Fall, 2009
- Speaker Birne
Binegar, Oklahoma State University
- Title W-Cells, Nilpotent Orbits, Primitive Ideals
and Weyl Group Representations
- Time/place 9/11/2009, Friday, 11:45 am in Hill 423
- Abstract Let $G$ be the real points of a connected
linear reductive
complex algebraic group defined over $\mathbb{R}$ and let $\widehat
{G}_{adm,\lambda}$ be the set of equivalences classes of irreducible
admissible representations of $G$ of infinitesimal character
$\lambda$,
which we assume to be regular and integral. The Atlas software
enumerates the
representations in $\widehat{G}_{adm,\lambda}$, and computes the
Kazhdan-Lusztig-Vogan polynomials $P_{x,y}\left( q\right) $ which
not only
prescribe the Jordan-H\"older decomposition of standard modules in
terms
of the irreducibles in $\widehat{G}_{\lambda,adm}$, the KLV
polnomials can
also be used to endow the set $\widehat{G}_{adm,\lambda}$ with the
structure
of a $W$-graph, a certain weighted directed graph. The strongly
connected
components of this $W$-graph are W-cells. In this talk I will
describe how
the weighted graph structure of an W-cell $\mathcal{C}$ allows one to
compute
the (common) associated variety of the annihilators of the
representations in
$\mathcal{C}$ and, moreover, allows one to determine exactly when two
representations $x,y\in\mathcal{C}$ share the same annihilator.
- Speaker
Christopher Sadowski, Rutgers University
- Title On a symmetry of the category of integrable
modules
(joint work with Bill Cook)
- Time/place 9/18/2009, Friday, 11:45 am in Hill 423
- Abstract This will be an introductory talk.
Haisheng Li showed that given a module (W, Y_W(\cdot, x))
for a vertex algebra (V, Y (\cdot, x)), one can obtain a new V-module
W^{\Delta}= (W, Y_W(\Delta(x)\cdot, x)) if \Delta(x) satisfies
certain natural conditions. Li presented a collection of such
\Delta-operators for V=L(k, 0) (a vertex operator algebra associated
with an affine Lie algebra, k a positive integer). In a joint paper
with Bill Cook,
for each irreducible L(k, 0)-module W, I find a highest weight
vector of W^{\Delta} when \Delta is associated with a minuscule
coweight. From this we completely determine the action of
these \Delta-operators on the set of isomorphism equivalence classes
of L(k, 0)-modules.
- Speaker
Lev Borisov, Rutgers University
- Title In search of families of dg-algebras related to
resolutions of Gorenstein toric singularities
- Time/place 10/2/2009, Friday, 11:45 am in Hill 425
- Abstract A Gorenstein toric singularity can be
described by simple
combinatorial data, namely a convex polytope $P$ in ${\bf Z}^n$ with
integer vertices. Different triangulations of $P$ with vertices given
by integer points of $P$ give rise to different resolutions of
the singularity. It has been shown that bounded derived categories of
coherent sheaves on these resolutions are equivalent. It is reasonable
to expect that there is in fact a continuous family of triangulated
categories that includes these categories as its limit points.
This is very much work in progress, and the main questions are still
wide open. It is my hope that by bringing this problem to your
attention
I can inspire someone to find such construction.
- Speaker Vladimir Retakh, Rutgers University
- Title Towards noncommutative cluster algebras
- Time/place 10/9/2009, Friday, 11:45 am in Hill 425
- Abstract Commutative cluster algebras were introduced
by Fomin and Zelevinsky in 2002. they appeared to be very useful
in many areas of representations theory. In my talk I will
discuss a number of examples that could lead to a theory
of noncommutative cluster algebras.
- Speaker Alex Feigold, Binghamton University, State
University of New York
- Title A New Perspective on the Frenkel-Zhu Fusion Rule
Theorem
- Time/place 10/16/2009, Friday, 11:45 am in Hill 425
- Abstract Fusion rules are analogous to tensor product
multiplicities,
and play an important role in conformal field theory. They are
dimensions
of spaces of intertwining operators determined by a triple of
irreducible
modules for a vertex operator algebra. An important class of examples,
known
in physics as Wess-Zumino-Witten models, comes from the theory of
affine
Kac-Moody Lie algebras, where the modules are the standard modules of
a fixed
non-negative integral level. This talk is an exposition of joint work
with Stefan
Fredenhagen (2008) in which we prove a formula for fusion coefficients
of affine
Kac-Moody algebras first conjectured by Walton (1994). It is a
reformulation of the
Frenkel-Zhu affine fusion rule theorem (1992), written so that it can
be
seen as a beautiful generalization of the classical Parasarathy-Ranga
Rao-Varadarajan
tensor product theorem (1967).
- Speaker Nigel Boston, University of Wisconsin
- Title Random Groups and Random Galois Groups
- Time/place 10/23/2009, Friday, 11:45 am in Hill 425
- Abstract In analogy to work of Dunfield and Thurston in
topology, we computed the probability that a random pro-p
presentation will yield a given p-group G. Now in joint work with
Jordan Ellenberg we give a heuristic for the probability that the
maximal pro-p extension of Q unramified outside a random set of
primes will have Galois group G. This is guided by the
Cohen-Lenstra heuristics and the theory of pro-p braid groups.
- Speaker Georgia Benkart, University of Wisconsin
- Title Quantum sl(2) and Temperley-Lieb-type
Combinatorics
- Time/place 10/30/2009, Friday, 11:45 am in Hill 425
- Abstract This talk will feature various algebras of
diagrams (some old,
some new) that have beautiful algebraic and combinatorial
properties and are related to the representation theory
of quantum sl(2).
- Speaker Dmitry Gourevitch, Institute for Advanced Study
- Speaker Zhenghan Wang, Microsoft and University of
California at Santa Barbara
- Title CFT, MTC, and FQH states
- Time/place 11/13/2009, Friday, 11:55 am in Hill 425 (Note
that the seminar will start at 11:55 am instead of 11:45 am)
- Abstract We will discuss the CFT approach to
groundstate wavefunctions of electrons in fractional quantum Hall
(FQH) liquids initiated by G. Moore and N. Read in 1991. In this
approach, groundstates of an electron liquid are given by
conformal blocks of a rational CFT, and the topological properties
of the quasi-particles
are described by the associated modular tensor category. Open
problems include the Moore-Read Holo=Mono conjecture, and
classifications of FQH states.
- Speaker Haisheng Li, Rutgers University, Camden
- Title Vertex algebras associated with elliptic affine
Lie algebras
- Time/place 12/4/2009, Friday, 11:45 am in Hill 425
- Abstract Elliptic affine Lie algebras, similar to the
usual affine Lie
algebras, are a family of infinite-dimensional Lie algebras associated
to finite-dimensional simple Lie algebras. It has been long known that
affine Lie algebras have a canonical association with vertex algebras
and their modules. In this talk, we will show how to associate
elliptic affine Lie algebras with what are called vertex
$\C((z))$-algebras and their modules in a certain category.
- Speaker Jochen Heinloth, University of Amsterdam (talk
canceled)
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