Spring, 2012
- Speaker Anthony Licata, Institute for Advanced Study
- Title Heisenberg algebras and vertex operators from categorification
- Time/place 2/17/2012, Friday, 11:45 am in Hill 705
- Abstract
Each affine Dynkin diagram gives rise to a calculus of diagrams in the
plane. We'll describe how, starting from these planar diagrams, one
can produce the Heisenberg algebra, its Fock space representation, and
the basic representation of the corresponding affine Lie algebra. We
will also describe applications to the geometry of Hilbert schemes and
geometric representation theory.
- Speaker Jinwei Yang, Rutgers University
- Title Logarithmic intertwining operators and associative algebras
- Time/place 2/24/2012, Friday, 11:45 am in Hill 705
- Abstract In this talk, I will present a joint work with
Prof. Yi-Zhi Huang. In this work, we establish a canonical isomorphism
between the space of logarithmic intertwining operators among
suitable generalized modules for a vertex operator algebra and
the space of homomorphisms between suitable modules for a
generalization of Zhu's algebra given by Dong-Li-Mason. This
result generalizes a result of Frenkel-Zhu and Li stating that
for a vertex operator algebra whose N-gradable weak modules are
all completely reducible, the space of intertwining operators
among irreducible modules is canonically isomorphic to the space
of homomorphisms between suitable modules for Zhu's algebra
associated to the vertex operator algebra.
- Speaker Siddhartha Sahi, Rutgers University
- Title The Capelli identity and its generalizations
(Suitable for visiting graduate students!)
- Time/place 3/2/2012, Friday, 11:45 am in Hill 705
- Abstract Let V be a finite dimensional real vector space.
The Weyl algebra W = W(V) is the algebra of all polynomial coefficient differential
operators on V. The classical Capelli identity [1887] is a certain
identity in W(V) where V is the space of n x n matrices. This identity
played a crucial role in 19th century invariant theory, and has continued
to find modern day applications, e.g. in the work of Atiyah-Bott-Patodi
on the index theorem, and more recently in representation theory and
mathematical physics. It has also been generalized in several directions.
In this talk we review the classical Capelli identity and some of its
generalizations, and describe some new results.
- Speaker Igor Kriz, Institute for Advanced Study and University of
Michigan
- Title On operads, vertex algebras and
algebraic approaches to conformal field
theory
- Time/place 3/23/2012, Friday, 11:45 am in Hill 705
- Abstract I will talk about my joint work
with R. Hortsch, A. Pultr and Y. Xiu on
attempting to axiomatize parts of
conformal field theory purely algebraically.
I will start with a reformulation of
the notion of vertex algebra using
algebraic graded co-operad algebra
structure on the module of correlation
functions. I will
then show how correlation functions
in genus 0 can be axiomatized using
the Riemann-Hilbert correspondence.
- Speaker Yiannis Sakellaridis, Rutgers University at Newark
- Title Scattering theory and the Plancherel formula for
spherical varieties
- Time/place 4/6/2012, Friday, 11:45 am in Hill 705
- Abstract Scattering theory is about "incoming" and "outgoing"
waves and the way they interact. In the context of harmonic analysis on a
G-homogeneous space X, the waves are the L^2 spaces of different G-orbits
in a compactification of X (or rather, their normal bundles), and their
scattering is the way they interact to form the continuous spectrum of
L^2(X). I will explain these ideas, which lead (under some conditions) to a
proof of the Plancherel formula, up to discrete spectra, for p-adic
spherical varieties. This is joint work with Akshay Venkatesh.
- Speaker Chris Sadowski, Rutgers University
- Title New examples of generalized twisted modules for a vertex operator
algebra in the affine Lie algebra case
- Time/place 4/13/2012, Friday, 11:45 am in Hill 705
- Abstract Twisted modules associated to a finite order automorphism
for a vertex operator algebra V are well known and have been studied
in detail. In a recent work, Yi-Zhi Huang introduced a more general
theory of twisted modules for a vertex operator algebra, namely those
associated to an arbitrary automorphism. Huang also gave a general
theorem for constructing these twisted modules. In this talk, I give
various examples of these constructions with certain desirable
properties in the affine Lie algebra case, as well as draw a
connection between these constructions and those in my previous work
with William Cook.
