Fall, 2023

Summer, 2024

  • Speaker Joseph Bernstein, Tel Aviv University
    • Title Groups, Groupoids, Stacks and Representation Theory
    • Time/place 6/11/2024, Tuesday, 11:00 am (Eastern Time), Hill 705 (in person)
    • Abstract Let G be a group and k some field. A representation of the group G over the field k is usually defined as a vector space V over k equipped with a linear action of the group G. One of important problems in Representation Theory is the study of the category Rep(G) of such representations.

      In my talk I will explain that there is another natural way to describe this category, namely, it can be described as a category of sheaves on some "geometric" object - the basic groupoid BG of the group G.

      I will explain that this more sophisticated categorical description is the more "correct" one. For example, this new description gives a more adequate description of the category Rep(G) in cases when we have additional structures (algebraic groups, some extra symmetries and so on).

      I will show that in many cases the category Rep(G) has a natural extension - the category M(G) - that has better behavior.

      My talk will partially follow my paper in arXiv:1410.0435.

  • Speaker Dmitry Gourevitch, Weizmann Institute
    • Title Hypergeometric orthogonal polynomial families
    • Time/place 6/11/2024, Tuesday, 1:00 pm (Eastern Time), Hill 705 (in person)
    • Abstract Motivated by the theory of hypergeometric orthogonal polynomials, we consider quasi-orthogonal polynomial families - those that are orthogonal with respect to a non-degenerate bilinear form defined by a linear functional - in which the ratio of successive coefficients is given by a rational function f(u,s) which is polynomial in u. We call this a family of Jacobi type.

      Our main result is that, up to rescaling and renormalization, there are only five families of Jacobi type. These are the classical families of Jacobi, Laguerre and Bessel polynomials, and two more one parameter families $E^c, F^c$. Each family arises as a specialization of some hypergeometric series. The last two families can also be expressed through Lommel polynomials, and they are orthogonal with respect to a positive measure on the real line for c>0 and c>-1 respectively.

      We also consider the more general rational families, i.e. quasi-orthogonal families in which the ratio f(u,s) of successive coefficients is allowed to be rational in u as well. I will formulate the two main theorems, one on Jacobi families and one on rational families, as well as the main ideas of the proofs. This is joint work with Joseph Bernstein and Siddhartha Sahi.

Spring, 2024

  • Speaker Shaobin Tan, Xiamen University
    • Title Toroidal Extended Affine Lie Algebras and Integrable Representations
    • Time/place 1/26/2024, Friday, 12:10 pm (Eastern Time), Hill 705 (in person)
    • Abstract The extended affine Lie algebras (EALAs for short) are generalization of the finite dimensional simple Lie algebras and affine Kac-Moody algebras over the field of complex numbers, and the toroidal EALAs are a class of the most important EALAs. In this talk, we deal with the classification of irreducible integrable representations for the elliptic Lie algebras, i.e. the toroidal EALAs of nullity two.

  • Speaker Terence Coelho, Rutgers University
    • Title New perspectives on fixed point subalgebras of affine Lie algebras corresponding to Dynkin diagram automorphisms
    • Time/place 3/20/2024, Wednesday, 11:45 am, Hill 705 (in person)
    • Abstract We present new realizations of the simply-laced affine Lie algebras that do not distinguish any node in the affine Dynkin diagram, which allows one to more easily study the fixed-point subalgebras corresponding to affine Dynkin diagram automorphisms. We show how these fixed-point subalgebras compare to the associated non-simply-laced affine Lie algebras, and present an alternative classification of affine root systems and affine Lie algebras from this perspective. Finally, we discuss the corresponding realizations of the level-one irreducible highest weight modules for a simply-laced affine Lie algebra and how the affine Dynkin diagram automorphisms can be lifted to their direct sum.

  • Speaker Lea Beneish, University of North Texas
    • Title Moonshine modules and a question of Griess
    • Time/place 3/22/2024, Friday, 12:10 pm (Eastern Time), Zoom link above (online)
    • Abstract Recent work on monstrous moonshine has shown that there are exact formulas for the multiplicities of the irreducible components of the moonshine modules, showing in particular that these multiplicities are asymptotically proportional to the dimensions. With the recent proof of the umbral moonshine conjecture it is natural to ask whether this distribution result extends to other instances of moonshine, including umbral moonshine. We consider the general situation in which a finite group acts on an infinite-dimensional graded module in such a way that the graded-trace functions are weakly holomorphic modular forms. Under some mild hypotheses we completely describe the asymptotic module structure of the homogeneous subspaces. As a consequence, we find that moonshine for a group gives rise to partial orderings on its irreducible representations. This serves as a first answer to a question posed by Griess. This talk is based on joint work with Victor Manuel Aricheta.

  • Speaker Hofie Hannesdottir, Institute for Aavanced Study
    • Joint with Mathematical Physics Seminar
    • Title Mathematical properties of scattering observables
    • Time/place 3/28/2024, Thursday, 12:10 pm (Eastern Time), Hill 705 (in person)
    • Abstract Scattering observables are at the core of discoveries in particle physics, providing a bridge between theoretical computations and experiments in particle colliders. In this talk, we will discuss recent advances on the mathematical properties of scattering amplitudes and other observables. In particular, we will explore intricate ways in which physical principles leave imprints on the complex analytic structure of these observables.

