**Michel Brion:
Stable reductive varieties** (joint with V. Alexeev)

We describe all degenerations of a connected reductive
group G, regarded as a variety with action of G×G. We also
construct a projective moduli space related to these degenerations.

**Peter Donovan:
Bounding the Cartan Invariants**

It is known that the Cartan invariants of
a p-block B of a (variable) finite group
with defect group isomorphic to fixed p-group
D are bounded by a function of D if D is
cyclic, dihedral, semi-dihedral or quaternion.
The conjecture that this holds in general remains
open. A geometrical approach that may render
the case where D is an elementary abelian
2-group accessible is described.

**Sergey Fomin:
Cluster algebras of finite type, part II**

This is the second of two talks on the ongoing joint project with
Andrei Zelevinsky. The talk will focus on concrete descriptions
of the cluster algebras of finite type, particularly on their
underlying combinatorial structure, captured by a ``cluster complex''.
We identify this complex as the generalized associahedron,
a spherical cell complex (indeed, a convex polytope) associated with
the corresponding root system.
Although strongly motivated by the material presented in Zelevinsky's
talk, this talk will be self-contained except at the very end.

**Dennis Gaitsgory:
A certain category of representations of a Kac-Moody algebra at the
critical level** (joint with E.Frenkel)

The talk will concentrate on a work in progress.
It is known that the category of representations of a
Kac-Moody algebra at the critical level possesses a
big center. We study the subcategory of representations,
whose central character belongs to a certain subvariety
of the center, and which are integrable with respect to
the action of the Iwahori subgroup.
We propose a conjecture that ties this category of
representations to the category of coherent sheaves on
the scheme of Miura opers with respect to te Langlands
dual group G.

**Ezra Getzler:
A Darboux theorem in the formal calculus of variations**

The Darboux theorem states that all symplectic structures on an affine
space are locally isomorphic. Hamiltonian operators are a
generalization of symplectic forms which play a central role in the
study of integrable hierarchies such as the KdV and KP equations; they
are a certain class of geometric structures on the jet-space of formal
curves in a manifold. In this talk, we prove a formal analogue of the
Darboux theorem for Hamiltonian operators.
The study of the moduli space of formal deformations is controlled by
a differential graded (dg) Lie algebra, which is an analogue of the
Schouten Lie algebra for the jet-space. We show that for an important
class of Hamiltonian operators, this dg Lie algebra is formal, in the
sense of homotopy theory.

**Victor Ginzburg:
Symplectic reflection algebras and geometry**and
**Representations of rational Cherednik algebras**

**Mark Goresky: Chern classes of modular varieties** (joint with W.
Pardon)

The Hirzebruch proportionality theorem states that the
Chern numbers of a compact modular variety X coincide (up to a
fixed constant of proportionality) with the Chern numbers of the
compact dual symmetric space. Mumford's generalization of this
result to noncompact X is the following. Although the canonical
(Baily Borel) compactification Y of X is singular, it has various
``toroidal'' resolutions Z. On each such toroidal resolution there
is a canonical vector bundle which extends the tangent bundle of
X. Mumford showed that the Chern numbers of this vector bundle
again agree (up to the same proportionality constant) with the
Chern numbers of the compact dual symmetric space.
Since these numbers are independent of the choice of resolution,
one might ask whether there is a similar statement for the Baily
Borel compactification Y, however the tangent bundle of X
(usually) has no extension to Y. Nevertheless, W. Pardon and I
were able to show that the Chern classes of X may be lifted in a
natural way to the cohomology of Y, and that the resulting Chern
numbers again satisfy the proportionality theorem. We are able
to use this result to show that the cohomology of Y contains the
full cohomology of the compact dual symmetric space.

**Ian Grojnowski:
Ramanujan sums and the Strong MacDonald conjecture**

**Mark Haiman:
Cores, quivers and n! conjectures**

It has occurred to many people that the recently established picture
connecting Macdonald polynomials, the Hilbert scheme, and the ``n!''
theorem should have a generalization in which some other group plays
the role of the symmetric group. The obvious candidate is a Weyl
group acting on two copies of its natural representation, but it
seems necessary restrict the problem in order to extend the program
in that direction.
I propose instead to consider the wreath product of the symmetric
group with a finite subgroup G of SL(2), replacing the Hilbert
scheme with a quiver variety corresponding to the fundamental affine
weight. It requires care to identify the correct conjecture in this
context, but I have a good candidate and some theoretical and
computational evidence to to support it.
When G=\mathbbZ/2\mathbbZ, the
wreath product is the Weyl group of type B_{n} or C_{n},
and it turns out that there are natural choices of quiver variety
giving satisfactory theories of types B and C, with interesting
connections to principal nilpotent pairs and the Springer
correspondence.

