This talk will discuss some new nonlinear generalizations of the 3+1 dimensional Yang-Mills equations that are related to wave maps ( i.e. nonlinear sigma models) for Lie group targets. Wave map equations arise naturally in many areas of mathematical physics as a geometrical nonlinear wave equation for a function on Minkowski space into a Riemannian target space. In the case of Lie group target spaces, the wave map equation has a dual formulation as a nonlinear abelian gauge field theory which, interestingly, allows a generalization to include various types of interactions with a non-abelian Yang-Mills gauge field. This yields a class of novel nonlinear geometrical field theories combining features of both wave map and Yang-Mills equations. In particular, investigation of the initial value problem, critical behavior of solutions, exact monopole solutions, coupling to gravity and black-hole/particle-like solutions are some topics of obvious interest.

This talk will describe how many features of Thurston's theory of Dehn surgery on hyperbolic 3-manifolds generalizes to (Riemannian) Einstein metrics in all higher dimensions. In particular, the construction produces large, infinite families of new Einstein metrics of uniformly bounded volume on compact manifolds. A key ingredient in the construction is the use of the AdS toral black hole metrics.Questions will be raised concerning the implications of the construction for the definition of the partition function in Euclidean quantum gravity.

The extended constraint equations arise as a special case of the conformal constraint equations that are satisfied by an initial data hypersurface Z in an asymptotically simple spacetime satisfying the vacuum conformal Einstein equations developed by H. Friedrich. The extended constraint equations consist of a quasi-linear system of partial differential equations for the induced metric, the second fundamental form and two other tensorial quantities defined on Z, and are equivalent to the usual constraint equations that Z satisfies as a spacelike hypersurface in a spacetime satisfying Einstein's vacuum equation. In this talk, I will present a method for finding perturbative, asymptotically flat solutions of the extended constraint equations in a neighbourhood of the flat solution on Euclidean space. This method is fundamentally different from the `classical' method of Lichnerowicz and York that is used to solve the usual constraint equations.

The uniformization theorem in two dimensions states that a closed orientable two dimensional manifold with handle number 0, 1, > 1 respectively admits uniquely the constant curvature geometry with positive, zero, or negative curvatures. This has proved a very powerful tool in two-dimensional physics, such as conformal field theories and string theory. In three dimensions there is not a uniformization theorem, but there is a conjecture due to W.P. Thurston that states that a three-manifold with a given topology has a canonical decomposition into a sum of `simple three-manifolds,' each of which admits one, and only one, of eight homogeneous geometries. The conjecture has not been completely proven, but considerable progress has been made by Thurston and recently there has been some progress in using parabolic-like flows to smooth arbitrary initial non-homogeneous geometries. We propose to broaden the above Ricci-Hamilton flow to include other fields defined on 3D space, in a manner suggested by the low energy limit of a bosonic string propogating in 3D space. In this talk I discuss some of the relevant properties of this 3D theory, and show that in one sector of the theory, the only solutions are six of the eight Thurston geometries, up to coordinate transformations. Finally I will discuss some of the properties of the flow, in particular the flow of locally homogeneous geometries to Thurston geometries.

This talk will describe two constructions of CR invariant differential operators on densities with leading part a power of the sublaplacian. These operators are the CR analogues of the so-called conformally invariant powers of the Laplacian. One construction proceeds via the conformal operators for the Fefferman conformal structure of the CR manifold. The second construction uses a CR tractor calculus.

This is joint work with Rod Gover.

Hypersurfaces of Euclidean space moving by mean curvature will typically smoothen out during the evolution and uniformise their curvature on the way, but they also develop some singularities in finite time. To extend the flow past such singularities by surgery in a controlled way it is necessary to obtain a priori estimates on the shape of the evolving surfaces and to classify the asymptotic behaviour of all possible singularities. The lecture explains the techniques involved in establishing a priori estimates and describes their use in the surgery construction. The lecture is selfcontained but closely related to the plenary lecture at this conference.

