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Seminars (GMR)

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Event When Speaker Title Presentation Material
GMR 9th August 2005
10:30 to 11:30
Gowdy Cosmologies: Theorems and Simulations Working Together
GMR 11th August 2005
10:30 to 12:00
The Bianchi system
GMR 15th August 2005
15:30 to 17:00
M Choptuik Numerical analysis and numerical relativity
GMR 16th August 2005
10:30 to 11:30
On the interaction of mathematical and numerical general relativity
GMR 17th August 2005
10:30 to 12:00
Hyperbolicity of second order in space systems of evolution equations
GMRW02 22nd August 2005
10:00 to 11:00
M Rees Current status of black hole observations
GMRW02 22nd August 2005
11:30 to 12:30
Harmonic coordinates and the stability of Minkowski space-time
GMRW02 22nd August 2005
14:00 to 15:00
Global properties of space-times with one Killing vector
GMRW02 22nd August 2005
15:30 to 16:30
Cosmic censorship in spherical symmetry
GMRW02 23rd August 2005
10:00 to 11:00
Singularity formation in non-linear wave equations
GMRW02 23rd August 2005
11:30 to 12:30
Radius of injectivity and curvature estimates
GMRW02 23rd August 2005
14:00 to 15:00
Numerical black hole simulations
GMRW02 23rd August 2005
15:30 to 16:30
M Anderson Conformal compactifications
GMRW02 24th August 2005
10:00 to 11:00
Some mathematical aspects of dynamical horizons
GMRW02 24th August 2005
11:30 to 12:30
Progress in nonlinear wave equations
GMRW02 25th August 2005
10:00 to 11:00
Quantum field theories in curved spacetimes, mathematical aspects
GMRW02 25th August 2005
11:30 to 12:30
Controlling global properties of solutions of the constraint equations
GMRW02 25th August 2005
14:00 to 15:00
J Isenberg Numerical relativity and mathematics
GMRW02 25th August 2005
15:30 to 16:30
Dynamical properties of cosmological models
GMRW02 26th August 2005
10:00 to 11:00
Black holes in higher dimensions
GMRW02 26th August 2005
11:30 to 12:30
Positivity of energy
GMRW02 26th August 2005
14:00 to 15:00
Asymptotic expansions for cosmological solutions of the Einstein equations
GMRW02 26th August 2005
15:30 to 16:30
V Moncrief Curvature propagation in general relativity - exploiting the Yang-Mills analogy
GMR 30th August 2005
15:30 to 16:30
Negative point mass singularities in general relativity

In this talk we will discuss a geometric inequality which is in the same spirit as the Positive Mass Theorem and the Penrose Inequality for black holes. Whereas the cases of equality of these first two theorems are respectively Minkowski space (which can be thought of as Schwarzschild with zero mass) and the Schwarzschild spacetime with positive mass, the case of equality for the inequality we will discuss is the Schwarzschild spacetime with negative mass.

Physically speaking, when positive amounts of energy are concentrated as much as possible, black holes results. However, when negative amounts of energy are "concentrated" as much as possible, it is in fact possible to form point singularities in each spacelike slice (which form a timelike curve of singularities in the spacetime).

As usual we will focus on maximal, spacelike slices of spacetimes as a first step. The assumption of nonnegative energy density on these slices implies that these Riemannian 3-manifolds have nonnegative scalar curvature. However, we will allow these 3-manifolds to have singularities which contribute negatively to the total mass. The standard example is the negative Schwarzschild metric on R^3 minus a ball of radius m/2, (1 - m/2r)^4 \delta_{ij}. This metric (which has total mass -m) has zero scalar curvature everywhere but has a singularity at r = m/2. We will propose a definition for the mass of a singularity, and prove a sharp lower bound on the ADM mass in terms of the masses of the singularities in the 3-manifold.

GMR 31st August 2005
15:30 to 16:30
Geons with spin and charge

An eternal black hole with a nondegenerate Killing horizon and suitable discrete isometries has a variant in which the spatial hypersurfaces are not wormhole-like but only have one asymptotic infinity. Such black holes are examples of Sorkin's topological geons, generalising into the black-hole context Wheeler's idea of a massive stable object built entirely out of gravitation. In this talk we construct geon black holes with angular momenta and gauge charges. We show in particular: 1) While Gauss's theorem precludes a conventional electromagnetic charge, there are charged geons with a suitably twisted Maxwell field; 2) Four-dimensional spherically symmetric SU(2) black holes have a straightforward geon variant; 3) There exist geon quotients of Myers-Perry black holes, continuously deformable to zero angular momentum, in all odd spacetime dimensions greater than 3 except 7.

GMRW05 1st September 2005
14:00 to 15:00
Information loss in black holes
GMRW05 1st September 2005
15:30 to 16:30
A new endpoint for Hawking evaporation

We show that certain charged black holes in string theory have a new endpoint for Hawking evaporation. As they evaporate toward extremality, there is a topology changing transition which removes the horizon and singularity and produces a Kaluza-Klein "bubble of nothing". This should lead to a completely nonsingular description of Hawking evaporation.

GMRW05 2nd September 2005
10:00 to 11:00
On scattering theory for field equations in the Kerr metric

We show asymptotic completeness for the massless Dirac field and the non- superradiant modes of the Klein-Gordon field in the Kerr metric.

In the first part we treat massless Dirac fields. We introduce a new Newman- Penrose tetrad in which the expression of the equation contains no artificial long-range perturbations. The main technique used is then a Mourre estimate. The geometry near the horizon requires us to apply a unitary transformation before we find ourselves in a situation where the generator of dilations is a good conjugate operator. The results are reinterpreted to provide a solution to the Goursat problem on the Penrose compactified exterior.

In the second part we treat Klein-Gordon fields. We start with an abstract Hilbert space result. From a Mourre estimate for a positive selfadjoint oparator one can deduce a Mourre estimate for its square root. Using this result and the techniques explained in the first part of the talk, we can establish an asymptotic completeness result for the non-superradiant modes of the Klein-Gordon field. Because of the mass of the field the wave operators have to be Dollard modified at infinity.

GMRW05 2nd September 2005
11:30 to 12:30
On the topology of black holes in higher dimensions

A basic result in the theory of black holes is Hawking's theorem on black hole topology which asserts that for 3+1 AF stationary black hole spacetimes obeying the dominant energy condition, cross sections S of the event must be spherical. The proof is a beautiful variational argument showing that if S is not spherical then it can be deformed to an outer trapped surface in the domain of outer communications, which is forbidden by basic results. The conclusion also holds, by a similar argument, for outermost apparent horizons in black hole spacetimes that are not necessarily stationary. Since Gauss-Bonnet is used, the results are restricted to 3+1. In this talk we present a recent result with Rick Schoen which extends Hawking's results to arbitrarily high dimensions, by showing that outermost apparent horizons must be of positive Yamabe type, i.e., must carry metrics of positive scalar curvature. In the time symmetric case, this follows from the minimal surface methodology of Schoen and Yau in their treatment of manifolds of positive scalar curvature. The present result, however, does not impose any restrictions on the extrinsic curvature of space. While the Jang equation is not used, a neat technique appearing in Schoen-Yau II proves useful.

