Fall, 2014

Fall, 2014

  • Speaker Christoph Keller, Rutgers University, Department of Physics and Astronomy
    • Title Non-rational VOAs and modular invariant partition functions
    • Time/place 9/5/2014, Friday, 12:00 in Hill 425 (note special room)
    • Abstract From the work of Zhu it is known that the characters of the representations of rational VOAs (under some weak additional assumptions) form representations of the modular group SL(2,Z). Physicists use this fact to obtain modular invariant functions, the so-called partition functions, as sesquilinear combinations of those characters. In the case of the Virasoro minimal models this leads to the ADE classification of modular invariant partition functions. I will discuss the situation for non-rational Virasoro theories and present some first steps towards a classification of partition functions in that case.

  • Speaker Shashank Kanade, Rutgers University
    • Title Recent developments in partition identities and the representation theory of vertex algebras
    • Time/place 9/12/2014, Friday, 12:00-12:30 (note special time) in Hill 705
    • Abstract Ever since Lepowsky-Wilson's remarkable vertex-operator-theoretic proof of the classical Rogers-Ramanujan identities, the area of ``algebraic combinatorics'' relating partition identities to the representation theory of vertex algebras has witnessed an explosion of ideas. In this introductory survey talk, I will discuss some recent and exciting developments in this area. Specifically, I will give an overview of some (new) generalizations of Andrews-Baxter's ``motivated proof'' of the Rogers-Ramanujan identities and their connections to the representations of certain vertex algebras. I will also explain how ``experimental mathematics'' is shaping the landscape. Most of the talk is based on a recent joint work with J. Lepowsky, M. C. Russell and A. V. Sills.

  • Speaker Francesco Fiordalisi, Rutgers University
    • Title Towards a proof of the modular invariance for logarithmic intertwining operators
    • Time/place 9/22/2014, Monday, 1:00 pm in graduate student lounge (note special time and room)
    • Abstract The full modular invariance conjecture of Moore and Seiberg for intertwining operators in the rational case was proved by Huang in 2003, An analogous conjecture in the logarithmic case was proposed by Huang a few years ago. In this talk, I will discuss my recent progress towards a proof of this conjecture.

  • Speaker Debajyoti Nandi, Rutgers University
    • Title Partition identities arising from the standard A2(2)-modules of level 4
    • Time/place 9/26/2014, Friday, 12:00 in Hill 705
    • Abstract In this talk, I will present a new set of (proposed) partition identities arising from the standard modules of level 4 for the affine Lie algebra A2(2) using a twisted vertex operator construction, and strong evidence for their validity. This talk is based on my recent work which is a continutation of a long line of research of investigating and discovering surprising interplay between representation theory and combinatorial/paritition identities using vertex-algebraic ideas and techniques. This was first exemplified by the vertex-operator-theoretic proof of the Rogers-Ramanujan-type identities using the standard A1(1)-modules by J. Lepowsky-R. Wilson. In his PhD thesis, S. Capparelli proposed new combinatorial identities using a twisted vertex operator construction of level 3 standard A2(2)-modules, which were later proved independently by G. Andrews, Capparelli, and M. Tamba-C. Xie. The level 4 case for A2(2) shows surprising new phenomena that were absent in previously known examples of this type.

  • Speaker Emily Leven, University of California at San Diego
    • Title Combinatorial Aspects of the rational Shuffle Conjecture
    • Note Joint Experimental Math/Lie Group/Quantum Math Seminar
    • Time/place 10/3/2014, Friday, 12:00 in Hill 705
    • Abstract In this talk, we will review the history and recent extensions of the Classical Shuffle Conjecture. This conjecture equates two symmetric polynomials, one of which is known to give the Frobenius characteristic of the space $DH_n$ of diagonal harmonics. The other side of the conjecture is purely combinatorial, showing the remarkable ability of certain symmetric function operators to control combinatorial objects, such as Dyck paths and parking functions. This branch of algebraic combinatorics was created to explore the representation-theoretical aspects of Macdonald polynomials. This led to the $n!$ conjecture and the introduction of the space of Diagonal Harmonics. A program outlined by Procesi led to the proof by Mark Haiman of the $n!$-conjecture by algebraic geometrical tools. There has recently been a flood of new operators and conjectures created, in our subject, by algebraic geometers. This talk covers some of the new results and conjectures obtained by a continuing effort to translate these developments back into the original algebraic-combinatorial setting. Our presentation should be accessible to the general mathematical audience.

  • Speaker Evan Wilson, Rutgers University
    • Title Tensor product decomposition of sl_n^-modules and generating series
    • Time/place 10/17/2014, Friday, 12:00 pm in Hill 705
    • Abstract In this talk, we describe recent joint work with Kailash Misra on decomposing the tensor product of two level one modules of the affine Kac-Moody algebra sl_n^, using the crystal basis of quantum U(sl_n^) of Misra and Miwa and some well-known graded dimension formulas. In the process, we uncover some generating series for partitions whose parts satisfy certain conditions.

  • Speaker Alex Kemarsky, Technion, Israel
    • Title Gamma factors of GL_n(R)-distinguished representations of GL_n(C)
    • Time/place 10/24/2014, Friday, 1:45 in Hill 425 (note special time and room)
    • Abstract An irreducible representation (\pi,V) of GL_n(C) is called GL_n(R)-distinguished if there exists a non-zero continuous GL_n(R)-invariant functional L:V \to C. In the talk we give a necessary condition for GL_n(R)-distinction. As a corollary, we prove that the Rankin-Selberg gamma factors of \pi \times \pi' at s=1/2 for \pi,\pi' distinguished representations of GL_m(C),GL_n(C) respectively equals 1.

  • Speaker Siddhartha Sahi, Rutgers University
    • Title The Macdonald conjectures for multivariate hypergeometric functions
    • Time/place 11/7/2014, Friday, 12:00 in Hill 705
    • Abstract TBA

  • Speaker Andrey Minchenko, Weizmann Institute, Rehovot, Israel
    • Title Finite multiplicity theorem for spherical pairs
    • Time/place 11/17/2014, Monday, 3:00 (note special day and time) in Hill 705
    • Abstract Let X be a spherical space for a real reductive group G. Recently, Kobayashi and Oshima, and independently Kroetz and Schlichtkrull have obtained results on boundedness of multiplicities of irreducible representations in the space of functions on X. We will consider another proof of these results, which seems to be shorter. One of the main steps is to show that the singular support of a certain distribution on G (spherical character) is a Lagrangian in the cotangent bundle of G. We will also use some non-trivial facts about D-modules and Springer resolution. Another advantage of this approach is the possibility for generalization to the p-adic case. The talk is based on a joint work of the speaker with A. Aizenbud and D. Gourevitch.

  • Speaker Hadi Salmasian, University of Ottawa
    • Title Spherical polynomials and the spectrum of invariant differential operators for the symmetric superpair GL(m,2n)/OSp(m,2n)
    • Time/place 12/5/2014, Friday, 12:00 in Hill 705
    • Abstract The algebra of invariant differential operators on a multiplicity-free representation of a reductive group has a concrete basis, usually referred to as the Capelli basis. The spectrum of the Capelli basis on spherical representations results in a family of symmetric polynomials (after \rho-shift) which has been studied extensively by Knop and Sahi since the early 90's. In this talk, we generalize some of the Knop-Sahi results to the symmetric superpair GL(m,2n)/OSp(m,2n). As a side result, we show that the qualitative Capelli problem (in the sense of Howe-Umeda) for this superpair has an affirmative answer. This talk is based on an ongoing project with Siddhartha Sahi.