# School of Mathematics

# Irish Geometry Conference 2006

#### May 19-20, 2006 at Trinity College Dublin

## Abstracts of the talks

**Hypersurfaces with constant principal curvatures**

Jürgen Berndt (UC Cork)
The classification of hypersurfaces with constant
principal curvatures is a classical problem in submanifold
geometry. In the talk I will present a survey about this problem
for Riemannian symmetric spaces of noncompact type.

**Metric space inversions**

Stephen Buckley(NUI Maynooth)
If we invert Euclidean space R^n through the unit sphere centred at some
point p, the associated pullback metric on R^n \ {p} is
i_p(x,y) = |x-y| / |x-p| |y-p|.
The analogue of i_p in a general metric space may not be a metric, but there
is always a comparable metric d_p, which we discuss. This is joint work
with David Herron and Xiangdong Xie.

**G2 spectral curves and Langlands duality**

Nigel Hitchin
(Oxford University)
G2 is an exceptional Lie group, but there is an unexceptional way of
looking at it as the stabilizer of a generic three-form in seven
dimensions. We use this to describe the abelian varieties which occur
as Lagrangian fibres in the integrable system of Higgs bundles with
structure group G2, and examine the duality relation amongst the
fibres. In doing so, we verify one of the predictions of the current
work of Kapustin and Witten on electro-magnetic duality and the
Langlands correspondence.

**Homogeneous Levi degenerate CR-manifolds**

Wilhelm Kaup
(University of Tübingen)
This is a report on joint work with G.Fels. We
give large classes of homogeneous CR-manifolds that are
Levi degenerate, but nondegenerate in a higher
order sense. In particular, all 2-nondegenerate
locally homogeneous CR-manifolds in the lowest possible
dimension (that is real dimension 5) are obtained.

**Degenerating vector bundles on elliptic curves and applications **

Bernd Kreussler
(MIC Limerick)
Fourier-Mukai transforms are useful tools to study vector bundles on
projective varieties. We use them to understand degenerations of vector
bundles on elliptic curves, whereby we allow the curves to pick up singularities.
As an application, the construction of solutions of the classical Yang-Baxter
equation and the study of degenerations of them will be discussed.
This talk reports on joint work with Igor Burban.

**Composition operators over bounded symmetric domains **

Michael Mackey (UC Dublin)
We look at topological structure of the composition operators
of bounded holomorphic functions on symmetric domains.

**Exotic Spheres and Curvature**

David Wraith (NUI Maynooth)
For the most part, this will be a survey about the
curvature of exotic spheres. However, we will also sketch some new
results.