May 16th, 3-4 p.m., Room E17-128
Geordie Williamson (MPIM, Bonn)
A reducible characteristic variety in type A
I will discuss an example of a reducible characteristic
variety of a simple highest weight module for sl_12. In other types it
has been known since the early 1980s (thanks to work of Kashiwara and
Tanisaki) that reducible characteristic varieties can occur.
Interestingly the singularity under Beilinson-Bernstein localization is
the same as that used by Kashiwara and Saito to demonstrate a reducible
characteristic cycle for IC sheaves on the flag variety of SL_8. I will
briefly discuss how the examples were found (using parity sheaves,
positivity and decomposition numbers for perverse sheaves).
April 11th, 3-5 p.m., Room E17-128
Nenad Manojlovic (U. of Lisbon)
Algebraic Bethe Ansatz for Gaudin model with triangular boundary
Following Sklyanin's proposal in the periodic case, we derive the
generating function for the Gaudin model with boundary. Our derivation is
based on the quasi-classical expansion of the linear combination of the
transfer matrix of the XXX chain and the central element, the so-called
Sklyanin determinant. The corresponding Gaudin Hamiltonians with boundary
terms are obtained as the residues of the generating function. We study the
relevant algebraic structure for the algebraic Bethe ansatz. In the case
when the boundary matrix is upper-triangular, we implement the algebraic
Bethe ansatz, obtaining the eigenvalues of the generating function and the
corresponding Bethe states.
April 4th, 3-5 p.m., Room E17-128
Chris Negron (U. of Washington)
Hochschild Cohomology of Koszul algebras
For any algebra A, the Hochschild cohomology is defined as the graded Ext group HH*(A)=Ext_A-bimod*(A,A). This
cohomology gives one a glimpse into the deformation theory of a
given algebra. As with any group of self-extensions, there is a
canonical product on the Hochschild cohomology. In this talk I will
describe how, given a Koszul algebra A, one can place a canonical
degree 1, square 0, derivation d_Kos on the tensor product of A with its Koszul dual so that the cohomology of the
resulting dg-algebra is equal to the Hochschild cohomology HH*(A) as an algebra. Some
examples will also be discussed.
March 14th, 3-5 p.m., Room E17-128
Joel Kamnitzer (U. of Toronto)
Representations of quotients of shifted Yangians
We study the representation theory of quotients of shifted Yangians. These algebras of interest because they quantize slices inside affine Grassmannians. We will give some conjectures and some results concerning finite-dimensional and Verma modules for these algebras.
February 14th, 3-4.30 p.m., Room E17-128
Thomas Church (Stanford)
Applications of representation stability in number theory
Representation stability for the mod-p homology of congruence subgroups, due to Putman, Church-Ellenberg-Farb-Nagpal, and Church-Ellenberg, was recently used in Calegari-Emerton's classification of stable mod-p Hecke eigenforms. The proof of representation stability rests on Church-Ellenberg's homological regularity theorem for FI-modules over Z, which guarantees that FI-modules have controlled resolution by "combinatorial" FI-modules (those coming from objects of Deligne's category Rep(S_t)). No background or details from last week's talk will be assumed.
February 7th, 3-5 p.m., Room E17-128
Thomas Church (Stanford)
Applications of representation stability in topology
I will give an introduction to representation stability, a
program describing stability in families of e.g. S_n-representations
as n goes to infinity, via a detailed look at two applications in
topology. First, I'll discuss configuration spaces of points on
manifolds; representation stability for their cohomology; and
differences in the stable behavior between open manifolds, closed
manifolds, and smooth projective varieties. Second, I will describe
representation stability for the mod-p homology of congruence
subgroups, and its use in Calegari-Emerton's recent classification of
stable mod-p Hecke eigenforms.