This short workshop is held by the
International Graduate College "Arithmetic and Geometry" and teh
Differential Geometry/Geometric Analysis Group at Humboldt University.
The aim is to discuss some recent result in Differential Geometry with
applications to Mathematical Physics, which are related to subjects of
the current seminares.
The workshop is supported by the
International Graduate College "Arithmetic and Geometry".
Location
The meeting will take place on Campus
Adlershof of the Humboldt University, Institut für Mathematik,
Rudower Chaussee 25, 12489 BErlin, Room I. 410.
Programm
16.30
- 17.30
(FS Geometric Analysis)
Frederik
Witt (Oxford University): Supersymmetric
maps and Hitchin's variational principle in dimensions 7 and 8, Part I
ABSTRACT: These
two lectures are based on recent work by Hitchin, variational geometry
and generalised structures. Variational geometry interprets special
geometries
as critical points of a certain functional defined over generic
homogeneous
forms. The best known instances of this are symplectic manifolds and
manifolds
with holonomy G2. The generalised setup is a canonic
construction
which naturally incorporates the action of a 2-form, the so-called
B-field,
and defines geometries through special even or odd forms. In
the two talks we will discuss aspects of our current research on
variational
geometries which give rise to a Riemannian structure. Our viewpoint,
however,
will center around a spinorial formulation of these geometries based on
the notion of a supersymmetric map. In the first talk we shall deal
with
PSU(3) structures which lead to the Rarita-Schwinger equation. The
second
talk focuses on (G2 x G2)- structures -- a
generalised
structure -- and how they relate to supergravity of type IIA/B.
Both talks are independent of each other.
________________________________________________________________________
Thursday, 15.07.04
ABSTRACT: Second talk on
the topic.
____________________________________________________________________________
11.30
- 12.30
Jose
Figeoroa O'Farrill (University of Edinburgh): Symmetry
and Supersymmetry in supergravity
ABSTRACT:
I will review the construction of the Killing superalgebra associated
to
a supergravity background and will prove that given enough
supersymmetry,
the background is automatically homogeneous. If time allows I will
discuss
homogeneous structures and discuss what happens to them in the plane
wave
limit. This is based on work in progress with Patrick Meessen and Simon
Philip.
____________________________________________________________________________
14.00
- 15.00
Wilhelm
Klingenberg (University of Durham): Reflection
of wave fronts on surfaces
ABSTRACT: Recent advances
in twistor theory of oriented lines are applied to geometric optics. In
particular, the wave front of a wave that is reflected on a surface is
given in terms of geometric data of the incoming wave and the
reflecting
surface.
____________________________________________________________________________
15.30
- 16.30
Brendan
Guilfoyle (University Tralee):
The
Casimir effect between non-parallel plates by geometric optics
ABSTRACT
: The Casimir effect is a small force found between surfaces held
very close together in a vaccuum. It is caused by the quantum
fluctuations
allowed by the uncertainty principle. Despite being predicted and
experimentally verified fifty years ago, it has remained extremely
difficult
to compute theoretically. Recently a geometric optics
approximation
has been constructed in which the Casimir energy
can be found by a sum
of geometric quantities integrated over multiply reflected paths
between the boundaries. Using the complex geometry of the space of
oriented
lines in R
3 and the technques explained in the previous talk
by W.K., the optical approximation can be regularised to find the
Casimir
force between non-parallel plates. How the technique can be used to
compute
the Casimir force in this and more general cases will be
explained
from a mathematical point of view.
____________________________________________________________________________
Friday, 16. 07.04
9.00
- 10.00
Felipe
Leitner (University Leipzig): Conformal
holonomy
ABSTRACT:
In conformal geometry there exists a canonical connection, which has as
structure the Moebius group. This connection gives rise to an invariant
notion of conformal holonomy. We will discuss certain properties of the
conformal holonomy on a smooth manifold with conformal structure. In
particular,
its relation to parallel tractors and the conformal Einstein condition.
For homogenous conformal spaces we find an explicit formula for the
holonomy
algebra. There might be time to make an explicit calculation for a
bi-invariant
metric on a Lie group.
____________________________________________________________________________
10.30 - 11.30
Thomas Leistner (University Adelaide): Transitivity
of Lorentzian holonomy systems and some generalisations of pp-waves
ABSTRACT: We present an attempt to
give
a short proof of the classification of Lorentzian holonomy groups by
using
the transitivity of holonomy systems due to Simons. This attempt leads
to a certain class of Lorentzian manifolds with recurrent vector field,
and with the property that the curvature tensor restricted to the
orthogonal
complement of the recurrent vector field vanishes. They can be seen as
generalisations of pp-waves and we study some of their curvature
properties.
If you are interested to come to the workshop, please contact
Prof. Helga Baum, Institut für Mathemtik, HU Berlin, Tel. 030 2093 1823
baum@mathematik.hu-berlin.de
Last Modification: 12.07.2004