GMRW05 2nd September 2005
14:00 to 15:00
Higher dimensional black holes

This talk will review black hole solutions of general relativity in more than four dimensions. Outline: motivation, vacuum solutions, supersymmetric solutions, open questions.

GMRW05 2nd September 2005
15:30 to 16:30
Decay of radiation in spacetimes with black holes

An important problem in general relativity is understanding ``radiation tails'' in the exterior regions of spacetimes containing black holes. The heuristic picture of what these tails should look like goes back to work of R. Price in 1972. These tails are related to stability properties of black hole exteriors, and also to the details of their inner structure, in particular, the generic presence of weak null singularities inside black holes. In this talk, I shall describe a rigorous proof of Price's power-law decay rates for the collapse of a spherically symmetric self-gravitating scalar field. Applications of these ideas to linear and non-linear wave equations on various fixed black hole backgrounds will also be discussed. This constitutes joint work with I. Rodnianski.

GMR 5th September 2005
14:00 to 15:00
Self-similar gravitational collapse and gravitational instantons

I will first describe some of my work with Hartnoll and Pope(hep-th/0208031) on the {\sl linear} (instability) of higher dimensional black holes, particularly those constructed from B\"ohm metrics on $S^5$. I shall then go on to relate this to the the fully {\ls non-linear} numerical studies of Bizon, Chmaj and Schmidt (gr-qc/0506074) on Bianchi IX Black holes in five spacetime dimensions. Certain gravitational instanstons, whose linear stability properties are known, figure as ultra-static solutions. I will present an exact time dependent solution found in (hep-th/0501117). Using Kaluza-Klein theory, I will make connections with spherically symmetric collapse of magnetically, and by electric-magnetic-magnetic duality, electrically charged black holes in four spacetime dimensions.

GMR 5th September 2005
15:30 to 16:30
On stability of higher dimensional static black holes

I discuss the stability of black holes in static, electro-vacuum spacetimes of higher-dimension. I first provide a master equation for gravitational perturbations of black holes in higher dimensional static spacetimes, which corresponds to Regge-Wheeler-Zerilli equation in 4-dimensional case. Then i study the stability against linear gravitational perturbations by examining whether the spatial derivative part of the master equation has a positive self-adjoint extension. In this method, for example, higher-dimensional version of Schwarzschild black holes are shown to be stable. Using similar method, I also discuss some other static solutions e.g., generalised black holes, negative mass naked singularities, and the issue of possible boundary conditions at infinities or singularities.

GMR 6th September 2005
15:30 to 16:30
Conformal scattering and the Goursat problem, main ideas and what perspectives for black hole space-times?

The general idea of conformal scattering theory is to replace the use of spectral techniques in the construction of a scattering theory by a geometric approach based on conformal compactification. The existence of a scattering operator is then interpreted as the well-posedness of the Goursat problem on null infinity. The advantage is that stationarity is no longer required for such constructions. When spacetimes contain energy, spacelike infinity is a singuarity of the conformal metric (the metric being not much better than Lipschitz there) and this requires techniques that allow us to deal with the Goursat problem in weak regularity.

We describe the essential ideas of the conformal scattering approach and their origin, and give the first results obtained in the framework of asymptotically simple spacetimes (from a joint work with Lionel Mason). For the resolution of the Goursat problem, we use a technique proposed by Lars Hormander for smooth metrics and extend it to metrics whose regularity is intermediate between ${\cal C}^1$ and Lipschitz (recent submitted work).

Then we turn to black hole space-times and describe the obstructions to such constructions in this case. These entail perspectives of further studies of the geometry of black hole space-times.

GMR 7th September 2005
15:30 to 16:30
Stability of marginally trapped surfaces and local existence of dynamical and trapping horizons

Stability properties of marginally outer trapped surfaces within some spacelike or null slice are discussed. In particular, they are related to the property of being boundaries for the regions containing trapped surfaces.

Moreover, given a spacetime with a smooth foliation and a strictly stably marginally trapped surface S on some initial leaf, we show that there is a smooth trapping horizon through S whose marginally trapped slices lie in the leafs of the given foliation.

GMR 12th September 2005
16:00 to 17:00
P Bizon Vacuum gravitational collapse in 4+1 dimensions

In my talk I will present a recent joint work with Chmaj and Schmidt on radially symmetric vacuum gravitational collapse in $4+1$ dimensions. The key idea which allows us to evade the Birkhoff theorem is to introduce dynamical degrees of freedom corresponding to deformations of the spatial three-sphere. I will discuss the process of convergence to the Schwarzschild black hole in this model and will demonstrate the discretely self-similar type II critical behavior at the threshold of black hole formation.

GMR 13th September 2005
16:00 to 17:00
Dynamics of Bianchi space-times

I will discuss strong cosmic censorship in the class of Bianchi IX spacetimes and also a result stating that for generic initial data (within the Bianchi IX class), the solution approaches an attractor. The work is motivated by the BKL conjecture in which the Bianchi IX (mixmaster) models play a very special role.

Related Links

GMR 15th September 2005
16:00 to 17:00
M Heinzle Asymptotic expansions and nonlinear stability of power-law inflation models

We show that homogeneous and isotropic solutions of the Einstein equations coupled to a nonlinear scalar field with a suitable exponential potential are stable under small nonlinear perturbations without any symmetry assumptions. Our proof makes use of recent results on the nonlinear stability of de Sitter spacetime and Kaluza-Klein reduction techniques. Ref.: Heinzle, J.M., Rendall, A.D., Power-law Inflation in Spacetimes without Symmetry, www.arxiv.org/gr-qc/0506134.

GMRW06 16th September 2005
09:00 to 10:00
Numerical studies of expanding $T2$-symmetric cosmologies

The goal of this talk is to illustrate the value of numerical exploration of Einstein's equations and the synergy with mathematical analysis. The initial discussion will discuss expanding (vacuum) Gowdy spacetimes and the nature of the surprising solutions found by Ringstr\"{o}m. The behavior of the Gowdy solutions will then be compared to that of (vacuum) expanding general $T2$-symmetric spacetimes. In all cases, the behavior is that of gravitational waves of decaying amplitude propagating in an averaged "background" homogeneous cosmology.

GMRW06 16th September 2005
10:00 to 11:00
WC Lim Dynamical systems approach to inhomogeneous cosmology

We discuss various aspects of the dynamics of inhomogeneous cosmologies: the approach to the initial singularity, spikes, close-to-FL dynamics and close-to-de Sitter dynamics. In analyzing these phenomena, we use approximation methods of a heuristic nature and numerical simulations.

GMRW06 16th September 2005
11:30 to 12:30
What have we learned and what can we learn from spatially homogenous cosmology?

