September 20th, 35 p.m., Room E17122
Leonid Rybnikov (HSE, Moscow)
Quantization of Drinfeld's zastava spaces
This is a joint work with M. Finkelberg. Quasimaps' space Z_d (also known as Drinfeld's zastava space) is a
remarkable compactification of the space of based degree d maps from
the projective line to the flag variety of type A. The space Z_d has a
natural Poisson structure,
which goes back to Atiyah and Hitchin. We describe
the Quasimaps' space as some quiver variety, and define the
AtiyahHitchin Poisson structure in quiver terms.
This gives a natural way to quantize this Poisson structure.
The quantization of the coordinate ring of the Quasimaps' space turns
to be some natural subquotient of the Yangian of type A.
I will also discuss some generalization of this result to the BCD types.
September 27th, 35 p.m., Room E17122
Galyna Dobrovolska (U. of Chicago)
Finite local systems in the DrinfeldLaumon construction
Motivated, on the one hand, by R. Bezrukavnikov's conjectures in the representation
theory of the rational Cherednik algebra in characteristic p, and, on the other hand, by the
geometric Langlands duality for the trivial local system, we prove the following theorem. The
result of the kth step of the DrinfeldLaumon construction applied to a local system
corresponding to the irreducible representation of the symmetric group S_n indexed by a
partition \mu=(n_1,n_2,...,n_m) (such that the n_i satisfy a set of inequalities) can be
described explicitly as follows. It is a perverse sheaf on the stack SCoh_{k+1} (of coherent
sheaves of rank k+1 with a section) which is supported on the closure of the preimage of
the HarderNarasimhan stratum in Coh_{k+1} with subquotients of rank one and of degrees
which are shifts of the n_i; the monodromy of the corresponding local system is the irreducible
representation of the symmetric group which is indexed by the partition obtained by deleting the
k largest parts of \mu.
October 25th, 35 p.m., Room E17122
Victor Kac
Representations of affine Lie superalgebras and mock theta functions
It is a wellknown fact that the characters of integrable
highest weight modules over affine Lie superalgebras can be expressed in
terms of classical theta functions, consequently they are modular
invariant. In the Lie superalgebra case the characters can be expressed in
terms of Appell's elliptic functions of the third kind, which are not
modular invariant. However it has been shown more recently by Zwegers
that Appell's functions can be modified by adding a mysterious real
analytic correction to restore modular invariance.
In a recent paper by Wakimoto and myself we show how to construct
modular invariant families of characters of affine Lie superalgebras,
using Zwegers' corrections. Applying the quantum Hamiltonian reductions
we find interesting modules over the N=2 superconformal
algebra (important in mirror symmetry) with central charge
c=3(1(2m+2)/M) (the case m=0 is the wellknown minimal series,
but m>0 is new).
All relevant material on Lie superalgebras and theta functions will be
explained.
November 1st, 35 p.m., Room E17122
Pierre Cartier (IHES and University ParisDiderot)
Lie groupoids, calculus of variations and Noether's theorems
The proper way to express symmetries in differential geometry is Lie groupoids rather than Lie groups . What replaces the Lie algebra of a group is the Lie algebroid of a groupoid . This construction lies at the foundations of calculus of variations . We shall present the classical theorems of Emmy Noether in this perspective , inspired by problems in mathematical physics (Einstein vs. Hilbert) .
November 22nd, 4:307:00 p.m., Room E17122 (NOTE TIME CHANGE!)
Brent Pym (McGill Univ.)
Poisson structures on Fano varieties and their quantizations
The degeneracy loci of a Poisson structure are the subspaces
on which the dimensions of its symplectic leaves drop. I will describe
joint work with Marco Gualtieri that is concerned with the local and
global properties of these degeneracy loci, and gives new evidence for a
conjecture of Bondal regarding Poisson structures on Fano varieties. I
will then explain how knowledge of Poisson structures on projective
threespace can be combined with deformation quantization to make
progress on an open problem in noncommutative algebra: the
classification of fourdimensional ArtinSchelter regular algebras.
