Spring, 2012

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.