Videos and presentation materials from other INI events are also available.
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 spacetime  
GMRW02 
22nd August 2005 14:00 to 15:00 
Global properties of spacetimes 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 nonlinear 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 YangMills 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 3manifolds have nonnegative scalar curvature. However, we will allow these 3manifolds 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 3manifold. 

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 wormholelike but only have one asymptotic infinity. Such black holes are examples of Sorkin's topological geons, generalising into the blackhole 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) Fourdimensional spherically symmetric SU(2) black holes have a straightforward geon variant; 3) There exist geon quotients of MyersPerry 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 KaluzaKlein "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 KleinGordon 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 longrange 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 KleinGordon 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 nonsuperradiant modes of the KleinGordon 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 GaussBonnet 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 SchoenYau 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 powerlaw decay rates for the collapse of a spherically symmetric selfgravitating scalar field. Applications of these ideas to linear and nonlinear 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 
Selfsimilar gravitational collapse and gravitational instantons I will first describe some of my work with Hartnoll and Pope(hepth/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 nonlinear} numerical studies of Bizon, Chmaj and Schmidt (grqc/0506074) on Bianchi IX Black holes in five spacetime dimensions. Certain gravitational instanstons, whose linear stability properties are known, figure as ultrastatic solutions. I will present an exact time dependent solution found in (hepth/0501117). Using KaluzaKlein theory, I will make connections with spherically symmetric collapse of magnetically, and by electricmagneticmagnetic 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, electrovacuum spacetimes of higherdimension. I first provide a master equation for gravitational perturbations of black holes in higher dimensional static spacetimes, which corresponds to ReggeWheelerZerilli equation in 4dimensional case. Then i study the stability against linear gravitational perturbations by examining whether the spatial derivative part of the master equation has a positive selfadjoint extension. In this method, for example, higherdimensional 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 spacetimes? 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 wellposedness 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 spacetimes and describe the obstructions to such constructions in this case. These entail perspectives of further studies of the geometry of black hole spacetimes. 

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 threesphere. I will discuss the process of convergence to the Schwarzschild black hole in this model and will demonstrate the discretely selfsimilar type II critical behavior at the threshold of black hole formation. 

GMR 
13th September 2005 16:00 to 17:00 
Dynamics of Bianchi spacetimes 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 powerlaw 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 KaluzaKlein reduction techniques. Ref.: Heinzle, J.M., Rendall, A.D., Powerlaw Inflation in Spacetimes without Symmetry, www.arxiv.org/grqc/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, closetoFL dynamics and closetode 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 spacetime symmetry properties, scale transformations, and the state space features these induce. 

GMRW06 
16th September 2005 14:30 to 15:30 
Cosmological solutions of the EinsteinVlasov system I will present some of the global results that have been proved in recent years for the EinsteinVlasov 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 wellunderstood 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 EinsteinVlasov 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 surfacesymmetric EinsteinVlasov system with positive cosmological constant In this talk we show that assuming the existence of a symmetry group with twodimensional spacelike orbits, the investigation of the EinsteinVlasov 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 NordstromVlasov system The NordströmVlasov system describes the kinetic evolution of selfgravitating 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 spacetimes, 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 3form, with special attention devoted to the 7 and 11dimensional 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 3manifolds We study constant mean curvature surfaces in asymptotically flat 3manifolds. We prove that, outside a given compact subset in an asymptotically flat 3manifold 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 3manifold 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 
DirichlettoNeumann map for PoincareEinstein metrics An analogue of a DirichlettoNeumann 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 selfdual PoincareEinstein metrics. 

GMRW07 
3rd October 2005 11:30 to 12:30 
Toric PoincareEinstein metrics  
GMRW07 
3rd October 2005 14:30 to 15:30 
The GaussBonnet theorem for PoincareEinstein metrics A useful tool in the study of the AdS/CFT correspondence is the renormalized volume. For four dimensional PoincareEinstein 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 GaussBonnet 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 1loop: 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 Bfield (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 spacetimes is introduced and interpreted. It is then shown that, with the exception of plane waves and spaces of constant curvature (and always for nonflat vacuum metrics),the sectional curvature function uniquely determines the spacetime 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 energymomentum tensor,(ii) wave surfaces and null geodesic congruences,(iii) the concept of a sectional curvaturepreserving 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 3Manifolds 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 3space. 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 noncocompact Fuchsian subgroup of O(2,1) admits proper affine deformations. (Work of FriedGoldman 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 quasilinear 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 blackholes We give a global (classical) description of the family of BTZ multi blackholes, including the spinning case, as quotients of open domains of the antide 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 quasifuchsian 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 AshtekarIsham 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 holonomyflux *algebra is introduced. It should be thought of as an algebra of quantum positionmomentum 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 WheelerDeWitt 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 stressenergy 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 AshtekarLewandowski 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 spacetime 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 spacetime 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} spacetime does not end at singularities. Quantum geometry can serve as a bridge between vast spacetime 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 
Radiationreaction on timelike surfaces in AdS This talk addresses the radiation backreaction problem for cosmological branes. A general framework is provided and results given for antide 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 postDirac 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 WheelerDeWitt equation which is free of ultraviolet divergences. However, ambiguities associated to the intermediate regularization procedure lead to an apparently infinite set of possible theories. The absence of an UV problemthe existence of well behaved regularization of the constraintsis 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 ``pointsplitting'' regularization of the non linear functionals of the connection. This ambiguity is labelled by a halfinteger $m$ 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 npoint function in nonperturbative quantum gravity and low energy limit of the loopspin foam formalism The boundary formulation of quantum field theory can be used to provide a backgroundindependent definition of npoint function. The application of this idea to a tentative loop/spinfoam theory defined via groupfieldtheory methods leads to a twopoint 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 npoint functions in nonperturbative 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 nontime 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 BowenYork initial data does not admit a smooth null infinity (if any). The relevance of these calculations on the light of the socalled Penrose's proposal is discussed. 

