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.