Fall, 2010
- Speaker David Radnell, American University of Sharjah
- Title Applications of Inifnite-dimensional Teichmueller
theory to conformal
field theory
- Time/place 8/25/2010, Wednesday, 2:00 pm in Hill 425
- Abstract Riemann surfaces with analytically
parametrized boundaries can be sewn
together using the parametrizations
to identify boundary points. The moduli space of these rigged
surfaces, the determinant line bundle and the sewing operation are
fundamental to two-dimensional conformal field theory (CFT).
Teichmueller theory is the natural way to
solve the (infinite-dimensional) complex analytic questions that
arise. For example, we have given the rigged moduli space a complex
Banach manifold strucure and proved that the sewing operation is
holomorphic.
By using conformal welding we generalize to quasisymmetric boundary
parametrizations. Because of the simplified
picture we obtain, it appears this is the natural setting for the
geometric objects in conformal field theory. Moreover, this allows the
rigged moduli space to be obtained as a quotient of the standard
infinite-dimensional Teichmueller space of bordered Riemann surfaces.
A clear two-way link between CFT and Teichmuller theory is thus
established. An overview of these recent results will be presented.
This is joint work with Eric Schippers.
- Speaker Eric Schippers, University of Manitoba
- Title Applications of conformal field theory to
infinite-dimensional
Teichmueller theory
- Time/place 8/26/2010, Thursday, 2 pm in Hill 425
- Abstract One of the central objects in two-dimensional
conformal field theory
is the "rigged moduli space" of Riemann surfaces with parameterized
boundary
curves. The "sewing operation" joins two parameterized Riemann
surfaces
along two boundaries, identifying points using the parameterizations.
David Radnell and I showed that the rigged moduli space of Riemann
surfaces of
a given type is a quotient of the Teichmueller space of bordered
surfaces of
the same type.
In this talk I will give applications of this correspondence to
Teichmueller
theory. It allows us to construct a natural fiber structure on
Teichmueller
space, which arises naturally from the puncture/local coordinates
model of the
space, which arises naturally from the puncture/local coordinates
model of the
rigged moduli space. As an application we can construct a new natural
coordinate
system on the infinite dimensional Teichmueller space of bordered
surfaces.
This is joint work with David Radnell.
- Speaker Siddhartha Sahi, Rutgers University
- Title Invariant distributions on Speh representations
- Time/place 10/1/2010, Friday, 11:45 am in Hill 425
- Abstract Speh representations are an interesting family
of unitary
representations of GL(2n,R), which were first discovered by Birgit
Speh in the residual spectrum. They are also important examples of
cohomologically induced representations constructed by Zuckerman.
They play a key role in the Tadic-Vogan classification of the
unitary dual of GL(n).
Recently, using automorphic techniques, Offen-Sayag proved
[unpublished] the somewhat surprising result that Speh
representation admit a (unique) distribution vector invariant
under the symplectic group Sp(2n,R), if and only if n is even!
In joint work with Gourevitch and Sayag, we prove a strengthening
of the Offen-Sayag result using (local) representation theoretic
techniques.
First using the Sahi-Stein realization of the Speh
representation we show, using analytic and geometric arguments
that, for even n, there is a unique distribution vector invariant
under the Siegel parabolic subgroup of Sp(2n,R). Next, using an
algebraic argument involving the K-type structure of the Speh
representations, we show that, for odd n, there are no
distribution vectors invariant under the unitary subgroup of Sp(2n,R).
- Speaker Vladimir Retakh, Rutgers University
- Title Noncommutative Toda Chains, Hankel
Quasideterminants And Painlev'e
II Equation
- Time/place 10/8/2010, Friday, 11:45 am in Hill 425
- Abstract We construct solutions of an infinite Toda
system and an
analogue of the Painlev'e II equation over noncommutative differential
division rings in terms of quasideterminants of Hankel matrices.
- Speaker Anders Buch, Rutgers University
- Title Quantum K-theory
- Time/place 10/15/2010, Friday, 11:45 am in Hill 425
- Abstract The quantum K-theory of a homogeneous space X
is a deformation of the
ordinary K-theory ring, where the structure constants are defined as
polynomial expressions in K-theoretic Gromov-Witten invariants. In
contrast to (ordinary) cohomological GW invariants, the K-theoretic
invariants can be non-zero in arbitrarily high degrees. As a
consequence, there might be infinitely many non-zero terms in the
product of two Schubert classes. When X is a Grassmannian of type A,
a Pieri rule proved by Mihalcea and the speaker implies that all
products are finite. I will speak about new work with Chaput,
Mihalcea, and Perrin that shows that the quantum K-theory of X is
finite when X is any cominuscule homogeneous space.
- Speaker Walter Freyn, University of Muenster
- Title Kac-Moody symmetric spaces
- Time/place 10/22/2010, Friday, 11:45 am in Hill 425
- Abstract Kac-Moody symmetric spaces are the natural
infinite
dimensional counterpart to finite dimensional Riemannian symmetric
spaces as studied by Cartan. In this talk, we explain why they are
natural and describe the interplay between geometric, algebraic and
functional analytic structures in their construction.
