Lie Group/Quantum Math Seminar

Lie Group/Quantum Mathematics Seminar

Organizers Lisa Carbone, Yi-Zhi Huang, Jim Lepowsky and Siddhartha Sahi.

Time Friday, 12:10 pm to 1:10 pm (Eastern time).

Place Hill 705 or online via Zoom (see below for the Zoom link and passcode).

YouTube channel Rutgers Lie Groups Quantum Math Seminar.

Starting from Spring, 2008, the Lie Group Seminar and Quantum Mathematics Seminar have merged together to a single seminar called the Lie Group/Quantum Mathematics Seminar. The information on seminar talks can also be found in the Seminars & Colloquia Calendar page in the department. For the Lie Group/Quantum Mathematics seminar in previous semesters, see this page. For talks in the Quantum Mathematics Seminar from Spring, 1998 to Fall, 2007, see this page. For a few years before 2008, the Quantum Mathematics Seminar shared the time and place with the Algebra Seminar. For talks in both the Algebra and Quantum Mathematics Seminars in these few semesters, see the page for the Previous Rutgers Algebra Seminars. For all the seminars and colloquia in the department, see the Seminars & Colloquia Calendar page.

Fall, 2025

In this semester, the seminar will be mostly in person. Occasionally there might be online talks using zoom. See the information below on each talk. For online talks, here is the information for the zoom meeting:

Zoom link: https://rutgers.zoom.us/j/93921465287

Meeting ID: 939 2146 5287

Passcode: 196884, the dimension of the weight 2 homogeneous subspace of the moonshine module

Some of the talks will be recorded and will be placed in the YouTube Channel for the seminar.

Upcoming talks

  • Speaker Vasily Krylov, Harvard University
    • Title From geometric realizations of affine Hecke algebras to character formulas
    • Time/place 12/12/2024, Friday, 1:10 pm (Eastern Time), Hill 705 (in person but also accessible online through Zoom)
      Zoom link: https://rutgers.zoom.us/j/93921465287
      Meeting ID: 939 2146 5287
      Passcode: 196884, the dimension of the weight 2 homogeneous subspace of the moonshine module
    • Abstract I will explain an approach to extract character formulas for irreducible \hat{g}-modules at the positive level from the geometry of some open subvarieties of Steinberg varieties. The approach uses Bezrukavnikov’s “exotic coherent” categorification of the canonical basis of the affine Hecke algebra for g. We will discuss the first nontrivial example and will see that it leads to explicit character formulas for all irreducible modules (with integral highest weights) in categories O for certain Vertex algebras coming from the 4D/2D correspondence. Based on joint works with Bezrukavnikov, Kac, and Suzuki.

All scheduled talks

  • Speaker Lisa Carbone, Rutgers University
    • Title AI tools for advanced mathematics
    • Time/place 9/24/2024, Wednesday, 12:00 pm (Eastern Time but note that it is on Wednesday, not Friday), Hill 705 (in person but also accessible online through Zoom)
      Zoom link: https://rutgers.zoom.us/j/93921465287
      Meeting ID: 939 2146 5287
      Passcode: 196884, the dimension of the weight 2 homogeneous subspace of the moonshine module
    • Abstract The main drawback of using generative AI for advanced mathematics via Large Language Models (LLM) is that they are probabilistic pattern-matchers, not logical reasoning engines. However, LLMs can pick up on patterns in higher mathematics that are difficult for humans to see. By putting the design of LLMs to their advantage, mathematicians may use them as powerful interactive assistants that can carry out laborious tasks, generate and debug code, check examples, formulate conjectures and more. We discuss how LLMs can be used to advance mathematics research by careful use of prompt engineering. We also discuss the integration of LLMs with a formal proof assistant such as Lean.
  • Speaker Yi-Zhi Huang, Rutgers University
    • Title C_1-cofiniteness and vertex tensor categories
    • Time/place 10/3/2024, Friday, 12:10 pm (Eastern Time), Hill 705 (in person)
    • Abstract I will discuss my most recent result that for an arbitrary vertex operator algebra, or more generally, a grading-restricted Möbius vertex algebra V, (logarithmic) intertwining operators among C_1-cofinite grading-restricted generalized V-modules satisfy the associativity property (operator product expansion) and the category of C_1-cofinite grading-restricted generalized V-modules has natural vertex and braided tensor category structures. The proof and construction in this work is based on a generalization of the Huang-Lepowsky-Zhang logarithmic tensor category theory to the case that the category might not be closed under the contragredient functor. The result follows after the assumptions to use this generalization are all verified.
    • Slides (not used in the talk because of a sign-in problem with Adobe Acrobat) pdf file.
    • Archive paper arXiv:2509.20737
  • Speaker Manish Patnaik, University of Alberta
    • Title Affine and Metaplectic variations on the Casselman-Shalika formula
    • Time/place 11/14/2024, Friday, 12:10 pm (Eastern Time), Hill 705 (in person)
    • Abstract The original formula for Casselman-Shalika (and Shintani) was a computation from the late 1970s of a p-adic Whittaker function in terms of the Weyl characters. While many of the early applications of this formula were to the global theory of Eisenstein series, we will consider variants of this formula in some new local contexts, namely for p-adic Kac-Moody groups and metaplectic covers. In the metaplectic context, we will explain a connection, predicted by geometric Langlands, to quantum groups at roots of unity. In the context of affine Kac-Moody theory, we will describe some (partly conjectural) links to a Kazhdan—Lusztig theory for double affine Hecke algebras.

