Fall, 2014
- Speaker Christoph Keller, Rutgers University,
Department of Physics and Astronomy
- Title Non-rational VOAs and modular invariant partition functions
- Time/place 9/5/2014, Friday, 12:00 in Hill 425 (note special room)
- Abstract From the work of Zhu it is known that the characters of the
representations of rational VOAs (under some weak additional
assumptions) form representations of the modular group SL(2,Z).
Physicists use this fact to obtain modular invariant functions, the
so-called partition functions, as sesquilinear combinations of those
characters. In the case of the Virasoro minimal models this leads to
the ADE classification of modular invariant partition functions. I
will discuss the situation for non-rational Virasoro theories and
present some first steps towards a classification of partition
functions in that case.
- Speaker Shashank Kanade, Rutgers University
- Title Recent developments in partition identities
and the representation theory of vertex algebras
- Time/place 9/12/2014, Friday, 12:00-12:30 (note special time)
in Hill 705
- Abstract Ever since Lepowsky-Wilson's remarkable
vertex-operator-theoretic
proof of the classical Rogers-Ramanujan identities, the area of
``algebraic combinatorics'' relating partition identities to the
representation theory of vertex algebras has witnessed an explosion of
ideas. In this introductory survey talk, I will discuss some recent
and exciting developments in this area. Specifically, I will give an
overview of some (new) generalizations of Andrews-Baxter's ``motivated
proof'' of the Rogers-Ramanujan identities and their connections to
the representations of certain vertex algebras. I will also explain
how ``experimental mathematics'' is shaping the landscape. Most of
the talk is based on a recent joint work with J. Lepowsky, M. C.
Russell and A. V. Sills.
- Speaker Francesco Fiordalisi, Rutgers University
- Title Towards a proof of the modular invariance for logarithmic
intertwining operators
- Time/place 9/22/2014, Monday, 1:00 pm
in graduate student lounge (note special time and room)
- Abstract The full modular invariance conjecture of Moore and Seiberg
for intertwining operators in the rational case was proved
by Huang in 2003, An analogous conjecture in the logarithmic
case was proposed by Huang a few years ago. In this talk, I
will discuss my recent progress towards a proof of this
conjecture.
- Speaker Debajyoti Nandi, Rutgers University
- Title Partition identities arising from the
standard A2(2)-modules of level 4
- Time/place 9/26/2014, Friday, 12:00
in Hill 705
- Abstract In this talk, I will present a new set of (proposed) partition
identities arising from the standard modules of level 4 for the affine
Lie algebra A2(2) using a twisted vertex operator construction, and
strong evidence for their validity. This talk is based on my recent
work which is a continutation of a long line of research of
investigating and discovering surprising interplay between
representation theory and combinatorial/paritition identities using
vertex-algebraic ideas and techniques. This was first exemplified by
the vertex-operator-theoretic proof of the Rogers-Ramanujan-type
identities using the standard A1(1)-modules by
J. Lepowsky-R. Wilson. In his PhD thesis, S. Capparelli proposed new
combinatorial identities using a twisted vertex operator construction
of level 3 standard A2(2)-modules, which were later proved
independently by G. Andrews, Capparelli, and M. Tamba-C. Xie. The
level 4 case for A2(2) shows surprising new phenomena that were absent
in previously known examples of this type.