- Speaker Robert McRae, Rutgers University
- Title Integral forms for lattice and affine Lie algebra vertex
operator algebras
- Time/place 4/20/2012, Friday, 11:45 am in Hill 705
- Abstract Abstract: Integral forms for vertex operator algebras are as natural
in vertex operator algebra theory as Chevalley bases are in Lie
algebra theory. Integral forms for vertex operator algebras based on
even lattices were first defined by Borcherds, and in this talk, I
will discuss my approach to constructing this integral form. I will
also define integral forms for affine Lie algebra vertex operator
algebras using the integral form of the universal enveloping algebra
of an affine Lie algebra constructed by Garland, and discuss some
properties of these integral forms.
- Speaker Po Hu, Institute for Advanced Study and Wayne State University
- Title Topological field theory and knot theory
- Time/place 4/27/2012, Friday, 11:45 am in Hill 705
- Abstract In this talk, I will discuss modular functor-like topological field
theories, and their homotopy-theoretical realizations. In the process, I
will introduce a new axiomatization of modular functors, and will briefly
discuss its advantages. I will also show how this new type of structure
leads to homotopy-theoretical refinements of Khovanov homology and its
variants, such as odd Khovanov homology. This is joint work with D. Kriz
and I. Kriz. I will discuss our results, as well as independent results by
Lipshitz and Sarkar.
- Speaker Dmitry Gourevitch, Weizmann Institute of Science, Israel
- TitleDerivatives for smooth representations of GL(n,R) and GL(n,C)
- Time/place 4/27/2012, Friday, 2:00 pm in Hill 705 (Note the special time)
- Abstract The notion of derivatives for smooth representations of
GL(n,Q_p) was defined by Bernstein and Zelevinsky. In the archimedean
case, an analog of the highest derivative was defined for irreducible
unitary representations by Sahi and called the ``adduced"
representation.
In our joint work with A. Aizenbud and S. Sahi we define derivatives
of all orders for smooth admissible Frechet representations (of
moderate growth). The real case is more problematic than the p-adic
case; for example arbitrary derivatives need not be admissible.
However, the highest derivative continues being admissible, and for
irreducible unitarizable representations coincides with the space of
smooth vectors of the adduced representation.
We prove exactness of the highest derivative functor, and compute
highest derivatives of all monomial representations. We apply those
results to finish the computation of adduced representations for all
unitary representations and to prove uniqueness of degenerate
Whittaker models for unitary representations.
- Speaker
Azat Gaynutdinov, Institut de Physique Theorique, CEA-Saclay
- Title
A limit of affine Temperley-Lieb algebras at a root of unity, the Lie
algebra sp_{infinity}, and bulk logarithmic CFT
- Time/place 5/30/2012, Wednesday, 2:00 in Hill 705
- Abstract
We study tensor-product representations of affine Temperley-Lieb (TL)
algebras and centralizer constructions in a root of unity case. An
analogue of Howe duality in such systems allows taking the direct
limit of these representations when the number of tensorands goes to
infinity. The limit of the representation spaces turns out to be a
logarithmic conformal field theory in the bulk, while the affine TL
algebra in the limit is expressed by a representation of an
infinite-dimensional Lie algebra extending sp_{infinity} and this
algebra contains a product of left and right Virasoro algebras as a
subalgebra.
- Speaker
Arne Meurman, Lund University
- Title
Ramanujan type formulas for 1/\pi
- Time/place 6/14/2012, Thursday, 2:00 in Hill 705
- Abstract
We present some recent generalizations by Z.-W. Sun, A. Aycock,
G. Almkvist of famous formulas by Ramanujan. This is connected to
hypergeometric functions and modular forms.
- Speaker
Thomas Edlund, Lund University
- Title
Explicit formulas for singular vectors in Verma modules
- Time/place 6/15/2012, Friday, 2:00 in Hill 705
- Abstract
A way to interpret monomials with complex exponents in the universal
enveloping algebra U(g) of a Lie algebra g, has been introduced in an
article by F. G. Malikov, B. L. Feigin and D. B. Fuchs. In the context
of symmetrizable Kac-Moody algebras, they show that if the exponents
are suitably choosen, the resulting expressions give rise to singular
vectors in Verma modules. The framework in which these calculations
occur is, however, somewhat involved. In this talk I will present a
rigorous setting, in which these ideas are realized in a different
manner. The construction includes Ore localization in U(g) and the
introduction of certain conjugation automorphisms.
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