      This is a report on joint works with Valentin Buciumas, Yanze Chen, and Anna Puskas.
  • Speaker Darlayne Addabbo, The State University of New York Polytechnic Institute
    • Title Modularity of Vertex Operator Algebra Correlators with Zero Modes
    • Time/place 11/21/2024, Friday, 12:10 pm (Eastern Time), Hill 705 (in person)
    • Abstract Zhu famously established that correlation functions of elements in rational C_2-cofinite vertex operator algebras (VOAs) transform nicely under the action of SL_2(Z). In this talk, we will discuss modular transformation properties of VOA correlation functions with zero modes inserted. In particular, we will describe recursion relations satisfied by these correlators and explain how to use these to establish modular transformation properties. (This talk is based on joint work with Christoph A. Keller.)
  • Speaker Daniel Tan, Rutgers University
    • Title C_n-cofinite twisted modules for C_2-cofinite vertex operator algebras
    • Time/place 12/5/2024, Friday, 12:10 pm (Eastern Time), Hill 705 (in person)
    • Abstract There is a well-known finiteness condition for vertex operator algebras and their modules called C_n-cofiniteness. It was shown by Geoffrey Buhl that C_2-cofiniteness of a CFT-type vertex operator algebra is strong enough to guarantee that all of its finitely-generated weak modules are C_n-cofinite for all n > 0. In this talk, we will discuss the extension of Buhl's result to the twisted case, i.e. the orbifold CFT setting. Given a vertex operator algebra V with a general automorphism g of V, we introduce a notion of C_n-cofiniteness for weak g-twisted V-modules. When V is C_2-cofinite and of CFT type, we show that all finitely-generated weak g-twisted V-modules are C_n-cofinite for all n > 0. This talk is based on my work https://arxiv.org/abs/2510.26657.
    • Archive paper arXiv:2510.26657
  • Speaker Jishen Du, Rutgers University
    • Title Twisted intertwining operators, tensor products of twisted modules, and their associativities
    • Time/place 12/5/2024, Friday, 1:10 pm (Eastern Time), Hill 705 (in person)
    • Abstract Let V be a VOA and G a subgroup of Aut(V). A project jointly with Yi-Zhi Huang and Daniel Tan aims at constructing a G-crossed vertex/braided tensor category structure on the category of g-twisted V-modules for all g in G. After having a construction of a tensor product bifunctor jointly with Yi-Zhi Huang, one main step is to construct a natural associativity isomorphism map. Under certain convergence and extension assumptions, I have constructed such an associativity isomorphism. To achieve this, a new notion of twisted intertwining operator has been introduced jointly with Yi-Zhi Huang. Then, the existence of the associativity isomorphism is equivalent to the associativity of twisted intertwining operators, which have been proved under the assumptions mentioned above.
  • Speaker Vasily Krylov, Harvard University
    • Title From geometric realizations of affine Hecke algebras to character formulas
    • Time/place 12/12/2024, Friday, 1:10 pm (Eastern Time), Hill 705 (in person but also accessible online through Zoom)
      Zoom link: https://rutgers.zoom.us/j/93921465287
      Meeting ID: 939 2146 5287
      Passcode: 196884, the dimension of the weight 2 homogeneous subspace of the moonshine module
    • Abstract I will explain an approach to extract character formulas for irreducible \hat{g}-modules at the positive level from the geometry of some open subvarieties of Steinberg varieties. The approach uses Bezrukavnikov’s “exotic coherent” categorification of the canonical basis of the affine Hecke algebra for g. We will discuss the first nontrivial example and will see that it leads to explicit character formulas for all irreducible modules (with integral highest weights) in categories O for certain Vertex algebras coming from the 4D/2D correspondence. Based on joint works with Bezrukavnikov, Kac, and Suzuki.

Previous semesters