During the last few decades there has been considerable progress as regards rigorous mathematical results about the dynamics of spatially homogeneous cosmological models. Rather than focusing on particular results the emphasis in this talk will be on the underlying reasons why any results at all have been found. It will be shown that the key reason is associated with a hierarchical structure that arises from space-time symmetry properties, scale transformations, and the state space features these induce.

GMRW06 16th September 2005
14:30 to 15:30
Cosmological solutions of the Einstein-Vlasov system

I will present some of the global results that have been proved in recent years for the Einstein-Vlasov system in the cosmological case. These results are all obtained under symmetry assumptions and require that spacetime admits an $N$-dimensional symmetry group where $N\geq 2.$ I will mainly focus on the cases $N=2$ and $N=3.$ These situations are rather different since only if $N=2,$ gravitational waves are admitted. The question of global existence is quite well-understood but other important global issues are still open. In the last couple of years studies including a cosmological constant $\Lambda$ or a scalar field have been carried out in cases where $N\geq 3.$ In particular, future geodesic completeness has been proved when $\Lambda>0,$ which is an open problem in the case $\Lambda=0.$

GMRW06 16th September 2005
16:00 to 17:00
Future asymptotics of cosmological spacetimes

I will review some results on the asymptotic behavior of cosmological models in the expanding direction. Among the issues I plan to discuss are nonlinear stability, asymptotic "geometrization", and the influence of matter on asymptotic behavior.

GMR 19th September 2005
16:00 to 17:00
H Lee Accelerated expanding models with Vlasov matter

I will talk about future asymptotic behaviours of the Einstein-Vlasov system having either a positive cosmological constant or a nonlinear scalar field as an accelerated expanding cosmological model. The talk will be focused on homogeneous solutions and some inhomogeneous cases will be mentioned.

GMR 20th September 2005
16:00 to 17:00
The surface-symmetric Einstein-Vlasov system with positive cosmological constant

In this talk we show that assuming the existence of a symmetry group with two-dimensional spacelike orbits, the investigation of the Einstein-Vlasov system with positive cosmological constant produces models with accelerated expansion.

GMR 22nd September 2005
16:00 to 17:00
S Calogero Global classical solutions to the 3D Nordstrom-Vlasov system

The Nordström-Vlasov system describes the kinetic evolution of self-gravitating collisionless matter in the framework of a relativistic scalar theory of gravitation. I prove global existence and uniqueness of classical solutions for the corresponding initial value problem in three dimensions when the initial data for the scalar field are smooth and the initial particle density is smooth with compact support.

GMR 27th September 2005
16:00 to 17:00
Killing spinors, from general relativity to supergravity

Killing spinors first appeared in an article by R. Penrose and M. Walker devoted to the search of quadratic first integrals of the geodesic flow on special space-times, i.e. in the context of General Relativity.

Later Killing spinors were recognized as the counterpart in supergeometry of Killing fields. This gave rise to an extensive study of the overdetermined system that defines them. A classification in the Riemannian case has been obtained, and their role in the limiting case for the lowest eigenvalue of the Dirac operator on compact manifolds.

Since Killing spinors are linked in some way to holonomy questions, it is not surprising to realize that the situation concerning them in Lorentzian and in Riemannian geometries are rather different.

Several generalisations of Killing spinors have been considered recently in relation to Supergravity models involving an exterior differential 3-form, with special attention devoted to the 7- and 11-dimensional cases.

GMR 28th September 2005
16:00 to 17:00
Eigenvalue extremality for the Dirac operator on the sphere

We show that the round sphere has the largest first eigenvalue of the Dirac operator amoung all metrics that are larger than it.

GMR 29th September 2005
16:00 to 17:00
On the uniqueness of the foliation of spheres of constant mean curvature in asymptotically flat 3-manifolds

We study constant mean curvature surfaces in asymptotically flat 3-manifolds. We prove that, outside a given compact subset in an asymptotically flat 3-manifold with positive mass, stable spheres of given constant mean curvature are unique. Therefore we are able to conclude that there is a unique foliation of stable spheres of constant mean curvature in an asymptotically flat 3-manifold with positive mass.

GMR 30th September 2005
16:00 to 17:00
Conserved quantities and rigidity for locally AdS spaces

In the Riemannian context, it is known that continuous groups of conformal isometries of the conformal infinity extend to groups of isometries of any bulk conformally compact Einstein metric with the given conformal infinity. This talk will focus on the analogous issue for Lorentzian Einstein metrics and the implications for the structure of such spaces.

GMRW07 3rd October 2005
10:00 to 11:00
Dirichlet-to-Neumann map for Poincare-Einstein metrics

An analogue of a Dirichlet-to-Neumann map for asymptotically hyperbolic Einstein metrics will be discussed. An explicit identification of the linearization of the map at the sphere will be given for even interior dimensions, together with applications to the structure of the map near the sphere and to the analysis of self-dual Poincare-Einstein metrics.

GMRW07 3rd October 2005
11:30 to 12:30
Toric Poincare-Einstein metrics
GMRW07 3rd October 2005
14:30 to 15:30
The Gauss-Bonnet theorem for Poincare-Einstein metrics

A useful tool in the study of the AdS/CFT correspondence is the renormalized volume. For four dimensional Poincare-Einstein manifolds, a theorem of Mike Anderson relates the renormalized volume and the Euler characteristic of the underlying manifold with boundary. We will discuss the extension of the renormalization procedure to curvature integrals on these manifolds and the proof of the corresponding Gauss-Bonnet theorem in all even dimensions.

GMRW07 3rd October 2005
16:00 to 17:00
K Skenderis Conserved charges and positivity of energy for asymptotically locally AdS spacetimes
GMR 4th October 2005
16:00 to 17:00
A gradient flow for the nonlinear sigma model at 1-loop: the physics of Perelman's entropy

Perelman has recently given a description of Ricci flow as a gradient flow on the space of Riemannian geometries. While this has received attention because of its application to the Poincar\'e and Thurston conjectures, his discovery clearly has other applications as well. In this talk, I will describe its application to a nonlinear sigma model (NLSM) that arises from string theory. It has long been recognized that Ricci flow is an approximation to a purely gravitational NLSM. In this talk, I will explain the necessary part of Perelman's formalism, present a gradient flow for an NLSM with B-field (and dilaton) as well as gravity, and use the Hessian to discuss the stability and rigidity of certain fixed points.

GMR 6th October 2005
16:00 to 17:00
Einstein metrics on product spaces
GMR 7th October 2005
16:00 to 17:00
G Hall Sectional curvature and general relativity

The geometrical idea of sectional curvature in space-times is introduced and interpreted. It is then shown that, with the exception of plane waves and spaces of constant curvature (and always for non-flat vacuum metrics),the sectional curvature function uniquely determines the space-time metric. Thus the suggestion is made that the sectional curvature function is a possible alternative variable for general relativity. Some of the properties of the sectional curvature function are then explored. These include (i) a certain critical point structure of this function and its relationship to the Petrov classification of the Weyl tensor and the Segre classification of the energy-momentum tensor,(ii) wave surfaces and null geodesic congruences,(iii) the concept of a sectional curvature-preserving vector field (iv) a generalisation of the Einstein space condition and a sectional curvature based concept of conformal flatness and (v) an alternative mathematical description of the sectional curvature function using quadric surfaces.

