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) | TBA | Dima |
| April 11 | Thomas Hameister (Boston College) | TBA | Josh/Dima |
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.