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.
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.
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.
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.
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 geometryand Representations of rational Cherednik algebras
Mark Goresky: Chern classes of modular varieties (joint with W.
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
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 Bn or Cn, 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 Vm/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 C2ÄC2Ä¼ÄC2 under SU2 ×SU2 ×¼×SU2 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.
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.
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.
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.