GMR 
15th November 2005 16:00 to 17:00 
Asymptotically AdSspacetimes I present a general definition of Hamiltonian generators of asymptotic symmetries in theories of gravity in asymptotically AdSspacetimes, within a covariant phase space formalism. It is explained how this definition is related to existing ones, including the "counterterm subtraction method", the "Weyltensor 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 AdSCFT 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} tr_*^{\frac{1}{2}}). This is sufficient to prove smalldata globalwellposedness 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 Bondienergy 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 spacetimes 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 coordinateindependent 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 spacetimes, 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 
Quasistationary routes to the Kerr black hole In this talk I shall discuss quasistationary 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: grqc/0205127, grqc/0405074, grqc/0506130 

GMRW09 
22nd November 2005 11:30 to 12:30 
Isomonodromic taufunctions on Hurwitz spaces and their applications We discuss JimboMiwa taufunctions corresponding to RiemannHilbert problems with quasipermutation monodromy groups; these taufunctions are sections of certain line bundles on Hurwitz spaces. We show how to compute these taufunctons explicitly in terms of thetafunctions 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 multidimensional dispersionless PDEs: the test for integrability A (d+1)dimensional dispersionless PDE is said to be integrable if it possesses infinitely many ncomponent hydrodynamic reductions parametrized by (d1)n arbitrary functions of one variable. Among the most important examples one should primarily mention the threedimensional dKP and the BoyerFinley equations, as well as the fourdimensional heavenly equation descriptive of selfdual Ricciflat 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 dbar and twistor approaches to multidimensional equations. 

GMRW09 
23rd November 2005 16:00 to 17:00 
Antiselfdual 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 shorttime existence and uniqueness for solutions close to a trivial solution. This trivial, or natural, solution corresponds to a stressfree body in rigid motion. The talk is based on joint work with M.WernigPichler 

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 /M^{2} > 10^{4}). Related Links 

GMR 
1st December 2005 16:00 to 17:00 
E Winstanley 
The abundant richness of EinsteinYangMills We explore some of the properties of static and stationary solutions of the EinsteinYangMills (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 counterexample for each step in the classic uniqueness proof for the KerrNewman metric in EinsteinMaxwell 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 antide Sitter EYM black holes. 

GMR 
2nd December 2005 16:00 to 17:00 
On the convergence of certain expansions at spacelike 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 falloff conditions at spacelike infinity these solutions admit real analytic conformal extensions in which spacelike 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) multipoles and a given sequence of multipoles 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 multipole 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, multiblackholes nohair 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, nonlinear, 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 quasilocal 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 PetrovGalerkin 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 PetrovGalerkin 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 YangMills 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 energymomentum extends smoothly to the entire phase space. These constructions are motivated by the variational definition of quasilocal mass. Related Links 

GMRW03 
12th December 2005 11:30 to 12:30 
Radial foliations of asymptotically flat 3manifolds  
GMRW03 
12th December 2005 14:30 to 15:30 
S Dain 
Spinmass 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 (nonstationary) 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 nonsingular 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 EinsteinScalar field system We present recent work (with J. Isenberg and Y. ChoquetBruhat) concerning the construction of solutions of the EinsteinScalar 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 Qcurvature, 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 underdeterminedelliptic points of view. We discuss several applications of these techniques, including the existence of asymptotically simple vacuum spacetimes, and a construction of multihorizon 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 quasilocal 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 spacetime 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 stressenergy tensor) and the total energy of a spacetime. 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 spacetime. 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 3dimensional manifolds which are asymptotic to a standard hyperbolic slice of antide Sitter spacetime in the following way: we define a sesquilinear form Q which is closely related to the energymomentum 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 antide Sitter. 

GMRW03 
16th December 2005 15:00 to 15:30 
N O'Murchadha 
Why we should not take the LiuYau quasilocal mass seriously The LiuYau mass is a true mass, it is frame independent. However, the LiuYau mass is bigger than the BrownYork 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 LiuYau mass and the BrownYork energy asymptote to the ADM mass (which is really an energy, the `0'th component of a Lorentz covariant 4vector). This means that in any asymptotically flat spacetime, I can find a 2surface with unboundedly large LiuYau 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 spacetimes 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 nonlinear scalar field I'm going to discuss Einstein's equations coupled to a nonlinear scalar field, the potential of which has a positive nondegenerate 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 wellknown 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. 