to Schwarzschild. The theorem applies in particular to Kerr and Kerr-Newman exterior backgrounds with a << M, Q << M.  This constitutes the first global result for general (i.e. without restriction of support in frequency and/or physical space) solutions to the wave equation on rotating black hole exteriors. In view of the geometric nature of the assumptions on the background spacetime, the results do not depend on fragile aspects of the Kerr geometry--like the separability of the wave equation--and thus may be useful for the ultimate goal of this analysis, namely the non-linear stability of the Kerr family. (This is joint workwith I. Rodnianski.)

  Erwann DELAY: Some initial data gluing for the vacuum Einstein equations

Abstract : In 2000, J. Corvino gave a construction of non trivial, exactly Schwarzschild, time symmetric initial data for the vacuum Einstein equations. We will discuss about the adaptation to the non time symmetric case and also, recently, to the non zero cosmological constant case. We will then focus our attention to the time symmetric, negative cosmologicalconstant case, where the result could probably  be improved.

Felix FINSTER: The dynamics of scalar waves in the Kerr geometry: superradiance and time independent Sobolev estimates

Abstract : After a brief review of the Kerr geometry, we explain mechanisms which allow to extract energy from a rotating black hole: the Penrose process for classical point particles and superradiance for scalar, electromagnetic or gravitational waves. These mechanisms can be understood from the fact that the physical energy can be negative inside the so-called ergosphere, an annular region outside the event horizon.
Reporting on joint work with Niky Kamran, Joel Smoller and Shing-Tung Yau, a rigorous treatment of superradiance is presented for scalar waves. The first step is an integral representation of the solution of the Cauchy problem. The next step is to derive time independent Sobolev estimates for the wave near spatial infinity. These estimates cannot be obtained with standard energy methods, and therefore we explain the used techniques in detail. Finally, superradiance is established by analyzing the long-time dynamics of wave packet initial data.

Klaus FREDENHAGEN: Locally covariant field theory on Lorentzian spacetimes

Abstract : The concept of local covariance provides a generalization
of the Haag-Kastler concept of algebras of local observables and admits a
generally covariant formulation of quantum field theory.

Helmut FRIEDRICH: The initial boundary value problem for Einstein's field equations and geometric uniqueness

Abstract : There can be formulated initial boundary value problems for Einstein's field equations for which the existence and the PDE-uniqueness of solutions as well as the preservation of the constraints and the gauge conditions can be shown. In general, it seems difficult, however, to show that the solutions are geometrically unique. We discuss the nature of this problem and some of its implications.

Philippe LEFLOCH:  Local foliation and optimal regularity of Einstein spacetimes

Abstract : Motivated by the study of the long-time behavior of solutions to the Einstein equations, we investigate the local optimal regularity of pointed Lorentzian manifolds, that is, time-oriented Lorentzian manifolds in which a point and a future-oriented, unit time-like vector (an observer) have been selected. Our main result covers the class of Einstein vacuum spacetimes. Under curvature and injectivity bounds only, we establish the existence of local coordinates charts that are defined in balls with definite size and in which the metric coefficients have optimal regularity. The proof is based on quantitative estimates for, on one hand, a constant mean curvature (CMC) foliation by spacelike hypersurfaces defined locally near the observer and, on the other hand, local CMC--harmonic coordinates which, by definition, are harmonic in space on each CMC slice. This is a joint work withB.-L. Chen (Guangzhou).

Lionel MASON: Graviton scattering on anti-self-dual backgrounds
and the twistor description of gravity

Abstract : The so called MHV gravity amplitudes describe a self-dual linear perturbation of the metric scattering off an anti-self-dual background.  These amplitudes have been calculated using ideas from string theory and guesswork, and more recently by use of certain recursion relations, although there remain certain gaps in the argument. We give a self-contained calculation ousing twistor methods.  These amplitudes are thought to generate the full perturbative scattering theory for gravity.  If so, their twistor formulation form the non-anti-self-dual part that extends a previous anti-self-dualtwistor action to an action for full gravity.

Alan RENDALL: Loss of regularity in solutions of the Einstein-Euler system

Abstract : It is well known that classical solutions of the Euler equations often lose regularity after a finite time. Physically this corresponds to the formation of shock waves. In this talk I describe work by Fredrik Ståhl and myself in which we study this phenomenon for a self-gravitating fluid in general relativity under the assumption of plane symmetry. The strategy is to first obtain a certain control of the geometry and the energy density of the fluid. This shows that the process of breakdown of classical solutions in the Einstein-Euler system is sufficiently similar to the analogous process in flat space to conclude that breakdown must occur.

Antonio SÁ BARRETO: Bounds for the resolvent in the
De Sitter-Schwarzschild metric.

Abstract : We prove polynomial bounds for the resolvent of an operator associated to the De Sitter-Schwarzschild metric on a strip about the real axis.This is part of joint work with A. Vasy and R. Melrose.

  Jérémie SZEFTEL: Bounded L² curvature conjecture in general relativity

Abstract : One of the main ingredient needed to prove the bounded L² curvature conjecture is the construction and the control of a parametrix for the solution of the wave equation, when one only assumes L² bounds on the curvature. We will present recent progress obtained on this problem. This is a joint work withSergiu Klainerman and Igor Rodnianski.

Juan VALIENTE-KROON: A conjecture on the behaviour of the development of time symmetric, conformally flat initial data at spatial infinity

Abstract :  The behaviour of the development of asymptotically Euclidean, time symmetric, conformally flat initial data for the Einstein vacuum field equations is discussed using the framework of the regular initial value problem near spatial infinity devised by H. Friedrich. This initial value problem, which relies on general properties of conformal structure, is such that the data and equations are regular, and spacelike and null infinity have a finite representation with their structure and location known a priori. For the class of initial data under consideration, explicit calculations have shown that a certain type of logarithmic divergences develop at the critical sets where null infinity "touches" spatial infinity. These singularities are an intrinsic part of the conformal structure of the solution to the Einstein field equations. An analysis to the conditions that have to be imposed on the data to avoid these logarithmic singularities has lead to conjecture that the only element in the class of initial data which is smooth at the critical sets isSchwarzschild data. Current work to provide a proof of this conjecture is described.