# The 2nd DIFFERENTIAL GEOMETRY DAY

## at LUND

## Wednesday the 17th of May 2004

### ABSTRACTS

**Steen Markvorsen** (Lyngby)
**Manifold Resistance and Transience**
Which Riemannian manifolds have finite resistance to infinity?
Does a random walk on a given manifold always return home - sooner or later?
These questions are related. If the manifold has finite resistance, then the walker is
not sure to return home and vice versa - the manifold (and the walk) is called transient.
Suppose P is a submanifold in an ambient Riemannian manifold N. The quest is then
to find the most general intrinsic geometric condition on N and the most general
extrinsic geometric condition on P which together will guarantee that P is transient.
This talk will explain some recent results from joint work with V. Palmer in this direction.
**Simon Chiossi** (Odense)
**Some interactions between exceptional geometry and nilmanifolds**
I shall describe how $SU(3)$ structures in dimension 6 are related to
7-dimensional $G_2$ manifolds in a couple of simple set-ups, and
show how nilmanifolds naturally enter in this picture.
**Andrew Swann** (Odense)
**Cone configurations and neutral metrics**
One of the geometries determined by a triple of symplectic structures is know
as hypersymplectic geometry. Such structures carry a Ricci-flat metric of
neutral signature. I will discuss how examples in dimension 4n with a T^n
symmetry may be constructed via moment map considerations and indicate how
features of the geometry and topology of the quotients may read off from
configurations of solid cones. This is joint work with Andrew Dancer.
**Francis Burstall** (Bath)
**Harmonic maps in unfashionable geometries**
Harmonic maps from surfaces to symmetric spaces comprise an integrable
system. This provides a unifying perspective on integrable aspects of
several problems in classical surface geometry via an appropriate notion
of Gauss map. In this talk, I shall give a survey of this topic and, in
particular, give a modern discussion of Thomsen's results (1925, 1928) on
analogues of the Willmore functional in projective and Lie sphere geometry.