Generalised associahedra
Tom Brady
Dublin City University
Generalised associahedra for crystallographic finite real reflection groups
were introduced by Fomin and Zelevinsky in 2003. The classical associahedron
corresponds to the case where the reflection group is a symmetric group. In
recent work with Colum Watt we give a different description of generalised
associahedra which is more tractible and extends to the non-crystallographic
cases.
Diophantine approximation on manifolds
Detta Dickinson
National University of Ireland,
Maynooth
A survey of the two main types of Diophantine approximation on Euclidean
manifolds will be given. For the case of dual approximation, almost all
results depend on the geometric properties of the manifold. On the other
hand, for simultaneous approximation, the results are split into two
different categories. First, if the error function is large enough the
approximation properties of the manifold seem to depend on its geometric
properties. However, for a small error function the arithmetical properties
begin to play a larger roll. This survey aims to give details of what
results are known and not known in the various cases.
The K(\pi,1) conjecture for non-spherical Artin groups
Graham Ellis
National University of Ireland,
Galway
The conjecture asserts the contractibility of a certain cellular
space associated to any generalised braid group. The space can be used to
compute the cohomology ring of the group in those cases where the conjecture
is known to hold. This talk will explain how Gromov's link condition for
non-positive curvature can be used to prove the conjecture in some new
cases.
The complex geometry of reflection and refraction
Brendan Guilfoyle
Institute of Technology,
Tralee
The space of oriented lines in Euclidean 3-space has a rich geometric
structure. In this talk we focus on the complex structure and how it
can be utilised to describe the simplest of optical phenomena,
reflection and refraction in a surface. We describe the scattering of
an arbitrary wave off a reflective surface in terms of this complex
geometry.
As an application, we show how a recently proposed geometric optics
approximation to the Casimir energy can be computed, and how, in
certain cases, convergence of the energy can be proved using analytic
techniques.
Curvature and D-differentiation
Donal Hurley
University College Cork
I will discuss the questions of tensorial curvature and the existence of
scalar curvature for a class of D-differentiation operators.
Variations on a theme of Klein
Anthony Small
National University of Ireland,
Maynooth
In the nineteenth century, a correspondence between lines in 3-dimensional
complex projective space and points on a certain quadric 4-fold was found.
We review this briefly and derive auxiliary correspondences which facilitate
the study of related variational phenomena. These include minimal surfaces
in Euclidean space, constant mean curvature 1 surfaces in 3-dimensional
hyperbolic space, and monopoles.
Geometric structures on fake lens spaces
Charles Thomas
University of Cambridge
A fake projective or lens space is defined by a free action by a finite
cyclic group on S^(2n-1). Up to homeomorphism such spaces are classified by
algebraic invariants, and we ask for conditions under which a contact form
can be defined. One necessary condition is smoothability, which already
gives rise to interesting arithmetic problems. A complete solution can be
given in dimension 5, and there are partial results in dimension 7. In both
cases these are nicely illustrated by cyclic group actions on Brieskorn
varieties. If time allows I shall say something about metric contact
structures, where the compatible metric has positive Ricci curvature. This
subject has the flavour of an odd-dimensional analogue to Calabi-Yau theory.
How degenerate can a CR structure be?
Dmitri Zaitsev
Trinity College Dublin
CR structure contains the full information remaining from the complex
structure after restricting it to a real submanifold. It turns out that its
properties heavily depend on how curved the real submanifold is with respect
to the ambient complex structure. The latter can be expressed in terms of
the degeneracy of the CR structure. I will discuss some recent research
directions and open problems.