**Hanspeter Kraft:Geometric Analogs to the First Fundamental Theorems**

Let V be a representation of a (complex) reductive group
G. A classical theorem due to Weyl says that the simultaneous
invariants (and covariants) of any number of copies of V can be
seen in n=dimV copies. This means that the invariants of more
than n copies are given by polarization, those for less by
restriction. (There are stronger results in case the representation
is orthogonal or symplectic.)
We are studying the question whether certain geometric objects
associated to a representation have a similar behavior. A trivial
example is the structure of orbits and their closures which can all
be seen in n=dimV copies. Another example is the rationality of
the quotients V^{m}/G in the sense of geometric invariant theory.
Again, it is easy to see that if the quotient is rational for m
copies then it is rational for m¢ copies for any m¢ ³ m.
In the talk we are mainly interested in the set of unstable vectors,
the so-called nullcone of the representation. We show that after a
certain number of copies, calculated from the weight system of V,
the number of irreducible components does not change anymore, and
that they all these components have a nice resolution of
singularities. An interesting question here is if the polarizations
define the nullcone, since this has important applications to the
computation of invariants. This, in turn, is related to the problem
of linear subspaces in the nullcone which is widely open, even in
very classical situations.
(This is joint work with Nolan Wallach from UCSD. It arose from the
study of the representations C^{2}ÄC^{2}Ä¼ÄC^{2} under SU_{2} ×SU_{2} ×¼×SU_{2} which play a
role in some mathematical aspects of quantum computing.)

**Robert D. MacPherson:
Springer representations through equivariant cohomology** and
**Why didn't Leray discover Perverse Sheaves?**

**Amnon Neeman: A non-commutative generalisation of
Thomason's localisation theorem**

Gersten showed us that vector bundles need not extend
from an open subset of a singular scheme. Thomason taught us
that Gersten's conterexample is minor. In a certain precise
sense, up to splitting idempotents, chain complexes of vector
bundles do extend.
The main results we will discuss, which are joint work with
Ranicki, show how to give a non-commutative analogue of
Thomason's result. At some level this is a result about
extending representations, the most interesting case being
that of group representations. The methods of the proof
involve
compactly generated triangulated categories. The main
application, and the reason Ranicki was interested, comes from
surgery. Roughly this is the case where the group is the
fundamental group of a suitable manifold.

**Viktor Ostrik:
Coherent sheaves on
the nilpotent cone and dominant weights**

I will describe new results and
conjectures related to Lusztig-Vogan bijection
between dominant weights and equivariant bundles
on the nilpotent orbits.

**Siye Wu:
Hermitian symmetric space, Shilov boundary, and Maslov index**

Given a non-compact Hermitian symmetric space (which can
be realized as a bounded domain), its Shilov boundary is part of the
topological boundary on which the maximum norm of any holomorphic
function is reached. Using the standard decompositions of Iwasawa,
Bruhat, Langlands, Levi, and Harish-Chandra of non-compact Lie groups,
we define an triple index on the Shilov boundary as an integral of
the Kahler form on a triangle bounded by the geodesics connecting the
points on the Shilov boundary. When the space is of tube type, the
index is an integer. In particular, it reduces to the Maslov triple
index of Kashiwara when the symmetric domain is of symplectic type.

**Andrei Zelevinsky:
Cluster algebras of finite type, part I**

This is the first of two talks on the ongoing project with Sergey Fomin.
We study a class of commutative rings called cluster algebras and
introduced two years ago as an attempt to design an algebraic framework
for the dual canonical bases in quantum groups and their
representations.
There is an appropriate notion of cluster algebras of finite type.
Their classification turns out to be one more instance of the
famous Cartan-Killing classification. I will present the main
definitions,
give a precise formulation of the classification result, and review the
main steps in its proof.

**R. B. Zhang: Quantum Supergroups and Link Invariants**

Quantum supergroups were originally introduced to describe
the `deformed' supersymmetries found in some integrable
models of 2-dimensional physics, but they also proved to
be quite useful for studying low dimensional topology.
This talk is a brief introduction to quantum supergroups.
We shall explain what quantum supergroups are,
and describe aspects of their representation theory.
We shall examine braidings of quantum supergroups,
and demonstrate how to construct topological
invariants of knots and 3-manifolds by using
their representation theory.

File translated from T

On 20 Jul 2002, 17:43.