I describe the thermodynamic properties of (d+1)-dimensional spacetimes with NUT charges. Such spacetimes are asymptotically locally anti de Sitter (or flat), with non-trivial topology in their spatial sections, and can have fixed point sets of the Euclidean time symmetry that are either (d-1)-dimensional (called "bolts") or of lower dimensionality (pure "NUTs"). I illustrate how to compute the free energy, conserved mass, and entropy for 4, 6, 8 and 10 dimensions for each, using both Noether charge methods and the AdS/CFT-inspired counterterm approach. These results can be generalized to arbitrary dimensionality. In 4k+2 dimensions that there are no regions in parameter space in the pure NUT case for which the entropy and specific heat are both positive, and so all such spacetimes are thermodynamically unstable. For the pure NUT case in 4k dimensions a region of stability exists in parameter space that decreases in size with increasing dimensionality. All bolt cases have some region of parameter space for which thermodynamic stability can be realized.

This talk will discuss the role of Cartan connections in understanding the asymptotic geometry of symmetric spaces. An important special case is that of an asymptotically hyperbolic space with an Einstein metric.

This talk will summarize work with Sumati Surya and Eric Woolgar, Phys. Rev. Lett. 89(2002) 121301, hep-th/0204198. It proves the positivity of mass in asymptotically anti-deSitter spacetime gravitational theories that are dual to conformal field theories on their conformal boundaries. The theorem assumes causality in the boundary theories rather than any local energy conditions in the bulk gravitational theories.

The topological censorship theorem implies the existence of eternal black holes for spacetimes with nontrivial fundamental group. However it does not indicate whether or not other topological structures collapse. Recent work shows that such collapse occurs for certain such structures; spacetimes in 5 or more dimensions with trivial fundamental group but non zero A-hat genera must be singular. This talk will discuss this and other work toward this issue and its implications for classical relativity in higher dimensions.

Particle physicists have long conjectured that there should exist a duality between certain and perhaps all gauge field theories and string theories. This duality holds the practical hope of yielding quantitative information about gauge theory in regimes which are otherwise inaccessible to analytic computations. During the past five years string theory research has found one explicit example of this kind of duality, known as the AdS/CFT correspondence. This lecture will give an overview of the basic ideas and recent results in this subject.

We consider a class of Lorentzian topology changing spacetimes, the so-called Morse spacetimes, and discuss the propagation of a massless scalar field in these spacetimes. We show that for a special class of causally continuous Morse neighbourhoods, the analysis does not lead to the kind of singular propagation associated with the 1+1 dimensional trousers spacetime, first demonstrated by Anderson and De Witt. On the other hand, their arguments can be shown to generalise to the higher dimensional causally discontinuous spacetimes for which the scalar field propagation is singular. We discuss these results in light of a conjecture due to Sorkin, which states that singular propagation of quantum fields occurs only in causally discontinuous spacetimes.

We look at the perturbative dynamics of an interacting scalar field theory living on a noncommutative sphere. As a quantum field theory, this shows a pathological mixing between low and high energy modes. We discuss various scaling limits that connect this theory to that on a noncommutative plane, emphasizing a particular limit that signals the transition between commutative to noncommutative regimes. We also study solitons in the non-perturbative sector of the theory, and the geometry of the moduli space of these solitons.

I will discuss the geometry of certain convex cocompact hyperbolic 3-manifolds from the perspective of ADS/CFT correspondence and present some results toward a geometric understanding of the renormalized volume.

A first step in characterizing the asymptotic dynamics of solutions to Einstein's Equation is finding a global foliation. There has been progress on this for solutions with two-dimensional, spatial, compact isometry group orbits. Recent work shows that in vacuum, the area of the group orbits tends to zero at the singularity if the spacetime is not flat.

Initial data sets with non-trivial topology are considered in n-dimensions. We focus on those which evolve into vacuum or electrovac (n+1)-dimensional spacetimes with non-zero angular momentum or charge respectively. The implications for black hole physics are considered.