GMR 10th October 2005
16:00 to 17:00
Complete Flat Lorentzian 3-Manifolds

Unlike Euclidean crystallographic groups, properly discontinuous groups of affine transformations need not be amenable. For example, a free group of rank two admits a properly discontinuous affine action on 3-space. Milnor imagined how one might construct such an action: deform a Schottky subgroup of O(2,1) inside the group of Lorentzian isometries of Minkowski space, although as he wrote in 1977, ``it seems difficult to decide whether the resulting group action is properly discontinuous.'' In 1983, Margulis, while trying to prove such groups don't exist, constructed the first examples. In his 1990 doctoral thesis, Drumm constructed explicit geometric examples from fundamental polyhedra, and showed that every non-cocompact Fuchsian subgroup of O(2,1) admits proper affine deformations. (Work of Fried-Goldman and Mess implies that these conditions are necessary.)

This talk will discuss the classification and construction of these manifolds, and the relation with deformations of hyperbolic structures.

GMR 11th October 2005
16:00 to 17:00
Barriers and CMC hypersurfaces

This talk will review the existence and regularity theory for spacelike hypersurfaces with prescribed mean curvature. These results rely on apriori gradient estimates and results on uniformly elliptic quasi-linear elliptic PDE. Conditions under which singular solutions arise as solutions to the associated variational problem will be described.

GMRW08 12th October 2005
10:00 to 11:00
AdS geometry and Mess' work
GMRW08 12th October 2005
11:30 to 12:30
Canonical Wick rotations in 3D gravity

We outline the main features of the theory (developed in arXiv.math.DG/0508485), by turning one's attention in particular to background and motivations (mostly in relation with the problem of constructing a 2+1 QFT pertinent to 3D gravity).

GMRW08 12th October 2005
14:30 to 15:30
A geometric insight into BTZ multi black-holes

We give a global (classical) description of the family of BTZ multi black-holes, including the spinning case, as quotients of open domains of the anti-de Sitter space (AdS) by discrete groups of isometries. Fixing the topology, and fixing the mass and angular momenta, this family is parametrized by the pairs of elements of the Teichm\"{u}ller space of a given Rieman surface with prescribed holonomy on the boundary.

GMRW08 12th October 2005
16:00 to 17:00
Minimal surfaces in singular constant curvature manifolds

We use minimal surface techniques to show that the set of quasi-fuchsian hyperbolic manifolds containing a closed surface with principal curvatures less than 1 is parametrized by a subset of the cotangent of Teichmüller space. This also yields a parametrization of the space of all GHMC AdS manifolds by the whole cotangent of Teichmüller space. The same techniques work for hyperbolic or AdS manifolds with singular curvatures (physically, particles) and provides a description in terms of the Teichmüller space with marked points which should be well adapted to quantization.

GMR 13th October 2005
16:00 to 17:00
On CMC foliations

We prove existence of CMC slicings for globally hyperbolic spatially compact spacetimes, of sectional constant curvature, in any dimension.

GMR 17th October 2005
16:00 to 17:00
Second variation in general relativity

The formula for the second variation of area of a hypersurface is well known in geometry, where it is used to study stability questions involving minimal surfaces. It is not so well known is that it plays a key role in several seemingly unrelated classical computations in general relativity. I'll describe some of these applications.

GMR 18th October 2005
16:00 to 17:00
J Lewandowski Mathematical introduction to loop quantum gravity

The aim of this talk is to introduce the basic definitions of Quantum Geometry and Loop Quantum Gravity. The starting point will be the Ashtekar-Isham holonomy C*-algebra and characterization of its Gel'fand spectrum - the space of quantum connections. The spectrum is endowed with a diffeomorphism invariant measure and plays the role of a (quantum) configuration space. Next, the holonomy-flux *-algebra is introduced. It should be thought of as an algebra of quantum position-momentum variables. There exists a unique diffeomorphism invariant positive functional on the algebra. The corresponding GNS representation is used to define operators of Quantum Geometry -- the kinematic quantum theory of initial data of gravitational field. The quantum Einstein vector constraints generate the group of diffeomomorphisms. The space of solutions is contained in the suitably defined dual vector space. The next step is introduction of the quantum scalar constraint defined by Thiemann. The constraint admits a large family of solutions. The derivations of all the quantum operators are free of infinities. Remaining ambiguities are reduced by various consistency conditions.

GMR 19th October 2005
11:00 to 12:30
Introduction to algebraic quantum field theory in curved spacetime

This talk is a pedagogical introduction to the ideas and methods of quantum field theory in curved spacetime using an approach based on abstract $*$-algebras of observables. In particular, the construction of the algebra used for the real scalar field will be discussed in detail, as will the GNS construction of Hilbert space representations.

GMR 19th October 2005
16:00 to 17:00
Comparison theory in Lorentzian geometry

An overview will be given of the development of comparison theory in differential geometry over the past 40 years. Aspects of cut points, Ricci curvature and geodesic lines, index theory methods and Riccati equation techniques, the Lorentzian splitting theorem and curvature rigidity, and volume comparison will be discussed.

GMR 20th October 2005
16:00 to 17:00
Loop quantum cosmology

Many techniques and results of quantum geometry also apply in the cosmological context, where they lead to crucial differences to a Wheeler-DeWitt quantization. This will serve as an illustration of those techniques and the resulting quantum representations and leads to several applications for very early stages of a universe.

GMR 21st October 2005
11:00 to 12:30
Energy conditions in quantum field theory

In classical general relativity, many key results are proved under the assumption that matter obeys one of the classical energy conditions (dominant, weak, strong, etc). However, it is impossible for any quantum field to obey these conditions! In this talk I will describe the construction of the stress-energy tensor in curved spacetime, give a simple argument to show that QFT necessarily violates the energy conditions, and then discuss Quantum Energy Inequalities, the remnant of the classical energy conditions after quantisation.

GMR 21st October 2005
16:00 to 17:00
F Nicolo A global solution for a characteristic problem for the Einstein vacuum equations with small initial data

In this work, using some techniques started by Christodoulou and Klainerman, we prove a small data global existence result for the Einstein vacuum equations with initial data assigned on the union of two null hypersurfaces. The two main issues are the initial data constraints and the control of appropriate energy norms to start the bootstrap machinery.

