June 18th at 10:00 AM CST - Greyson Potter Title: The Witten-Reshetikhin-Turaev TQFT and Quantum Computing
Abstract: The Witten-Reshetikhin-Turaev (WRT) topological quantum field theory has deep connections with quantum topology, the representation theory of quantum groups, conformal field theory, geometric quantization of moduli spaces, 2+1-dimensional quantum gravity, and quantum computing. I will begin with an overview of its origin and various constructions, including Witten’s interpretation of the Jones polynomial through quantum Chern-Simons theory, Atiyah’s axiomatic formulation of TQFT, the combinatorial construction of Reshetikhin and Turaev using quantum groups, and the geometric-quantization description of the WRT state spaces in terms of moduli spaces of flat connections, developed by Axelrod–Della Pietra–Witten and Hitchin.
I will then explain the role of WRT TQFT in topological quantum computation. Following Kitaev’s proposal for fault-tolerant computation with anyons, Freedman, Larsen, and Wang showed that suitable WRT modular functors yield universal quantum computation, with information encoded in fusion spaces and gates implemented by braiding. Finally, I will describe how this relationship can be reversed: rather than using a TQFT to build a quantum computer, one can use the standard quantum circuit model to develop quantum algorithms for approximating WRT invariants.
May 28th at 10:00 AM CST - David Sun Title: p-adic integration and BPS-invariants of character stacks
Abstract: In this talk, I will give a brief overview of p-adic integration, motivated by a series of results of Groechenig, Wyss, and Ziegler. In particular, I will introduce their BPS-invariants for “nice” moduli stacks, which is connected to Donaldson–Thomas theory. I will then describe a method for evaluating BPS-invariants of G-character stacks for reductive groups G. Finally, I will present some small computational results, including the BPS-invariant of the GL2-character stack matches the E-polynomial of the twisted GL2-character variety computed by Hausel and Rodríguez-Villegas, as well as a possible mirror symmetry statement relating the SL2-character stack and the twisted PGL2-character stack.
May 14 at 10:00 AM CST - Wennan Zhang Title: Tangency Operations in Free Loop spaces
Abstract: In this talk, I will discuss a string-topology construction on the free loop space LM of a smooth manifold M, using a chosen nowhere-vanishing vector field v. The basic idea is to study loops whose tangent direction agrees with the direction of M. I will first explain the ordinary tangency condition, then introduce the positive tangency refinement.
This construction is partly motivated by open holomorphic curve theory, where boundary loops in a Lagrangian interact with chosen nowhere-vanishing vector fields in four-chain and linking constructions. I will describe the resulting chain-level operations and explain why the positive tangency operator becomes a chain map when M is odd-dimensional.
April 30th at 10:00AM CST - Kuntal Banerjee Title: Co-Higgs Bundles of Schwarzenberger Type and the Determinant Morphism
Abstract: We identify images of the determinant morphism of trace-free co-Higgs bundles modelled on rank 2 Schwarzenberger bundles.
April 16th at 10:00AM CST - Shomrik Bhattacharya Title: Fundamental Group of the Complement of the Wobbly Divisor in the Intersection of Two Quadrics
Abstract: We study the fundamental group of the complement of the ‘wobbly’ divisor in the moduli of semistable vector bundles of rank two and degree one on a genus two curve. We construct an exact sequence that will help us to calculate the fundamental group. This exact sequence of fundamental groups concerns spaces in any arbitrary dimension. We devise a method to calculate the fundamental group in the least possible dimension, i.e., quadrics in dimension 3. Here, it is easy to calculate all the fundamental groups involved. Then, we extrapolate this method to higher dimensions. I briefly recall some facts about braid groups, which naturally arise as the fundamental groups of the spaces involved. I prove that the fundamental group of one of the spaces involved is a mixed braid group.
April 2nd at 9:00AM CST - Eric Boulter Title: Co-Higgs bundles and Poisson structures
Abstract: In 1997, Polischuk described a family of holomorphic Poisson manifolds that can be constructed from rank-2 co-Higgs bundles on some compact complex manifold $X$. We will describe this construction, investigate the structure of symplectic foliations on these Poisson manifolds, and use the classification of rank-2 co-Higgs bundles on a Hopf surface to study some examples. Partially based on joint work with Ruxandra Moraru.