- Speaker Stuart Margolis, Bar-Ilan University and CAISS,
CCNY
- Title On the Monoids Associated to the Coxeter Complex
and the Bruhat Order of a Coxeter Group
- Time/place 10/29/2010, Friday, 11:30 am in Hill 425 (or in
another room to be determined)
- Abstract Coxeter groups are ubiquitous in mathematics
arising in many parts of algebra, geometry, topology and other
fields, providing deep and surprising connections between diverse
areas of mathematics and computer science. There are a number of
geometric and combinatorial objects associated to a Coxeter group,
two of the most important being the Coxeter complex and the Bruhat
order. While these have played an extremely important role in the
theory, it was only within the last years that it was realized
that each of these has the natural structure of a monoid.
The monoid structures have both been .frequently rediscovered. and
have only slowly been realized to give very important information
about the Coxeter group itself, that is not available from a purely
group theoretic perspective. For example, the first edition of Kenneth
Brown.s book Buildings does not mention the monoid structure on the
Coxeter Complex at all, whereas the monoid plays a prominent role in
the second edition. Also, while no finite group has a non-trivial
multiplicative partial order, Bruhat order turns out to be a
multiplicative order on the associated monoid.
The purpose of this talk is to give an introduction to these monoids
for both group and semigroup theorists. I will discuss their structure
and how they fit into two very important classes of monoids, the so
called left regular bands and J -trivial monoids. I will look at their
representation theory as well as how they are used to give deeper
information on the associated Coxeter group. Surprisingly, the
algebras of these monoids are intimately related to the Solomon
Descent Algebra and the 0-Hecke algebra of the Coxeter groups as
well.
- Speaker Leonardo Mihalcea, Baylor University
- Title Spaces of rational curves in flag manifolds and
the quantum Chevalley formula
- Time/place 11/5/2010, Friday, 11:30 am in Hill 425
- Abstract Given Omega a Schubert variety in a flag
manifold, one can
consider two spaces: the moduli space GW_d(\Omega) of rational
curves of
fixed degree d passing through Omega (a subvariety of the moduli
space of stable maps), and the space Gamma_d(\Omega) obtained by
taking
the union of these curves (a subvariety of the flag manifold). I will
show how some considerations about the geometry of these spaces leads
to a new, natural, proof of the equivariant quantum Chevalley formula
proved earlier by Fulton and Woodward and by the speaker. This is
joint
work with A. Buch.
- Speaker Dmitry Gourevitch, Rutgers University
- Title On associated variety, Whittaker functionals,
derivatives and rank
for representations of GL(n,R)
- Time/place 11/12/2010, Friday, 11:45 am in Hill 425
- Abstract There are several ways
of measuring size of representations of
reductive
groups. One of them is to attach to a representation a certain
subvariety
of the nilpotent cone of the group. Another way is to ask whether the
representation posseses functionals equivariant with respect to some
subgroups, in particular the nilradical of the minimal parabolic. Such
functionals are called Whittaker functionals. I will discuss those
notions
and report a recent result of ours that connects those notions in the
case
G=GL(n,R).
For unitary representations of GL(n,R) we also compute the associated
variety of the highest derivative or a representation. Based on our
results we suggest a new notion of rank of a representation,
which extends Howe's notion of rank.
This is work in progress, joint with Siddhartha Sahi.
- Speaker Stewart Wilcox, MIT
- Title Representations of rational Cherednik algebras
- Time/place 11/30/2010, Tuesday, 1:40 pm in Hill 124 (note
the special date, time and room)
- Abstract The rational Cherednik algebra was introduced
by Etingof and Ginzburg as a
degeneration of Cherednik's ``double affine Hecke algebra". It may
also be
viewed as a universal deformation of an algebra of differential
operators,
and
the type A rational Cherednik algebra is closely related to the
Calogero-Moser
space.
In this talk, after briefly introducing the algebra along the above
lines, I
will discuss results concerning the support sets of its
representations,
especially in the type A case.
- Speaker Avraham Aizenbud, MIT
- Title A quantum analogue of Kostant's theorem for the
general linear group
- Time/place 12/3/2010, Friday, 11:45 am in Hill 425
- Abstract A fundamental result in representation theory
is Kostant's
theorem which describes the algebra of polynomials on a reductive
Lie algebra as a module over its invariants.
We will review this result and prove a quantum analogue
of it for the general linear group.
- Speaker David Jordan, MIT
- Title Quantization of multiplicative quiver varieties
- Time/place 12/3/2010, Friday, 2:00 pm in Hill 425 (note the
special time)
- Abstract Quiver varieties, as introduced by Lusztig and
Nakajima, are
related to such diverse topics as Hilbert schemes, representations of
Hecke
algebras and semi-simple Lie algebras, canonical bases, and character
varieties of Riemann surfaces. Recently, Crawley-Boevey and Shaw
solved
the Deligne-Simpson problem by introducing so-called multiplicative
quiver
varieties and studying their algebro-geometric properties.
In this talk, we construct certain algebras $A^\lambda_d$, which
quantize
the symplectic structure on multiplicative quiver varieties. Our
construction involves Hamiltonian reduction by quantum group actions.
As
applications, we give a new description of the representation category
of
the spherical double affine Hecke algebra of type $A_{n-1}$ with
formal
parameters.
- Speaker Tony Milas, SUNY-Albany
|