GMR 24th October 2005
11:00 to 12:30
Discussions on recent developments in mathematical quantum gravity
GMR 25th October 2005
15:30 to 16:30
Uniqueness of the kinematical representation of loop quantum gravity

One of the corner stones of loop quantum gravity (LQG) is the Ashtekar-Lewandowski representation, a Hilbert space representation of the basic kinematical variables of the theory. It is constructed without using any background geometric structure, and hence is diffeomorphism invariant. On the one hand, much of the subsequent developments in LQG depend on this representation. On the other hand it is well know from quantum field theory that generically there exist many inequivalent, and hence physically different, representations of a given algebra of basic variables. It is therefor an important question wether there exist other representations in the case of LQG. Surprisingly, one can show that the AL representation is the only background independent representation of the algebra of basic variables of LQG. In the talk I will review motivation, precise formulation, and idea of proof, of this uniqeness result.

GMR 26th October 2005
11:00 to 12:30
Discussions on recent developments in mathematical quantum gravity
GMR 27th October 2005
16:00 to 17:00
Quantum geometry and space-time singularities

General relativity provides a subtle and powerful interplay between gravity and geometry, thereby opening numerous possibilities for novel phenomena. However, this interplay also implies that the space-time itself ends when the gravitational field becomes singular. In loop quantum gravity, the interplay is elevated to the quantum regime through quantum geometry. I will present examples which suggest that the {\it physical} space-time does not end at singularities. Quantum geometry can serve as a bridge between vast space-time regions which are classically unrelated. Thus, contrary to one's initial intuition, ramifications of quantum geometry can reach far beyond the Planck regime.

GMR 28th October 2005
11:00 to 12:30
I Moss & D Jennings Radiation-reaction on timelike surfaces in AdS

This talk addresses the radiation back-reaction problem for cosmological branes. A general framework is provided and results given for anti-de Sitter backgrounds. The talk also discusses the question-'can a brane radiate away the cosmological constant?' This discussion session includes a short contribution from David Jennings of the CMS on 'The Unruh effect in AdS'.

GMR 1st November 2005
16:00 to 17:00
Constrained systems: Dirac and post-Dirac quantisation

I will first review Dirac's procedure of quantising (first class) constrained systems, discussing the mathematical structure that a quantisation should provide. I will then specialise to systems whose gauge group is a Lie group and address group averaging as a method of implementing the quantisation. The case of a noncompact gauge group, and the open questions therein, will be emphasised in view of its relevance for quantum gravity.

GMR 2nd November 2005
11:00 to 12:30
The Hamiltonian constraint in loop quantum gravity
GMR 3rd November 2005
16:00 to 17:00
Colombeau algebras in GR

It is widely believed that Laurent Schwartz showed that it was impossible to multiply distributions. However in the 1980's J F Colombeau constructed a commutative and associative differential algebra for which there was a canonical embedding of the space of distributions as a linear subspace and a canonical embedding of the space of smooth functions as a subalgebra. I will start by outlining this construction and explain how it gets round the Schwartz "impossibility" result. Unfortunately the Colombeau construction relies on the linear structure of R^n. I will show how (in joint work with a group of mathematicians in Vienna) it was possible to reformulate the theory to allow the multiplication of distributions on manifolds. I will also explain how the theory has been extended to provide a theory of distributional differential geometry. I will end by giving some applications of these ideas to general relativity, firstly to give a description of weak singularities and secondly to obtain solutions of the wave equation on singular spacetimes.

GMR 4th November 2005
11:00 to 12:30
The master constraint programme for loop quantum gravity
GMRW04 7th November 2005
14:00 to 15:00
Quantum riemannian geometry and its ramifications
GMRW04 7th November 2005
15:00 to 16:00
Gravitational wave astronomy: The large detectors are going into operation!
GMRW04 7th November 2005
16:30 to 17:30
Before the big bang? A new perspective on the Weyl curvature hypothesis
GMR 8th November 2005
16:00 to 17:00
Spin foam quantum gravity

I will review the idea of the spim foam approach to quantum gravity and explain some of the properties of popular models

GMR 9th November 2005
11:00 to 12:30
The representation ambiguity in loop quantum gravity

One of the main achievements of loop quantum gravity is the consistent quantization of the analog of the Wheeler-DeWitt equation which is free of ultra-violet divergences. However, ambiguities associated to the intermediate regularization procedure lead to an apparently infinite set of possible theories. The absence of an UV problem---the existence of well behaved regularization of the constraints---is intimatelly linked with the ambiguities arising in the quantum theory. Among these ambiguities there is the one associated to the $SU(2)$ unitary representation used in the diffeomorphism covariant ``point-splitting'' regularization of the non linear functionals of the connection. This ambiguity is labelled by a half-integer $m$ and, here, it is referred to as the {\em $m$-ambiguity}. I will ellaborate on this issue and show some results that suggest that the degree of ambiguity is reduced when considering the dynamics in the corresponding theory.

GMR 10th November 2005
16:00 to 17:00
Definition of n-point function in nonperturbative quantum gravity and low energy limit of the loop-spin foam formalism

The boundary formulation of quantum field theory can be used to provide a background-independent definition of n-point function. The application of this idea to a tentative loop/spinfoam theory defined via group-field-theory methods leads to a two-point function that, at a preliminary analysis appears to be consistent with the propagator of general relativity, and therefore with the Newton law.

GMR 11th November 2005
11:00 to 12:30
Discussions on n-point functions in non-perturbative quantum gravity
GMR 14th November 2005
16:00 to 17:00
Time asymmetric spacetimes near null and spatial infinity

This talk is concerned with the implementation, in a non-time symmetric setting, of Friedrich's regular value problem near spatial infinity for the Conformal Einstein equations. Computer algebra methods are used to calculate a particular type of asymptotic expansions and to deduce a hierarchy of obstructions to the smoothness of null infinity. These calculations show, for example, that the development of Bowen-York initial data does not admit a smooth null infinity (if any). The relevance of these calculations on the light of the so-called Penrose's proposal is discussed.

GMR 15th November 2005
16:00 to 17:00
Asymptotically AdS-spacetimes

I present a general definition of Hamiltonian generators of asymptotic symmetries in theories of gravity in asymptotically AdS-spacetimes, within a covariant phase space formalism. It is explained how this definition is related to existing ones, including the "counterterm subtraction method", the "Weyl-tensor definition", and the spinor definition. Of particular interest are theories admitting several asymptotically AdS boundary conditions, specified by a certain function W. I show that the energy is bounded from below in such theories if W is bounded from below, and that solutions minimizing the energy have to be static. The relevance and relation of these results to the AdS-CFT correspondence is briefly explained.

GMR 16th November 2005
16:00 to 17:00
Decay and the conformal energy of waves around the Schwarzschild black hole

We provide decay estimates for solutions to the decoupled, inhomogeneous wave equation around a Schwarzschild black hole. In Euclidean space, the conformal charge is a conserved quantity which is used to prove the decay of the local energy. The analogue around a Schwarzschild black hole can grow because of trapping near the photon sphere at r=3M. The trapping terms can be controlled by a Mourre estimate. However, compared to the Euclidean case, this requires more angular differentiability and allows the local energy to decay more slowly. One refinement of this method reduces the loss of angular differentiability, and another recovers the Euclidean rate of decay for the local energy. From the faster decay result, solutions decay like |\phi|=O(r^{-1} |t-|r_*||^{-\frac{1}{2}}). This is sufficient to prove small-data global-well-posedness for certain non linear problems. The initial data can be general in the sense that it is not composed of finitely spherical harmonic modes.