March 19th at 11:00AM CST - Yuki Matsubara Title: On a certain tamely ramified geometric Langlands correspondence
Abstract: The geometric Langlands correspondence (GLC) is a geometric analogue of the Langlands conjecture in number theory, relating algebraic geometry, representation theory, and many other areas. Since A. Kapustin and E. Witten pointed out the relation between GLC and mirror symmetry, there have been various studies of GLC from both physics and mathematics.
Dima Arinkin's 2001 result established the geometric Langlands correspondence for the case G = SL2 on the complex projective line P1 with four fixed regular singularities.
When one attempts to extend this to five or more singularities, it turns out to be more natural to decompose the correspondence into a Radon transform-type correspondence and a "GLC-like" correspondence. In this talk, I will explain the calculations of cohomology that support the proof of this GLC-like correspondence in the P1 with five fixed regular singularities case.
This talk is based on my paper Cohomology of vector bundles on the moduli space of parabolic connections on P1 minus 5 points (https://arxiv.org/abs/2510.12578).
March 5th at 9AM CST - Aidan Lindberg Title: Hochschild homology for derived stacks
Abstract: Hochschild and cyclic homology are invariants of stable categories which generalize the notion of differential forms and de Rham cohomology to noncommutative geometry. In this talk I will recall the classical definition of cyclic homology (and related invariants) and present a generalization of this definition which makes sense for derived stacks. As an application of this construction, I will demonstrate a new way to calculate the de Rham cohomology of a higher Lie groupoid.
February 19th at 9AM CST - Robert Cornea Title: Stable Wild Vafa-Witten Pairs on P^2
Abstract: In this talk, we consider wild Vafa-Witten Pairs, a generalization of Vafa-Witten-Higgs bundles on Kahler surfaces. Over P^2, such a pair (E,\Phi) consists of a holomorphic vector bundle E and a morphism \Phi : E \to E \otimes O(d) for some d \geq 0 called a Higgs field. We focus on the case where E is a Schwarzenberger bundle of type \ell. After describing these bundles for \ell=1,2, we show that for d=\ell, via the spectral correspondence, any stable wild Vafa-Witten pair (E,\Phi) with smooth spectral cover must have E a Schwarzenberger bundle of type \ell. Moreover, for d=\ell=1, the moduli space of such pairs is smooth with expected dimension. If time permits, I will discuss the structure of the component of the moduli space of stable wild Vafa-Witten pairs for arbitrary d>0 containing Schwarzenberger bundles of type \ell=1, showing that it remains smooth with the expected dimension.
February 5th at 11AM CST - Paul Cusson Title: Spectral curves of Euclidean SU(N)-monopoles
Abstract: Monopoles over Euclidean R^3 with gauge group SU(N), originally analytic objects, can be studied using the algebro-geometric properties of their spectral curves. We will first do an overview of the subject, then discuss how the spectral curves depend on the Higgs field of the monopole. We will also describe a correspondence between a certain moduli space of monopoles and flag bundles over CP^1. Time permitting, we will quickly go over some elementary results restricting the degrees of the spectral curves when we impose symmetries on these monopoles from finite subgroups of SO(3).
January 22nd at 11AM CST - Mahmud Azam Title: Non-Abelian Hodge Theory for Moduli Stacks
Abstract: The complex non-Abelian Hodge correspondence takes two main forms: a homeomorphism between
the moduli spaces of polystable Higgs bundles and of semisimple flat connections, and a
quasiequivalence between the differential graded categories of semistable Higgs bundles and of flat
connections. In this talk, we will discuss some work in progress towards unifying these results
through moduli stacks. First, we will define moduli stacks of the respective objects over the site
of smooth manifolds to enable the analysis needed for non-Abelian Hodge theory. Next, in analogy to
our recent work on quiver connections (arXiv:2512.12188) as well as work of Porta and Sala (section
4.1 in arXiv:1903.07253), we will define, in the smooth setting, moduli stacks parametrizing
diagrams of Higgs bundles and those of flat connections. Taking the stacks parametrizing diagrams
indexed by the standard simplices give category stacks (or, 2--Segal objects in the derived setting)
of the respective objects. Finally, we will discuss a tentative non-Abelian Hodge correspondence for
these category stacks or 2--Segal objects. All original work is joint with Dr. Steven Rayan.