Fall, 2010

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
    • Title Some distinguished vertex subalgebras of lattice vertex operator algebras
    • Time/place 12/10/2010, Friday, 11:00 pm in Hill 423 (note the special time and the special room)
    • Abstract I'll introduce certain subalgebras of lattice vertex operator algebras and discuss their structure, including combinatorial bases of modules. This part of the talk is based on M. Penn's PhD thesis at SUNY-Albany. In the second part we shall focus primarily on multiples of root lattices. Some (q,x)-series identities will be presented.

      This talk is primarily aimed at graduate students.