- Speaker Emily Leven, University of California at San Diego
- Title Combinatorial Aspects of the rational Shuffle Conjecture
- Note Joint Experimental Math/Lie Group/Quantum Math Seminar
- Time/place 10/3/2014, Friday, 12:00
in Hill 705
- Abstract In this talk, we will review the history and recent
extensions of the Classical Shuffle Conjecture. This conjecture equates two
symmetric polynomials, one of which is known to give the Frobenius characteristic
of the space $DH_n$ of diagonal harmonics. The other side of the conjecture is
purely combinatorial, showing the remarkable ability of certain symmetric
function operators to control combinatorial objects, such as Dyck paths and
parking functions. This branch of algebraic combinatorics was created to explore
the representation-theoretical aspects of Macdonald polynomials. This led to the
$n!$ conjecture and the introduction of the space of
Diagonal Harmonics. A program outlined by Procesi led to the proof
by Mark Haiman of the $n!$-conjecture by algebraic geometrical tools. There
has recently been a flood of new operators and conjectures created, in our
subject, by algebraic geometers. This talk covers some of the new results
and conjectures obtained by a continuing effort to translate these developments
back into the original algebraic-combinatorial setting. Our presentation should
be accessible to the general mathematical audience.
- Speaker Evan Wilson, Rutgers University
- Title Tensor product decomposition of sl_n^-modules and generating series
- Time/place 10/17/2014, Friday, 12:00 pm in Hill 705
- Abstract
In this talk, we describe recent joint work with Kailash Misra on
decomposing the tensor product of two level one modules of the affine
Kac-Moody algebra sl_n^, using the crystal basis of quantum U(sl_n^)
of Misra and Miwa and some well-known graded dimension formulas. In
the process, we uncover some generating series for partitions whose
parts satisfy certain conditions.
- Speaker Alex Kemarsky, Technion, Israel
- Title Gamma factors of GL_n(R)-distinguished
representations of GL_n(C)
- Time/place 10/24/2014, Friday, 1:45 in Hill 425
(note special time and room)
- Abstract
An irreducible representation (\pi,V) of GL_n(C) is
called GL_n(R)-distinguished if there exists a non-zero continuous
GL_n(R)-invariant functional L:V \to C. In the talk we give a necessary
condition for GL_n(R)-distinction. As a corollary, we prove that the
Rankin-Selberg gamma factors of \pi \times \pi' at s=1/2 for \pi,\pi'
distinguished representations of GL_m(C),GL_n(C) respectively equals 1.
- Speaker Siddhartha Sahi, Rutgers University
- Title The Macdonald conjectures for multivariate
hypergeometric functions
- Time/place 11/7/2014, Friday, 12:00 in Hill 705
- Abstract TBA
- Speaker Andrey Minchenko, Weizmann Institute, Rehovot, Israel
- Title Finite multiplicity theorem for spherical pairs
- Time/place 11/17/2014, Monday, 3:00 (note special day and
time) in Hill 705
- Abstract
Let X be a spherical space for a real reductive group G. Recently,
Kobayashi and Oshima, and independently Kroetz and Schlichtkrull have
obtained results on boundedness of multiplicities of irreducible
representations in the space of functions on X. We will consider
another proof of these results, which seems to be shorter. One of the
main steps is to show that the singular support of a certain
distribution on G (spherical character) is a Lagrangian in the
cotangent bundle of G. We will also use some non-trivial facts about
D-modules and Springer resolution. Another advantage of this approach
is the possibility for generalization to the p-adic case. The talk is
based on a joint work of the speaker with A. Aizenbud and
D. Gourevitch.
- Speaker Hadi Salmasian, University of Ottawa
- Title
Spherical polynomials and the spectrum of invariant differential
operators for the symmetric superpair GL(m,2n)/OSp(m,2n)
- Time/place 12/5/2014, Friday, 12:00 in Hill 705
- Abstract
The algebra of invariant differential operators on a multiplicity-free
representation of a reductive group has a concrete basis, usually
referred to as the Capelli basis. The spectrum of the Capelli basis on
spherical representations results in a family of symmetric polynomials
(after \rho-shift) which has been studied extensively by Knop and Sahi
since the early 90's. In this talk, we generalize some of the
Knop-Sahi results to the symmetric superpair GL(m,2n)/OSp(m,2n). As a
side result, we show that the qualitative Capelli problem (in the
sense of Howe-Umeda) for this superpair has an affirmative
answer. This talk is based on an ongoing project with Siddhartha Sahi.
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