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SS 2024

17.04.2024 um 15:00 Uhr in 69/125

Dr. Anna Marouschka Viergever (Leibniz Universität Hannover)

The $\mathbb{A}^1$-Euler characteristic of the symmetric powers of curves and linear varieties.
Abstract: To any smooth projective scheme over a field which is not of characteristic two, one can assign its $\mathbb{A}^1$-Euler characteristic, which is a quadratic form constructed using motivic homotopy theory. These forms carry a lot of information inside of them and are often used in the fast-growing field of refined enumerative geometry. Work of Arcila-Maya, Bethea, Opie, Wickelgren and Zakharevich extends the $\mathbb{A}^1$-Euler characteristic to all varieties over the base field in characteristic zero. It is in general hard to compute $\mathbb{A}^1$-Euler characteristics, and at the moment, there is no general formula for quotient schemes under a group action. In this talk, I will discuss some recent progress on the case of a symmetric power. I will talk about a joint work with Lukas Br\"oring, in which we calculate the $\mathbb{A}^1$-Euler characteristic of the symmetric powers of curves using the motivic Gauss-Bonnet Theorem of Levine-Raksit. I will also discuss joint work with Jesse Pajwani and Herman Rohrbach, in which we show that in characteristic zero, one can calculate the symmetric powers of a large class of varieties (which we call ``linear varieties") using the power structure which was introduced by Pajwani and Pál.

19.06.2024 um 14:00 Uhr in 69/E23

Egor Zolotarev (LMU München)

Geometric part of the algebraic special linear cobordism

In his ICM address on motivic homotopy theory Voevodsky introduced the algebraic cobordism spectrum MGL, mimicking construction of the Thom spectrum in topology. He also proposed a conjecture that the geometric diagonal of the coefficient ring of this spectrum should have a nice description as the coefficient ring of the universal formal group law. This was proven over fields of characteristic zero by Morel, Hopkins, and Hoyois. In this talk I will explain computation of the analogous part of the coefficient ring of the Panin-Walter algebraic special linear cobordism spectrum MSL, which can be interpreted as “oriented algebraic cobordism”. For these purposes, I will construct another Thom spectrum that provides a geometric model for the cofiber of multiplication by the Hopf element on MSL. Topological analogs of this spectrum have been studied by Wall, Conner, and Floyd.