Algebra and Algebraic Geometry Seminar Spring 2025

The seminar normally meets 2:30-3:30pm on Fridays, in the room Van Vleck B325.

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Spring 2025 Schedule

date	speaker	title	host/link to talk
January 31	Jakub Koncki (Warsaw)	SSM Thom polynomials of multisingularities	Laurentiu
February 28	Tamanna Chatterjee (Notre Dame)	Mautner’s cleanness conjecture	Josh
April 4	Sam Grushevsky (Stony Brook)	Maximal compact subvarieties of moduli spaces	Dima
April 11	Thomas Hameister (Boston College)	Relative Duality for Hitchin Systems	Josh/Dima
April 25	Toni Annala (UChicago)		Dreams and Nightmares in Intersection Theory	Josh

Note that on April 17-18 there is Singularities in the Midwest IX hosted by Max.

Abstracts

Jakub Koncki

SSM Thom polynomials of multisingularities

Thom polynomials are a tool used for understanding the geometry of singular loci of maps. To a singularity germ \eta we associate a polynomial in infinitely many variables. Upon substituting these variables with the Chern classes of the relative tangent bundle of a stable map, we obtain the fundamental class of the \eta-singular loci of the given map. Thom polynomials have several generalizations, including extension to multisingularities, and polynomials that compute other cohomological properties of the singular loci, such as the Segre-Schwartz-MacPherson class.

In the talk, I will review these concepts and focus on the SSM-Thom polynomials of multisingularities. I will present a structure theorem for them that generalizes earlier results of Kazarian.

The talk is based on a joint project with R. Rimányi.

Tamanna Chatterjee

Mautner’s cleanness conjecture

Let $G$ be a complex, connected, reductive algebraic group and $\mathcal{N}$ be the nilpotent cone of $G$. In 1985-1986, Lusztig proved a remarkable property of cuspidal perverse sheaves on $\mathcal{N}$. He proved when the sheaf coefficient has characteristic $0$, then the cuspidal perverse sheaves are “clean,” meaning that their stalks vanish outside a single orbit. This property is crucial in making character sheaves computable by an algorithm, and it plays an important role in “block decompositions” of the generalized Springer correspondence. About 10 years ago, Mautner conjectured that cuspidal perverse sheaves remain clean when the sheaf coefficient has positive characteristic (with some exceptions for small $p$). In this talk, I will discuss the history and context of the cleanness phenomenon, along with recent progress on Mautner’s conjecture. This is joint work with P. N. Achar.

Sam Grushevsky

Maximal compact subvarieties of moduli spaces

We present results on the maximal dimension of compact subvarieties of the moduli space of abelian varieties and of moduli of complex curves of compact type. Equivalently, this is the maximal dimension of a compact complex parameter space for a maximally varying family of abelian varieties/curves, etc. Based on joint work with Mondello, Salvati Manni, Tsimerman.

Thomas Hameister

Relative Duality for Hitchin Systems

The Hitchin fibration for a reductive group G is a certain generalized abelian fibration, which admits remarkable duality properties. Most famously, the Hitchin fibrations for G and the Langlands dual group G^\vee are generically dual abelian varieties. In this talk, we will explore a relative form of this duality, in the sense of Ben-Zvi, Sakellaridis, and Venkatesh. Namely,  we associate to a spherical variety or symplectic representation particular sheaves on the Hitchin moduli spaces for G and G^\vee which we conjecture to be Fourier-Mukai dual, generalizing a conjecture of Hitchin. We will show how this duality reduces to simple calculations in invariant theory. This is based on joint work with Zhilin Luo and Benedict Morrissey.

Toni Annala

Dreams and Nightmares in Intersection Theory

Intersection theory studies the Chow ring, an algebraic structure whose elements correspond to subvarieties of a variety X, and whose product corresponds to the intersection of subvarieties in X. Thanks to the foundational work of Fulton and MacPherson, this theory is well-understood when X is smooth. However, extending it to singular varieties remained mysterious for a long time. In this talk, I will explain the recent resolution of this problem, due to the work of Elmanto–Morrow and Kelly–Saito, and highlight some of the surprising obstacles encountered when approaching the problem naively.