GMR 17th November 2005
16:00 to 17:00
Asymptotic flatness at null infinity in higher dimensional gravity

We give a geometrical definition of the asymptotic flatness at null infinity in higher (even) dimensions within the framework of conformal infinity. We discuss the stability of our definition against perturbations to linear order. Then, we derive a higher (even) dimensional version of the Bondi-energy within the Hamiltonian framework. We discuss why our definitions and constructions do not work in odd spacetime dimensions.

GMR 18th November 2005
16:00 to 17:00
A new development of the causal boundary of spacetimes

The boundary construction for spacetimes suggested by Geroch, Kronheimer and Penrose in the early seventies presents some important obstacles when both, the past and the future boundaries are considered simultaneously. Since then, different authors have tried to solve these questions with new approaches to the causal boundary, without totally satisfactory results. In this talk I suggest a new development of the GKP construction intended to provide a solution to these problems. We will analyze the properties of this new approach, compare it with previous constructions and compute it in some physical examples.

GMRW09 21st November 2005
10:00 to 11:00
LJ Mason The twistor theory of the Ernst Equation
GMRW09 21st November 2005
11:30 to 12:30
Integrable reductions of Einstein's field equations: monodromy transform and the linear integral equation methods

For each of the known today integrable reductions of Einstein's field equations for space-times with two commuting isometries, the monodromy transform (similarly to the well known Inverse Scattering Transform applied successfully for many other completely integrable equations) provides us with a unified and convenient mapping of the complete space of local solutions of the symmetry reduced field equations in terms of a finite set of unconstrained coordinate-independent functions of the spectral parameter (analogous to the scattering data). These set of functions arises as the monodromy data for the fundamental solution of associated linear systems (``spectral problems'') and they can serve as free independent ``coordinates'' in the infinite dimensional space of the local solutions. The direct and inverse problems of such ``coordinate transformation'', (monodromy transform), i.e. the problems of calculation of the monodromy data for given solution of the field equations and of calculation of the solution, corresponding to given monodromy data, possess unique solutions. In principle, the monodromy data functions can be calcul ated also from some boundary, or initial, or characteristic initial data for the fields, and many physical properties of solutions are simply ``encoded'' in the analytical structures of these functions. However, to find the solutions of the mentioned above direct and inverse problems, we have to solve explicitly the systems of ordinary differential and linear singular integral equations respectively, that can occur a difficult problems in many cases.

In the introduction we give a short survey of various integrable symmetry reductions of Einstein's field equations and mention some interrelationships between various developed linear integral equation methods. We describe also in a unified manner the common structure of various integrable reductions of Einstein's field equations -- the (generalized) hyperbolic and elliptic Ernst equations for vacuum and electrovacuum space-times, for Einstein - Maxwell - Weyl fields, for stiff matter fluids as well as their matrix generalizations for some string gravity models with coupled gravity and dilaton, axion and Abelian vector fields. The structure of the direct problem of the monodromy transform and general construction of the linear singular integral equation solving the inverse problem will be considered and some applications of this approach for construction of infinite hierarchies of exact solutions will be presented. In this context we present also another linear integral equation forms of integrable hyperbolic symmetry reductions of Einstein's field equations which provides a solution (viz. linearization) of the characteristic initial value problems for colliding waves and for evolution of inhomogeneous cosmological models.

GMRW09 21st November 2005
14:30 to 15:30
Quasi-stationary routes to the Kerr black hole

In this talk I shall discuss quasi-stationary transitions from rotating equilibrium configurations of normal matter to rotating black holes via the extreme Kerr metric. For the idealized model of a rotating disc of dust, rigorous results derived by means of the 'inverse scattering method' are available. They are supplemented by numerical results for rotating fluid rings with various equations of state.

References: gr-qc/0205127, gr-qc/0405074, gr-qc/0506130

GMRW09 22nd November 2005
11:30 to 12:30
Isomonodromic tau-functions on Hurwitz spaces and their applications

We discuss Jimbo-Miwa tau-functions corresponding to Riemann-Hilbert problems with quasi-permutation monodromy groups; these tau-functions are sections of certain line bundles on Hurwitz spaces. We show how to compute these tau-functons explicitly in terms of theta-functions and discuss their applications in several areas: large N expansion in Hermitian matrix models, Frobenius manifolds, determinants of laplacians over Riemann surfaces and conformal factor of Ernst equation.

GMRW09 22nd November 2005
14:30 to 15:30
Periodic instantons \& monopoles in gauge theory (and gravity)
GMRW09 22nd November 2005
16:00 to 17:00
Hydrodynamic reductions of multi-dimensional dispersionless PDEs: the test for integrability

A (d+1)-dimensional dispersionless PDE is said to be integrable if it possesses infinitely many n-component hydrodynamic reductions parametrized by (d-1)n arbitrary functions of one variable. Among the most important examples one should primarily mention the three-dimensional dKP and the Boyer-Finley equations, as well as the four-dimensional heavenly equation descriptive of self-dual Ricci-flat metrics. It was observed that the integrability in the sense of hydrodynamic reductions is equivalent to the existence of a scalar pseudopotential playing the role of dispersionless Lax pair. Lax pairs of this type constitute a basis of the dispersionless d-bar and twistor approaches to multi-dimensional equations.

GMRW09 23rd November 2005
16:00 to 17:00
Anti-self-dual conformal structures with null Killing vectors
GMR 28th November 2005
16:00 to 17:00
On the motion of a compact elastic body

We study the problem of motion of a relativistic, ideal elastic solid with free surface boundary by casting the equations in material form ("Lagrangian coordinates"). By applying a basic theorem due to Koch, we prove short-time existence and uniqueness for solutions close to a trivial solution. This trivial, or natural, solution corresponds to a stress-free body in rigid motion. The talk is based on joint work with M.Wernig-Pichler

GMR 29th November 2005
14:30 to 15:30
JM Heinzle Static perfect fluid models - a dynamical systems approach

We present a dynamical systems approach to the analysis of relativistic and Newtonian static perfect fluid solutions. By recasting the field equations into a regular autonomous system of differential equations on a compact state space we are able to apply dynamical systems techniques to study the qualitative features of perfect fluid solutions associated with large classes of equations of state. We also show that the methods generalize to other matter models such as Vlasov matter.

