Purdue Mathematical Physics Seminar
In Spring 2026, the Purdue Mathematical Physics Seminar will be held on Tuesdays 10:30am - 11:30am Eastern time in MATH 215 (when we meet in person) unless otherwise noted. For Zoom meetings, use the link
https://purdue-edu.zoom.us/j/98588621708?pwd=9a21yDxedyTWrzbFhZPqbigKosV9Qc.1 (passcode: 808095)
SPRING 2026
March 3 (online)
Duncan Laurie (University of Edinburgh)
Zoom: to be provided
Title: Quantum toroidal algebras and their representations
Abstract: Quantum toroidal algebras Uq(g_tor) are the ‘double affine’ objects within the quantum setting. After reviewing and motivating the more well-studied finite and affine type quantum groups, we’ll introduce these algebras and outline a collection of new results. First, we shall construct double affine braid group and modular GL(2,Z) actions on Uq(g_tor). Then, using these, we will investigate the representation theory of quantum toroidal algebras, obtaining well-defined tensor products and (R-matrix) braidings for their principal module categories. Time permitting, I’ll mention work in progress with Théo Pinet which equips certain module subcategories with cluster algebra structures in type A, establishing generalised T-systems along the way.
March 13 (in-person)
Daniil Klyuev (Northwestern University)
Unusual time and location: MATH 631, 11am-noon
Title: q-characters in Yangian category O via the sphere trace
Abstract: I will talk about twisted traces on quantized quiver Coulomb branches (truncated shifted Yangians.) Any Verma module over a quantized Coulomb branch gives rise to a twisted trace. This trace contains the information about the q-character of the shifted Yangian acting on this module. For conical Coulomb branches there is also a trace introduced by Gaiotto and Okazaki ("sphere trace"). I will show how the sphere trace allows us to compute the q-character of certain simple objects in Yangian category O. Based on joint work in progress with Vasily Krylov.
March 24 (online)
Keyu Wang (University of Vienna)
Title: Langlands branching rule for representations of quantum affine algebras
Abstract: Two simple Lie algebras are called Langlands dual if their Cartan matrices are transposes of each other, because they appear as a pair in the Langlands correspondence. We are interested in the relationship between the representation theories of Langlands dual Lie algebras. In this talk, we first recall the character theory and introduce McGerty’s Langlands branching rule for finite-dimensional Lie algebras. This question naturally leads us to quantum groups. We then introduce the Frenkel–Hernandez conjecture for quantum affine Lie algebras. Finally, we briefly summarize results proved in joint work with Jingmin Guo and Jian-Rong Li. This talk assumes no background in quantum groups. Familiarity with Lie algebras will be helpful.
April 3 (in-person)
Matthew Harper (Michigan State University)
Unusual time and location: SCHM 103, 11am-noon
Title: Hopf ideals in quantum groups at roots of 1
Abstract: Quantum groups are a deformation of enveloping algebras of Lie algebras depending on a parameter q. When q is generic, their representation theory is very similar to that of classical Lie algebras, whereas at roots of unity it is closer to Lie algebras in finite characteristic. Motivated by questions of uniqueness and well-definedness of Reshetikhin--Turaev knot invariants for quantum groups at roots of unity, in this talk we investigate the structure of the ideals and subalgebras generated by power elements in the quantum group. We show that the choices made in their construction, specifically those related to choices of word in the Weyl group, depend only on the underlying group element and not the chosen word. These results extend some work of De Concini-Kac-Procesi on quantum groups at odd roots of unity. We discuss types A_n and B_2 as specific examples of the general theory and to formulate broader conjectures. If time permits, we discuss the quantum invariant associated to sl_3 at a fourth root of unity. An introduction to root systems, Weyl groups, and quantum groups will be given. Main results of this talk are based on joint work with Thomas Kerler.
April 7 (in-person)
Aatmun Baxi (Texas A&M)
Title: On a kind of local reflection product of modular tensor categories
Abstract: Modular tensor categories (MTCs) taxonomize the conjectured quasiparticles of topological quantum computing, where braiding is a fundamental unit of computation. In the project of classifying them, braided zesting and the tensor product over symmetric fusion categories have been used to realize new modular data—a numerical “shadow” of MTCs. Physically, the latter corresponds to layering topological phases (MTCs) with a local symmetry (a symmetric fusion subcategory), then breaking the larger symmetry by condensation. We discuss how this process relates to zesting and its success in producing new MTCs. In preliminary work, we introduce a relaxation of this construction, where local symmetries are replaced with an arbitrary local phase, i.e. a braided fusion subcategory, with particular interest on the abelian case.
April 14 (online)
Stefan Kolb (Newcastle University)
Title: Short star products for quantum symmetric pairs
Abstract: The theory of quantum symmetric pairs concerns coideal subalgebras of Drinfeld-Jimbo quantum groups which are quantum group analogs of Lie subalgebras fixed under an involution. A crucial ingredient in this theory is the quasi K-matrix which is an analog of the quasi R-matrix in the theory of quantum groups. The quasi K-matrix plays an important role in the construction of canonical bases, universal K-matrices and braid group actions for quantum symmetric pairs. In this talk I will give an introduction to the theory of quantum symmetric pairs. I will explain how the underlying coideal subalgebras can be interpretated as short star-product filtered quantizations. This perspective leads to a new construction of the quasi K-matrix. The talk is based on joint work with Milen Yakimov.
April 28 (in-person)
Pallav Goyal (UC Riverside)