Spring, 2008
- Speaker Antun Milas, SUNY-Albany
- Title W-algebras, quantum groups and combinatorial
identities
- Time/place Tuesday, 2/5/2008, 2:15 pm in Hill 124
- Note Special time and place
- Abstract I will discuss a conjectural relationship
between certain quantum
W-algebras (vertex algebras) and finite-dimensional quantum groups
associated to $sl_2$ (Hopf algebras). In the process we shall
encounter interesting multisum identities.
- Speaker Kevin McGerty, Chicago and IAS
- Title Hall algebras and the Quantum Frobenius
- Time/place Friday, 2/15/2008, 11:45am in Hill 423
- Abstract See the preprint math.QA/0601150.
- Speaker Olivier Schiffman, Paris and IAS
- Title Macdonald polynomials and Eisenstein series on
elliptic curves
- Time/place Friday, 2/29/2008, 11:45am in Hill 423
- Abstract
- Speaker Tom Robinson, Rutgers
- Title The automorphism property, differential
representations and classical combinatorial identities
- Time/place Friday, 3/7/2008, 11:45 am in Hill 423
- Abstract We shall first recall the automorphism
property of exponentiated
derivations, and then discuss representing two variable derivations
(formal partial differential operators) as one variable derivations.
Then we shall show how one may use these two algebraic ingredients to
compute simple classical identities involving special hyperbinomial
numbers like the Stirling numbers.
- Speaker Tom Robinson, Rutgers
- Title Formal differential representations, Faa di Bruno
and the Riordan Group
- Time/place Friday, 3/14/2008, 11:55 am in Hill 425
- Note Adjusted time and room
- Abstract First I will show explicitly how a calculation
in Frenkel-Lepowsky-Meurman's book on vertex operator algebras,
which I
will in its essentials redo, can be viewed as an application of a
formal representation of exponentiated derivations. The outcome of
the calculation is Faa di Bruno's formula for the higher derivatives
of a composite function. Then building on this result I will show how
another application of an easy class of formal differential
representation leads to the Riordan Group. No prerequisites
necessary.
- Speaker David Ben-Zvi, Univ. of Texas, Austin and IAS
- Title Real Groups and Topological Field Theory
- Time/place Friday, 3/28/2008, 11:45 am in Hill 423
- Abstract I will explain current joint work with David
Nadler, in which
the representation theory of real reductive Lie groups is
examined through the lens of topological field theory
and the geometric Langlands program. Our main results show
how to recover the representation theory of real forms
of a complex group G from the representation theory
of G, and how to deduce a Langlands dual
description of the representation theory
(a form of Soergel's conjecture, generalizing
results of Vogan and Langlands).
- Speaker Liang Kong, Max Planck Institute for Mathematics at
Bonn
- Title A tensor-categorical study of open-closed
rational conformal field theory
- Time/place Friday, 4/4/2008, 11:45 am in Hill 423
- Abstract I propose a reformulation of open-closed
rational conformal field theory in terms of certain algebras in
modular tensor categories. I will explain where it comes
from. Then I will give a somewhat dual formulation. I will also
discuss the so-called open-closed duality in this framework.
- Speaker Hadi Salmasian, Alberta
- Title On the structure and geometry of
infinite-dimensional classical Lie groups
- Time/place Friday, 4/11/2008, 11:45 am in Hill 423
- Abstract Although structure theory and representations of
finite-dimensional classical Lie groups/Lie algebras have been studied
for about a hundred years, only recently have infinite-dimensional
classical Lie groups/Lie algebras been investigated
systematically. In this talk I begin by surveying recent
progress on classification of simple locally finite Lie algebras,
their Cartan and Borel subalgebras, and the geometry of
their associated flag manifolds. Next I will
focus on certain B-stable subvarieties of these flag
manifolds which probably deserve to be called
infinite-dimensional Schubert varieties. The talk
will conclude with a result on finiteness of weight
multiplicities of modules
canonically associated to line bundles of these varieties.
- Speaker Robert Wilson, Rutgers
- Title Andrews' multisum identities and affine Lie
algebras
- Time/place Friday, 4/18/2008, 11:45 am in Hill 423
- Abstract Certain important power series identities (in
particular, the Rogers-Ramanujan identities and their
combinatorial generalizations by Andrews, Gordon and Bressoud)
have been explained in terms of filtrations of standard modules
for affine Lie algebras (in particular, A_1^{(1)}). Andrews has
given another generalization of the Rogers-Ramanujan identities in
which the sum side is replaced by a multisum. We give a Lie
theoretic interpretation of this expression using the operators
X^{(i)} introduced by Meurman and Primc on a standard
A_1^{(1)}-module.
- Speaker Daniel Sage, LSU and NYU
- Title The Springer correspondence, unipotent
characters, and
perverse coherent sheaves
- Time/place Friday, 4/25/2008, 11:45 am in Hill 423
- Abstract
In this talk, I will discuss a combinatorial coincidence in
representation theory as well as some hints at its geometric
interpretation. Let G be a connected reductive algebraic group with
Weyl group W. The Springer correspondence assigns to each irreducible
representation of W an irreducible equivariant local system on a
unipotent class of G. A major ingredient of Lusztig's parametrization
of unipotent characters of finite reductive groups (or of unipotent
character sheaves) has a similar flavor, with each irreducible
representation of W associated to an element of a certain finite set
which may be thought of as irreducible equivariant local systems on
certain finite groups. In this talk, I will describe a remarkable
compatibility between these two assignments of local systems to
irreducible representations of W. This result is related to a
conjecture of Lusztig on the geometry of special pieces in the
unipotent variety. I will explain how an extension of the
Deligne-Bezrukavnikov theory of perverse coherent sheaves can be used
to address this conjecture. This work is joint with Pramod Achar.
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