Spring, 2015

Spring, 2015

  • Speaker Yi-Zhi Huang, Rutgers University
    • Title Vertex operator algebras, fractional quantum Hall states and topological orders
    • Time/place 1/30/2015, Friday, 12:00 in Hill 705
    • Abstract In Landau's symmetry-breaking theory, different orders in states of matter correspond to different symmetries (or more precisely, the breaking of symmetries). Fractional quantum Hall states give a new type of orders called topological orders that cannot be described by symmetries or symmetry-breaking. In this talk, using the constructions of wavefunctions for fractional quantum Hall states by Moore and Read as examples, I will propose that topological orders in $2+1$ dimension can in fact also be viewed as orders corresponding to "symmetries" that are described not by groups but instead by vertex operator algebras. In particular, all the conceptual and powerful mathematical methods and results in the representation theory of vertex operator algebras can be applied to the study of fractional quantum Hall states and topological orders in $2+1$ dimension.

  • Speaker Vladimir Retakh, Rutgers University
    • Title Noncommutative triangulations
    • Time/place 2/6/2015, Friday, 12:00 in Hill 705
    • Abstract The celebrated Ptolemy relation plays an important role in various studies of triangulated surfaces including hyperbolic geometry, geometrical applications of cluster algebras and so on. We will discuss a noncommutative version of the relation which can be seen as a "categorification" of the classical one. This leads to new noncommutative invariants of the surfaces and provides several examples of the noncommutative Laurent phenomenon answering some questions by Kontsevich. (Joint work with Arkady Berenstein from University of Oregon)

  • Speaker Semeon Artamonov, Rutgers University
    • Title Noncommutative Inverse Scattering Method for the Kontsevich system
    • Time/place 2/13/2015, Friday, 12:00 in Hill 705
    • Abstract In my talk I will formulate an analog of Inverse Scattering Method for integrable systems on noncommutative associative algebras. In particular I will define Hamilton flows, Casimir elements and noncommutative analog of the Lax matrix. The noncommutative Lax element generates infinite family of commuting Hamilton flows on an associative algebra. The proposed approach to integrable systems on associative algebras satisfy certain universal property, in particular it incorporates both classical and quantum integrable systems as well as provides a basis for further generalization.

      The motivation for definition will be given by explicit construction of noncommutative analog of Lax matrix for a system of differential equations on associative algebra recently proposed by Kontsevich. First these equations will be presented in the Hamilton form by defining a bracket of Loday type on the group algebra of the free group with two generators. To make the definition more constructive I will utilize (with certain generalizations) the Van den Bergh approach to Loday brackets via double Poisson brackets. Finally, it will be shown that there exists an infinite family of commuting flows generated by the noncommutative Lax element.

  • Speaker Siddhartha Sahi, Rutgers University
    • Title Generalized Whittaker functionals for real reductive groups
    • Time/place 2/20/2015, Friday, 12:00 in Hill 705
    • Abstract Let (\pi,V) be a smooth representation of a real reductive group G. A generalized Whittaker functional is a linear functional on V that transforms by a character of a suitable unipotent subgroup of G. Such functionals play an important role in various applications, especially in the study of automorphic forms. In this talk I will describe a number of recent results obtained in this direction.

  • Speaker Siddhartha Sahi, Rutgers University
    • Title Generalized Whittaker functionals for real reductive groups, Part II
    • Time/place 2/27/2015, Friday, 12:00 in Hill 705
    • Abstract Let (\pi,V) be a smooth representation of a real reductive group G. A generalized Whittaker functional is a linear functional on V that transforms by a character of a suitable unipotent subgroup of G. Such functionals play an important role in various applications, especially in the study of automorphic forms. In this second talk I will continue to describe a number of recent results obtained in this direction.

  • Speaker Robert McRae, Peking University
    • Title On the braided tensor category of standard affine Lie algebra modules at fixed non-negative integral level
    • Time/place 3/6/2015, Friday, 12:00 in Hill 705
    • Abstract For a simple Lie algebra g, Huang and Lepowsky showed that the category of standard modules for the affine Lie algebra g^ at a fixed non-negative integral level is a braided tensor category. This category of g^-modules is the category of modules for a certain simple vertex operator algebra, and Frenkel and Zhu showed that this category is equivalent to the category of finite-dimensional modules for a certain quotient of U(g). In this talk, I will explain this equivalence between the category of fixed-level standard affine Lie algebra modules and the corresponding subcategory of finite-dimensional g-modules, and I will describe how the subcategory of finite-dimensional g-modules inherits braided tensor category structure from the category of affine Lie algebra modules. The most interesting problem is to describe the (non-trivial) associativity isomorphisms on triple tensor products of g-modules; these isomorphisms turn out to be related to "Drinfeld associator" isomorphisms coming from the KZ equations.

  • Speaker Pierre Cartier, IHES
    • Title From the exponential mapping to Witt vectors through jet spaces; a tale of categories
    • Time/place 3/6/2015, Friday, 2:30 in Hill 705 (note the special time)
    • Abstract

  • Speaker Shashank Kanade, Rutgers University
    • Title Some Results on the Representation Theory of Vertex Operator Algebras and Integer Partition Identities
    • Time/place 4/9/2015, Thursday, 2:00 in Hill 525 (note the special time)
    • Abstract Ph.D. thesis defense.

  • Speaker Francesco Fiordalisi, Rutgers University
    • Title Logarithmic Intertwining Operator and Genus-One Correlation Functions
    • Time/place 4/9/2015, Thursday, 3:30 in Hill 525 (note the special time)
    • Abstract Ph.D. thesis defense.

  • Speaker Alex Kontorovich, Rutgers University
    • Title On Lusztig's Conjecture
    • Time/place 4/10/2015, Friday, 12:00 in Hill 705
    • Abstract We will attempt to describe the problem in the title, and some recent work on the problem by Geordie Williamson.

  • Speaker Misha Ershov, University of Virginia
    • Title Tarski numbers of groups
    • Time/place 5/1/2015, Friday, 12:00 in Hill 705
    • Abstract A group G is said to admit a paradoxical decomposition if it can be represented as a disjoint union of finitely many subsets A_1,.., A_n and B_1,..., B_m such that for some elements g_1,..., g_n and h_1,...,h_m of G, the translates g_i A_i cover the entire G, and the same is true for the translates h_j B_j. Such decompositions are ``responsible'' for the Banach-Tarski paradox and other related phenomena. It is well known that G admits a paradoxical decomposition if and only if G is non-amenable, in which case the smallest possible number of pieces in such a decomposition of G is called the Tarski number of G.

  • Speaker Eugene Gorsky, Columbia University
    • Title Stable Khovanov-Rozansky homology of torus knots
    • Time/place 5/5/2015, Tuesdayday, 12:00 in Hill 705 (note special date)
    • Abstract The Jones and HOMFLY-PT polynomials of torus knots were computed by Rosso and Jones in the beginning of 90's. The computation of their "categorified" versions (Khovanov and Khovanov-Rozansky homology) remains an important open problem in knot theory. In the talk, I will explain some recent results and conjectures describing the stable limit of these homologies in terms of a certain explicit Koszul complex. If both parameters of a torus knot tend to infinity, there is a surprising relation to generalized Gordon and Rogers-Ramanujan identities.