


Physics and Geometry / Physique et Géométrie (Org: Maung MinOo and/et Eric Woolgar)
 STEPHEN ANCO, Brock
Nonlinear deformations of the YangMills equations

This talk will discuss some new nonlinear generalizations of the 3+1
dimensional YangMills 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 nonabelian YangMills
gauge field. This yields a class of novel nonlinear geometrical field
theories combining features of both wave map and YangMills equations.
In particular, investigation of the initial value problem, critical
behavior of solutions, exact monopole solutions, coupling to gravity
and blackhole/particlelike solutions are some topics of obvious
interest.
 MICHAEL ANDERSON, SUNY at Stony Brook, Stony Brook, New York 11794, USA
Dehn surgery construction of Einstein metrics

This talk will describe how many features of Thurston's theory of Dehn
surgery on hyperbolic 3manifolds 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.
 ADRIAN BUTSCHER, University of Toronto at Scarborough, Toronto,
Ontario M1C 1A4
The conformal constraint equations

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 quasilinear 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.
 JACK GEGENBERG, University of New Brunswick, Fredericton, New Brunswick E3B
5A3
Using 3D stringy gavity to understand the Thurston conjecture

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 twodimensional 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 threemanifold with a given topology has a
canonical decomposition into a sum of `simple threemanifolds,' 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 paraboliclike flows to smooth arbitrary initial
nonhomogeneous geometries. We propose to broaden the above
RicciHamilton 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.
 C. ROBIN GRAHAM, Department of Mathematics, University of Washington,
Seattle, Washington 98195, USA
CR invariant powers of the sublaplacian

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 socalled 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.
 GERHARD HUISKEN, AEI Golm
Analytical aspects of mean curvature flow with singularties

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.
 ROBERT MANN, University of Waterloo, Waterloo, Ontario N2L 3G1
Nutty thermodynamics and the AdS/CFT correspondence

I describe the thermodynamic properties of (d+1)dimensional
spacetimes with NUT charges. Such spacetimes are asymptotically locally
anti de Sitter (or flat), with nontrivial topology in their spatial
sections, and can have fixed point sets of the Euclidean time symmetry
that are either (d1)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/CFTinspired
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.
 M. MINOO, McMaster University, Hamilton, Ontario
Asymptotically symmetric spaces

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.
 DON PAGE, University of Alberta, Edmonton, Alaberta T6G 2J1
Positive mass without local energy conditions

This talk will summarize work with Sumati Surya and Eric Woolgar,
Phys. Rev. Lett. 89(2002) 121301, hepth/0204198. It proves the
positivity of mass in asymptotically antideSitter 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.
 KRISTIN SCHLEICH, UBC
Topological censorship and beyond: black holes and singularities
from topological structures in d > 4

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 Ahat genera must be singular. This
talk will discuss this and other work toward this issue and its
implications for classical relativity in higher dimensions.
 GORDON SEMENOFF, University of British Columbia, Vancouver, British Columbia
Gauge fields, strings and gravity

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.
 SUMATI SURYA, University of Alberta, 412, Avadh Bhatia Labs, Edmonton,
Alberta T6G 2J1
Singular propagation of quantum fields in topology changing
spacetimes

We consider a class of Lorentzian topology changing spacetimes, the
socalled 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.
 SACHIN VAIDYA, University of California, Davis
Perturbative and nonperturbative dynamics in noncommutative
spaces

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 nonperturbative sector of the theory, and the geometry
of the moduli space of these solitons.
 XIANDONG WANG, MIT Mathematics 2244, Cambridge, Massachusetts 02139, USA
Holography and geometry of some convex cocompact hyperbolic
3manifolds

I will discuss the geometry of certain convex cocompact hyperbolic
3manifolds from the perspective of ADS/CFT correspondence and
present some results toward a geometric understanding of the
renormalized volume.
 MARCIA WEAVER, Department of Physics, University of Alberta, Edmonton,
Alberta
Global foliation of spatially inhomogeneous solutions to
Einstein's Equation

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 twodimensional, 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.
 DON WITT, UBC
Charged and spinning initial data sets

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