GMR 29th November 2005
16:00 to 17:00
The extreme distortion of black holes due to matter

In this talk, I will begin by sketching out the highly accurate numerical methods that allow us to study axially symmetric, stationary spacetimes containing a Black Hole surrounded by a fluid ring. A definition of mass and angular momentum is introduced for each of the two objects. I then discuss the strong influence of matter on the properties of the Black Hole, including the fact that the ratio of the absolute value of the Black Hole's angular momentum to the square of its mass can exceed ten thousand (|J| /M2 > 104).

Related Links

GMR 1st December 2005
16:00 to 17:00
E Winstanley The abundant richness of Einstein-Yang-Mills

We explore some of the properties of static and stationary solutions of the Einstein-Yang-Mills (EYM) equations. The EYM system has been studied for over 15 years, yielding many surprises along the way. We will review the current state of knowledge of soliton and black hole solutions in asymptotically flat space, where there is a counter-example for each step in the classic uniqueness proof for the Kerr-Newman metric in Einstein-Maxwell theory. We will then discuss recent work on EYM with a negative cosmological constant, where the space of solutions is even richer. We will also describe how the isolated horizons formalism describes the asymptotically flat EYM black holes, and outline how an analogous description might be developed for asymptotically anti-de Sitter EYM black holes.

GMR 2nd December 2005
16:00 to 17:00
On the convergence of certain expansions at space-like infinity of asymptotically flat, static vacuum solutions

We study asymptotically flat, static solutions to Einstein's vacuum field equations. It is known that under weak fall-off conditions at space-like infinity these solutions admit real analytic conformal extensions in which space-like infinity is represented by a regular point i. Furthermore, the expansion of the field at i is determined uniquely by the sequence of the (suitably defined) multi-poles and a given sequence of multi-poles determines via the conformal static field equations a unique formal expansion of the field at $i$. We address the question of the convergence of a related type of expansion in a particular conformal gauge. If the conformal fields are extended near i holomorphically into the complex domain they induce certain `null data' on the complex null cone at i. The expansion coefficients of these null data, which are related in a 1:1 fashion to the multi-pole moments, also determine a unique formal expansion of the conformal fields. We give estimates on the null data under which these formal expansions define analytic solutions to the static field equations.

GMR 6th December 2005
16:00 to 16:30
P Chrusciel The classification of static electrovacuum black holes

I will show how to remove the last restriction in the static, electrovacuum, multi-black-holes no-hair theory. (The restriction is that all degenerate components of the event horizon carry charges of the same sign.) The talk is based on joint work with Paul Tod, done during the GMR programme.

GMR 6th December 2005
16:30 to 17:00
J Isenberg Spherically symmetric dynamical horizons
GMR 8th December 2005
16:00 to 17:00
M Mars Second order perturbations of rotating bodies in equilibrium; the exterior vacuum problem

We study the exterior vacuum problem for first and second order stationary and axially symmetric perturbations of static bodies. The boundary conditions and their compatibility for the existence of an asymptotically flat exterior solution are discussed. Some ideas on how these results could be applied for the full, non-linear, problem will be mentioned.

GMR 9th December 2005
11:00 to 12:00
QL Mass definitions with positivity proofs

there are now several definitions of quasi-local mass which pass the minimal test of positivity, perhaps assuming some additional conditions. This talk will concentrate on the proof techniques involved.

GMR 9th December 2005
16:00 to 17:00
Nonlinear Approximation Techniques in General Relativity and Geometric Analysis

In this lecture, we consider nonlinear approximation techniques for treating singularities in geometric analysis and general relativity. We first review approximation theory for Petrov-Galerkin and Galerkin techniques for nonlinear variational problems. We then examine the use of a posteriori error estimation for adaptive construction of discrete (finite element, wavelet, spectral) spaces for deriving nonlinear approximation techniques; these techniques attempt to meet a target approximation quality using discrete spaces of minimal dimension, and are of increasing importance in modeling and computational science.

We then turn to nonlinear elliptic problems in geometric analysis, and focus on the constraints in Einstein flow. We look briefly at weak solution theory on manifolds with boundary, and various Lp and Sobolev estimates for the constraints which are required to develop approximation theory. We then derive a priori and a posteriori error estimates for Petrov-Galerkin approximations to the constraints, and develop some nonlinear approximation algorithms based on adaptive multilevel finite element methods. We illustrate some of the approximation techniques using the Finite Element ToolKit (FEtk).

If time permits, we will describe the use of the nonlinear approximation techniques to enforce constraints during numerical integration of evolution systems such as the Yang-Mills and Einstein equations, by the use of variational techniques. These techniques yield discrete solutions which exactly satisfy the (discrete) constraints at each discrete moment in time, yet a very simple argument shows that the solutions retain the accuracy of standard time integration methods which do not enforce constraints.

GMRW03 12th December 2005
10:00 to 11:00
Hilbert structure on the ADM phase space

The Einstein equations can be formulated as a densely defined flow on a phase space modelled on the Hilbert space H2 x H1 with appropriate decay conditions. I will show that the constraint system determines a smooth Hilbert submanifold and the ADM energy-momentum extends smoothly to the entire phase space. These constructions are motivated by the variational definition of quasi-local mass.

Related Links

GMRW03 12th December 2005
11:30 to 12:30
Radial foliations of asymptotically flat 3-manifolds
GMRW03 12th December 2005
14:30 to 15:30
S Dain Spin-mass inequality for axisymmetric black holes

Abstract: In this talk I will discuss the physical relevance of the inequality J < m^2, where m and J are the total mass and angular momentum, for axially symmetric (non-stationary) black holes. In particular I will show that for any vacuum, maximal, complete, asymptotically flat, axisymmetric initial data close to extreme Kerr data, this inequality is satisfied. The proof consists in showing that extreme Kerr is a local minimum of the mass.

Related Links

GMRW03 12th December 2005
16:00 to 16:30
Nonsingular stationary metrics with a negative cosmological constant

In a joint work with Piotr Chrusciel, we construct infinite dimensional families of non-singular stationary space times, solutions of the vacuum Einstein equations with a negative cosmological constant.

GMRW03 13th December 2005
10:00 to 11:00
M Khuri Global bounds and new existence theorems for the Yamabe problem
GMRW03 13th December 2005
11:30 to 12:30
A Bahri A variational approach to the Yamabe problem
GMRW03 13th December 2005
14:30 to 15:30
F Pacard Singular solutions of the Yamabe equation

The existence of conformal metrics with constant (positive) scalar curvature on subdomains of the sphere is related to the existence of singular solutions for some semilinear elliptic equation.

I will review the sufficient conditions which are known to ensure the existence of singular solutions for this equation.

GMRW03 13th December 2005
16:00 to 16:30
Constructing solutions of the constraint equations with sources: the Einstein-Scalar field system

We present recent work (with J. Isenberg and Y. Choquet-Bruhat) concerning the construction of solutions of the Einstein-Scalar field constraint equations via the conformal method.

