Schedule

CBMS Conference on Quiver Varieties and Quantum Affine Algebras

May 25 Tuesday (All talks are in Auditorium, SAS 2203, 2nd FL; Coffee breaks are in Math Lounge, 4th FL)

8:00-9:00 Registration/reimbursement, Coffee

9:00-9:15 Welcome remarks
by Loek Helminck, Head

9:15-10:15 Nakaijima (RIMS, Kyoto)

10:30-11:30 Nakajima (RIMS, Kyoto)

2:00-3:00 Mikhail Khovanov (Clolumbia)
Categorification of quantum groups

3:00-3:30 Tea, Math Lounge (4th FL)

3:30-4:30 Yiqiang Li (Virginia Tech)
On geometric realizations of q-Schur algebras of type A and their canonical bases

May 26 Wednesday

8:30-9:00 Coffee

9:00-10:00 Nakaijima (RIMS, Kyoto)

10:15-11:15 Nakajima (RIMS, Kyoto)

2:00-3:00 Weiqiang Wang (Virginia)
Spin Kostka polynomials and spin invariants for the symmetric group

3:00-3:30 Tea, Math Lounge (4th FL)

3:30-4:30 Pavlo Pylyavskyy (Michigan)
Geometric crystals and total positivity in loop groups

May 27 Thursday

8:30-9:00 Coffee

9:00-10:00 Nakaijima (RIMS, Kyoto)

10:15-11:15 Nakajima (RIMS, Kyoto)

2:00-3:00 Shrawan Kumar (UNC-Chapel Hill)
A generalization of Fulton's conjecture for arbitrary groups

3:00-3:30 Tea, Math Lounge (4th FL)

3:30-4:30 Antun Milas (SUNY-Albany)
Vertex Algebras and W-algebras

6:00--, Banquet ($20) at C&T Restaurant (Chinese/Thai food)

May 28 Friday

8:30-9:00 Coffee

9:00-10:00 Nakaijima (RIMS, Kyoto)

10:15-11:15 Nakajima (RIMS, Kyoto)

2:00-3:00 Zongzhu Lin (KSU)
Frobenius twisted conjugations and representations of weighted quivers

3:00-3:30 Tea, Math Lounge (4th FL)

3:30-4:30 Jiping Zhang (Peking)
A noter on Defect groups and TI sets

May 29 Saturday

8:30-9:00 Coffee

9:00-10:00 Nakaijima (RIMS, Kyoto)

10:15-11:15 Nakajima (RIMS, Kyoto)



Abstracts:

M. Khovanov, Categorification of quantum groups

Abstract: The positive half of the quantum group U(g)
decomposes into the sum of finite-dimensional weight spaces.
We'll exhibit a family of algebras that categorify these
weight spaces (the Grothendieck groups of these algebras
are integral lattices spanning weight spaces) and come with
induction and restriction functors that categorify multiplication
and comultiplication operations. This is a joint work with
Aaron Lauda.

Y. Li, On geometric realizations of q-Schur algebras of type A and their
canonical bases

Abstract: A geometric realization of q-Schur algebras of type A and their canonical bases is
given by using certain localized equivariant derived categories of double framed
representation varieties of quivers.
The construction is shown to be compatible with the classical construction in
~\cite{BLM90} by derived categories of double partial flag varieties.
A conjectural generalization to other types will then be discussed.

W. Wang, Spin Kostka polynomials and spin invariants for the symmetric group

Abstract: Kostka polynomials have played a fundamental role in algebraic
combinatorics and representation theory.
We formulate a new notion called spin Kostka polynomials, and explain
its favorable properties
from several viewpoints: symmetric functions, (co)invariants, spin
symmetric groups, and Lie theory.
This is a joint work with Jinkui Wan.

P. Pylyavskyy, Geometric crystals and total positivity in loop groups.

Abstract: We study products of the affine geometric crystal of type A
corresponding to symmetric powers of the standard representation. The
quotient of this product by the R-matrix action is constructed inside
the unipotent loop group. We give an explicit subtraction-free formula
for the associated energy function. It is shown to be a stretched
staircase shape loop Schur function, as introduced by the authors in
the study of total positivity in loop groups. The connection with
wiring diagrams on a torus will be made. This is joint work with
Thomas Lam.

S. Kumar, A generalization of Fulton's conjecture for
arbitrary groups

A. Milas, Vertex Algebras and W-algebras

Abstract: We first review the construction of affine W-algebras via (quantum)
Drinfeld-Sokolov
reduction. Then we introduce two families of vertex algebras based on
lattice vertex algebras and on Wakimoto modules. Some conjectures about their
structure will be discussed.

Z. Lin, Frobenius twisted conjugations and representations of weighted
quivers.

Abstract: The motivation of the question is from classifying all
restricted Lie algebra structures on a finite dimensional Lie algebra over
a field of positive characteristic. In case the Lie algebra is abelian,
the classification question is exactly the classification of Frobenius
twisted conjugacy classes of the general linear group in the spaces of all
n by n matrices. This question now leads to representations of weighted
quivers, which are quivers with an integer associated to each arrow. I
will discuss the questions of classifying indecomposable representations
and some geometry associated to them.

J. Zhang, A noter on Defect groups and TI sets

Abstract: It is wellknown that a defect group $D$ of a given $p$-block
of a finite group $G$ is a Sylow $p$-intersection, that is $D-P\cap P^x$
for some $x\in G$, where $P\in Syl_p(G)$. We will prove that $P$ can be
replaced by any $p$-subgroups of $G$ containing $D$. In order to do so, we
introduce the weak TI sets (WTI sets), It is quite Surprizing that WTI=TI.
There might be some group theoretical implications.