  • Speaker Hong Chen, Rutgers University
    • Title Symmetric polynomials and interpolation polynomials
    • Time/place 4/3/2024, Wednesday, 11:45 am, Computing Research & Education Building (CoRE), Room 433 (note the special day, time and room)
    • Abstract Symmetric polynomials, for example, Schur, Jack, and Macdonald polynomials, are classical objects in the study of algebra, representation theory, and combinatorics. Interpolation polynomials are certain inhomogeneous versions of the classical Jack and Macdonald polynomials. In this talk, I will first review some basics of symmetric polynomials, introduce the interpolation polynomials, and discuss our recent work. As an application, I will talk about a characterization of the containment partial order in terms of Schur positivity or Jack positivity. This result parallels the works of Cuttler--Greene--Skandera, Sra, and Khare--Tao, which characterize two other partial orders in terms of Schur positivity.

      This work is joint with my advisor, Prof. Siddhartha Sahi, and available on arXiv:2403.02490.

  • Speaker Eric Schippers, University of Manitoba
    • Title The sewing operation for the determinant line bundle over the Segal moduli space
    • Time/place 4/5/2024, Friday, 12:10 pm (Eastern Time), Hill 705 (in person)
    • Abstract Segal considered a determinant line bundle on Riemann surfaces with boundary parametrizations, which plays an important role in two-dimensional conformal field theory. This is the determinant line bundle of the operator pi which takes a holomorphic function on the Riemann surface to the negative Fourier modes of the boundary values of this function. Segal sketched a definition of a sewing operation for this determinant line.

      This talk is about some of the interactions between geometry and analysis -- particularly Teichmuller theory -- and the determinant line bundle together with its sewing operation. We also discuss open problems.

  • Speaker Hao Zhang, Tsinghua University
    • Title Conformal blocks for C_2-cofinite vertex operator algebras
    • Time/place 4/19/2024, Friday, 12:10 pm (Eastern Time), Hill 705 (in person)
    • Abstract Vertex operator algebras (VOAs) mathematically characterize 2d conformal field theories. For C_2-cofinite and rational VOAs (plus some minor assumptions), Yi-Zhi Huang proved that their representation categories are rigid modular tensor categories. For C_2-cofinite (not necessarily rational) VOAs, Huang-Lepowsky-Zhang proved that their representation categories are braided tensor categories, which mean that the rigidity and modularity are still open problems. Geometrically, one of the main reasons is that, for C_2-cofinite VOAs, their conformal blocks associated to compact Riemann surfaces of genera 1 were not clear at that time. In 2023, Yi-Zhi Huang proved that such conformal blocks can be expressed by pseudotraces of genus 0 conformal blocks (intertwining operators). This result is expected to be used to prove rigidity (and hence modularity).

      In this talk, I will introduce a systematic approach towards higher genus conformal blocks for C_2-cofinite VOAs and explain the relationship between our work and Yi-Zhi Huang’s results above. This is based on an ongoing project (arXiv: 2305.10180) joint with Bin Gui.

  • Speaker Dennis Hou, Rutgers University
    • Joint with Algebra Seminar
    • Title Formal Colombeau theory
    • Time/place 4/24/2024, Wednesday, 2:00 pm (Eastern Time), Hill 705 (in person)
    • Abstract The vector space of doubly infinite formal Laurent series does not form a ring under the usual multiplication. In the theory of vertex operator algebras, a partially defined associative product is developed in terms of formal calculus. We consider a nonstandard approach to embedding this space into a differential algebra.

  • Speaker Jishen Du, Rutgers University
    • Title Twisted intertwining operators and tensor products of (generalized) twisted modules
    • Time/place 4/26/2024, Friday, 12:10 pm (Eastern Time), Hill 705 (in person)
    • Abstract It is known that the notion of vertex/intertwining operator defined using formal variables and Jacobi identity has an equivalent definition using complex analytic approach. When studying orbifold theory, even for one twisted intertwining operator among twisted modules, it is necessary to use the complex analytic approach. Using the complex analytic approach, we introduce a more general notion of twisted intertwining operator, and prove basic properties. We introduce a notion of P(z)-tensor product of two twisted modules and give a construction of such a P(z)-tensor product under suitable assumptions. We formulate a P(z)-compatibility condition and a P(z)-grading-restriction condition and used these conditions to give another construction of the P(z)-tensor product, which is expected to be useful to construct a G-crossed braided tensor category from a VOA and a group G of automorphisms of the VOA.

      This work is joint with my advisor, Prof. Yi-Zhi Huang.

  • Speaker Forrest Thurman, Rutgers University
    • Title Integration on flags of lattices using Maass-Selberg relations
    • Time/place 4/26/2024, Friday, 2:00 pm (Eastern Time), Hill 705 (in person)
    • Abstract We use the Maass-Selberg relations for the Borel Eisenstein series to study integrals of functions of covolumes of flags in random lattices. This work can be seen as a generalization of Siegel's integration formula which says that a function on R^n, n>1 can be integrated by taking the expected value of the sum of the function over the lattice points of a random covolume one lattice in R^n. Since the Maass-Selberg relations hold for arbitrary split real lie groups, we are able to compute integration results for functions of flags of random symplectic lattices as well.