Impromptu Differential Geometry Workshop

Dates:  Friday 18th August and Friday 25th August 2000
Place:  EMG07 (Ground Floor: Engineering and Mathematical Sciences Building)

Invited speakers: Andrea D'Agnolo (Paris VI)
                  Reyer Sjamaar (Cornell)
                  Jonathan Weitsman (U of California, Santa Cruz)

Friday 18th August

11:10am Reyer Sjamaar:     Linear inequalities and Schubert cycles

12:10am Michael Murray:    Central extensions of the loop group
                           Postponed until next week

 1:10pm Lunch

 2:10pm Andrea D'Agnolo:   Twistor duality and D-modules

 3:10pm Mike Eastwood:     Homogeneous hypersurfaces

 4:10pm Jonathan Weitsman: Surjectivity for q-Hamiltonian G-spaces

 5:10pm Drinks

Friday 25th August 2000

10:10am Paul Norbury:    Holomorphic constructions related to monopoles
        EMG06 (Ground Floor: Engineering and Mathematical Sciences Building)

11:10am Peter Bouwknegt: String Theory and noncommutative geometry
        G08 (Ground Floor: Main Mathematics Building (using Powerpont))

12:10pm Lunch

 1:10pm Reyer Sjamaar:   Moment maps and Riemannian symmetric pairs
        EMG07 (Ground Floor: Engineering and Mathematical Sciences Building)

 2:10pm Michael Murray:  Central extensions of the loop group
        EMG07 (Ground Floor: Engineering and Mathematical Sciences Building)

Abstracts

Sjamaar I: This is joint work with Arkady Berenstein. Consider a compact Lie
group and a closed subgroup. Generalizing a result of Klyachko, we give a
necessary and sufficient criterion for a coadjoint orbit of the subgroup to be
contained in the projection of a given coadjoint orbit of the ambient group.
The criterion is couched in terms of the "relative" Schubert calculus of the
flag varieties of the two groups.

D'Agnolo: Sheaf theory and D-module theory give a framework for the study of
integral geometry. In particular, we obtain a general adjunction formula,
describing the intertwining operators. We will illustrate these issues by
considering the example of the twistor duality, where we can treat in a unified
manner results as Martineau duality, projective, affine, or conformal Radon
transforms, and Murray's correspondence. If time permits, we will also
consider gerbes of twisted sheaves, and show how the "twisted" twistor duality
boils down to the classical Cauchy-Fantappi\`e formula.

Eastwood: A homogeneous hypersurface in affine space is one that everywhere
looks the same from the affine point of view. I shall talk about the
classification of these surfaces in 2, 3, and 4 dimensions. This is joint work
with Vladimir Ezhov. The methods are elementary and I shall start from scratch.

Weitsman: Kirwan's surjectivity theorem for Hamiltonian G-spaces is the
fundamental result which allows the computation of the cohomology rings of
symplectic quotients in terms of the cohomology of the original manifold. In
joint work with R. Bott and S. Tolman we prove an analogous result for the
quasi-Hamiltonian G-spaces of Alexeev, Malkin, and Meinrenken.

Norbury: In this talk I will survey holomorphic aspects of the study of
monopoles and introduce a new holomorphic construction for monopoles.

Bouwknegt: String Theory suggests that noncommutative geometry is the correct
framework to formulate a theory of quantum gravity. I will briefly review how
noncommutative geometry emerges from String Theory, and I will discuss some
recent ideas on how one might formulate a manifestly background independent
open string field theory.

Sjamaar II: This is joint work with Luis O'Shea. Our main result is a "real
form" of Kirwan's convexity theorem, which in the abelian case was proved
earlier by Duistermaat. We apply our result to flag varieties of real
semisimple groups and obtain eigenvalue inequalities, which generalize
inequalities found by Weyl, Ky Fan, Kostant, Klyachko, and many others.

Murray: The loop group, the group of smooth maps of the circle into a compact
Lie group, has a central extension by the circle group. This central extension
is topologically non-trivial and therefore difficult to construct. This talk
will give a general construction which unifies a number of earlier
constructions. If time permits I will explain how this relates to work on
string structures. This is joint work with Danny Stevenson.

Sponsored by Adeliade University Institute for Geometry and its Applications.
Organised by Mike Eastwood and Siye Wu.