Alan Carey: Index theorems with non-commutative symbol
In joint work with Fyodor Sukochev and John Phillips we discovered how to
prove index theorems for generalised Toeplitz operators using the methods of
noncommutative geometry. In one sense this is not surprising since the Toeplitz
operator theory as developed by Helton and Howe formed one of the motivating
examples for Connes. The new ingredient that we discovered is that there is a
general abstract index theorem for operators on certain (Breuer)-Fredholm
modules that has, as special cases, the Toeplitz operator examples.
Michael Cowling: Automatic continuity
A bijection of R^2 which maps lines to lines is automatically continuous.
A bijection of S^2 which sends circles to circles is also automatically
continuous. The proofs that I know of one of these facts is algebraic, and of
the other is topological. These and some related results will be presented, and
the audience's collective brain will be picked for further results of this
nature.
Tony Dooley: Orbital convolutions for Lie groups and symmetric spaces
A description of the convolution structure of adjoint and coadjoint orbits
of Lie groups and its relationship with the group convolution of their images
under the exponential map is in some senses a "predual" of the Kirrilov
character formula. The theory works particularly well for compact Lie groups,
and has recently been extended to semi-direct products. In fact, a version also
holds for compact symmetric spaces, using the e-functions of Rouviere. Much of
this is joint work with Norm Wildberger.
Michael Eastwood: Symmetries of the Laplacian
Which linear differential operators preserve harmonic functions?
Klaus Ecker: Mean value formulas for evolution equations
It is well-known that harmonic functions (solutions of Laplace's
equation) can be characterised by a mean value property which states that
the value of the function at any point equals the average of the function
over any ball (or sphere) centred at that point.
It is less widely known that an analogous formula holds for
solutions of the heat equation. Here, spheres (which can be thought of
as level sets of the fundamental solution for Laplace's equation) are
replaced by the level sets in space and time of the standard backward heat
kernel. This formula was discovered by Fulks and by Watson and later
extended by Evans and Gariepy. In some sense such a mean value formula can
be thought of as a local version of the standard heat kernel representation
formula for solutions of the heat equation.
In this talk, we present versions of local mean value formulas for a
number of nonlinear evolution equations such as the mean curvature flow
and harmonic map heat flow and discuss applications.
Min-Chun Hong: Heat flow for the Yang-Mills-Higgs field
For a parameter lambda > 0, we study a type of vortex equations, which
generalize the well-known Hermitian-Einstein equations, for a connection and
a section phi of a holomorphic vector bundle E over a Kaehler manifold. We
establish global existence of smooth solutions to heat flow for a self-dual
Yang-Mills-Higgs field on E. Assuming the lambda-stability of (E,phi), we prove
the existence of an Hermitian Yang-Mills-Higgs metric on the holomorphic
bundle E by studying the limiting behaviour of the heat flow.
Gen Komatsu: Reproducing kernels for strictly pseudoconvex domains
in C^n
Lecture 1: An introduction to the Bergman kernel
This lecture introduces the Bergman kernel of a strictly pseudoconvex
domain in C^n and describes how it may be applied in studying the geometry of
the domain and its boundary.
Lecture 2: Sobolev-Bergman kernels of strictly pseudoconvex domains
This is an attempt to generalize the Bergman kernel of strictly
pseudoconvex domains from the viewpoint of the invariant theory. We consider
reproducing kernels of the spaces of holomorphic functions contained in L^2
Sobolev spaces, where the kernels satisfy transformation laws under
biholomorphic mappings.
Michael Murray: The information metric
Recently, ideas from physics about so-called holography and the AdS-CFT
correspondence have rekindled an interest in the information metric (see for
example hep-th/0108122). I will explain the information metric, what examples
we can know about, its relationship with the work of Graham and Lee, and what
is in the preprint above.
Hyam Rubinstein: Ideal triangulations of 3-manifolds (joint work with
Ensil Kang, Korean National University)
For atoroidal 3-manifolds with tori boundary (like complements of knots and
links), one can define ideal triangulations, where the vertices are `at
infinity' i.e are placed on the boundary tori. Thurston, in his theory of
hyperbolic metrics on 3-manifolds, made extensive use of these. In particular,
he introduced a system of gluing equations, where a solution represents a
collection of shapes of the tetrahedra leading to a compatible hyperbolic
metric. Casson around 1995 attempted a new proof of Thurston's uniformisation
theorem in this case, by proving that there is always such a solution, by
modifying the triangulation, if necessary. Our main result is a partial answer
to Casson's program - we show that if the triangulation is `taut', then there
is a solution of the gluing equations to give an angle structure, which is a
weak version of a hyperbolic metric. Lackenby has shown by a direct method that
all such 3-manifolds have taut triangulations. We also have preliminary results
on the global theory of spun and ordinary normal surfaces in ideal
triangulations. These are like complete minimal surfaces and closed minimal
surfaces, respectively.
Paul Tod: Causality and Legendrian linking
In (n+1)-dimensional Minkowski space events are causally related if and only if
their corresponding "skies", which are Legendrian (n-1)-spheres in the
(2n-1)-dimensional contact manifold of unscaled null geodesics, either meet or
are (topologically) linked. For n=2, the skies are circles in S^1xS^2 and one
can draw pictures. Now go to curved space: is it still true? Definitely not for
n=3 but apparently for n=2. How to save the day? Conjecture: ask for Legendrian
linking i.e. can't be separated while staying Legendrian. There are only
preliminary results so far (employing Legendrian knot polynomials).
Neil Trudinger: Analytical perspectives in affine differential geometry
Joseph Wolf: Cycle spaces
Lecture 1: Overview: geometry and representation theory
Many representations of semisimple Lie groups have a straightforward geometric
construction, where the analysis and the geometry work smoothly together.
I'll describe this. Cycle spaces and a certain double fibration (complex
Penrose) transform give an alternative picture that, hopefully, will give a
geometric construction of the remaining representations. I'll attempt to
describe this.
Lecture 2: Structure of the cycle space: current state
The structure of the cycle space is the key to the "alternative picture" of the
first lecture. There has been a lot of progress lately, but much remains to be
done. I'll describe the current state of the matter.