Related Links

GMRW03 14th December 2005
10:00 to 11:00
Global convergence of the Yamabe flow
GMRW03 14th December 2005
11:30 to 12:30
Rough initial data

The story of constant mean curvature $H^s$ solutions of the constraint equations with $s>3/2$ has largely been completed, both for asymptotically Euclidean and compact manifolds. It turns out that the standard existence results for smooth solutions extend fully and naturally to the low regularity setting. In this talk I will describe how these results were obtained. One point of interest, even for smooth solutions, is that the rough theory leads to a unified and simpler approach for working with the various cases of the CMC conformal method on compact manifolds.

GMRW03 15th December 2005
10:00 to 11:00
Numerical construction of the solutions of the constraint equations
GMRW03 15th December 2005
11:30 to 12:30
Optimal constraint projection in general relativity
GMRW03 15th December 2005
14:30 to 15:30
Global conformal invariants and their applications

We discuss our recent partial confirmation of a conjecture of Deser and Schwimmer regarding the structure of "global conformal invariants". These are scalar quantites whose integrals over compacr manifolds remain invariant under conformal changes of the underlying metric. We also discuss the implications that the full conjecture would have regarding the notions of Q-curvature, and of the renormalized volume and conformal anomalies of conformally compact Einstein manifolds

GMRW03 15th December 2005
16:00 to 16:30
Some applications of scalar curvature deformation in general relativity

The past several years have seen much activity in constructing solutions of the constraint equations by using geometric gluing techniques. These results require an understanding of the scalar curvature operator (and more generally the constraint operator), from the conformal as well as the underdetermined-elliptic points of view. We discuss several applications of these techniques, including the existence of asymptotically simple vacuum spacetimes, and a construction of multi-horizon initial data with trivial topology.

GMRW03 16th December 2005
10:00 to 11:00
GJ Galloway Rigidity and positivity of mass for asymptotically hyperbolic manifolds

We discuss an approach to the proof of positivity of mass without spin assumption, for asymptotically hyperbolic Riemannian manifolds, based on the general methodology of Schoen and Yau. Our approach makes use of the "BPS brane action" introduced by Witten and Yau in their work on the AdS/CFT correspondence, and takes hints from work of Lohkamp. This is joint work with Lars Andersson and Mingliang Cai.

GMRW03 16th December 2005
11:30 to 12:30
On problems related to Bartnik's definition of quasi-local mass (sponsored by CQG)
GMRW03 16th December 2005
14:30 to 15:00
Positive energy theorem for asymptotically hyperbolic manifolds

General Relativity is a geometrical theory of gravity which asserts that the geometry of space-time is closely related to matter. There exists some consistent definition for total energy (and momentum) of isolated systems which by definition are manifolds whose metric approaches a background metric (Euclidean or hyperbolic). The positive mass theorem can be considered as attempts at understanding the relationship between the local energy density (namely the stress-energy tensor) and the total energy of a space-time. On one hand P. T. Chrusciel and G. Nagy rigorously defined in a recent work notions of mass and momentum for manifolds which are asymptotic to a standard hyperbolic slice of Minkowski space-time. On the other hand P. T. Chrusciel and M. Herzlich proved a positive mass theorem for Riemannian asymptotically hyperbolic manifolds. My work extends this result for orientable 3-dimensional manifolds which are asymptotic to a standard hyperbolic slice of anti-de Sitter space-time in the following way: we define a sesquilinear form Q which is closely related to the energy-momentum and prove, under the relevant energy condition, that Q is a nonnegative Hermitian form which is in fact definite unless our manifold is isometrically embeddable in anti-de Sitter.

GMRW03 16th December 2005
15:00 to 15:30
N O'Murchadha Why we should not take the Liu-Yau quasi-local mass seriously

The Liu-Yau mass is a true mass, it is frame independent. However, the Liu-Yau mass is bigger than the Brown-York energy on any surface for which both can be defined. Further, if I take a sequence of `coordinate spheres' on any spacelike slice, both the Liu-Yau mass and the Brown-York energy asymptote to the ADM mass (which is really an energy, the `0'-th component of a Lorentz covariant 4-vector). This means that in any asymptotically flat spacetime, I can find a 2-surface with unboundedly large Liu-Yau mass. This is even true for Minkowski space.

GMRW10 10th October 2006
11:00 to 12:00
The problem of stability for black hole spacetimes

I review recent results on the behaviour of linear fields on black hole spacetime backgrounds with zero, positive, and negative cosmological constant and discuss the relation of this with the problem of stability for black hole spacetimes.

GMRW10 11th October 2006
16:00 to 17:00
On the Dirichlet problem for the Einstein equations

We show that the space of solutions to the (Riemannian)Einstein equations on a bounded domain is either empty or an infinite dimensional Banach manifold for which the map to the metric on the boundary is Fredholm, of index 0. The same result holds for metrics with compact "inner" boundary with (for instance) asymptotically flat ends. It also holds for the Einstein equations coupled to general matter fields, and in all dimensions. Applied to the static (or stationary) vacuum Einstein equations, the result is relevant to Bartnik's static extension conjecture, and generalizes results of P. Miao.

GMRW10 12th October 2006
16:00 to 17:00
Uniqueness of solutions to wave equations with data on the horizon of a black hole

It is well known that the problem of prescribing initial data on the boundary of a domain of dependence $\DD$ of solutions to a wave equation is not well posed in the complement of $\DD$. It is expected, however, that one still has uniqueness. In collaboration with Alexandru Ionescu we have been recently able to prove some uniqueness results both in the Minkowski space, as well as for the Schwarzschild and Kerr space-times in the domain of outer communication.

GMRW10 16th October 2006
16:00 to 17:00
On stability of cosmological solutions to Einstein's equations coupled to a non-linear scalar field

I'm going to discuss Einstein's equations coupled to a non-linear scalar field, the potential of which has a positive non-degenerate minimum, in the cosmological context. The question I will address is that of stability in the expanding direction.

GMRW10 17th October 2006
16:00 to 17:00
Formation of singularities for the wave map equation in 2+1 dimensions
GMRW10 18th October 2006
16:00 to 17:00
Stability of solutions of the Einstein equations
GMRW10 19th October 2006
16:00 to 17:00
J Isenberg Black hole rigidity in higher dimensions
GMRW10 20th October 2006
16:15 to 17:15
GJ Galloway Rigidity of outermost MOTS and the topology of higher dimensional black holes

In a talk given here last fall I presented joint work with Rick Schoen, in which we obtained a generalization to higher dimensions of a classical result of Hawking concerning the topology of black holes. We proved, for example, that, apart from certain exceptional circumstances, cross sections of the event horizon in stationary black hole spacetimes obeying a standard energy condition are of positive Yamabe type. This implies many well-known restrictions on the topology, and is consistent with recent examples of five dimensional stationary black hole spacetimes with horizon topology $S^2 \times S^1$. In this talk I show how to rule out in this setting the possibility of any such exceptional circumstances (which might have permitted, e.g., toroidal cross sections). This follows from the main result to be discussed, which is a rigidity result for suitably outermost marginally outer trapped surfaces that are not of positive Yamabe type.

University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons