Seminar Algebra und Topologie

Das Seminar findet normalerweise am Mittwoch in I.13.71 (!), 16:30 - 17:30 statt.
Beteiligte Dozenten: Jens Hornbostel, Sascha Orlik, Kay Rülling, Tobias Schmidt, Britta Späth, Matthias Wendt.

Henrik Russell: The geometric fundamental group of the affine line over a finite field

The affine line A¹ over a finite field F is taken as a benchmark for the problem of describing geometric étale fundamental groups. Using a reformulation of Tannaka duality we construct a (non-commutative) universal affine group Lu_{A¹_F}, such that any finite and étale Galois covering of A¹ over F is a pull-back of a Galois covering of Lu_{A¹_F}. Then the geometric fundamental group of A¹_F is a completion of the k-points of Lu_{A¹_F}, where k is an algebraic closure of F, and we obtain an explicit description of Lu_{A¹_F}.

Fabian Hebestreit: Basic higher almost ring theory

(with P.Scholze) For a flat and idempotent ideal I in a commutative ring R the localisation of Mod(R) at the maps whose kernel and cokernel is annihilated by I is Falting's category of almost R-modules. Examples of the set-up arise from perfectoid fields, and almost modules prominently feature in the tilting equivalences relating their étale information in characteristic 0 and p. The possibly simplest example has R obtained from K[X] by adjoining all roots of X which together span I. Now, the category of almost R-modules derives many of its good properties from the fact that R/I is derived tensor-idempotent R-algebra on account of the assumptions on I. The flatness assumption is, however, often not satisfied in higher dimensional situations, eg if R and I are similarly obtained from K[X,Y], and effort has been expended into weakening or removing it. In the talk I will explain how this can be achieved rather easily by directly passing to derived categories and replacing the ordinary ring R/I by a certain new animated commutative ring.

Jack Davies: Geometric norms on equivariant elliptic cohomology

A notion of multiplicative structure on equivariant cohomology theories will be called a ``geometric norm structure''. We will then introduce equivariant elliptic cohomology, as defined by Lurie and Gepner-Meier, and discuss how it naturally comes with a geometric norm structure. We will also touch on a few applications of these structures: constructions of C_p-normed algebra structures and integral models for Behrens' Q(N) spectra. This is joint work-in-progress with William Balderrama and Sil Linskens.

Stefan Weinzierl: Feynman Integrals

Feynman integrals are the central objects in the perturbative expansion within quantum field theory. They are indispensible for precision calculation in many areas of fundamental physics. In this talk I will introduce these objects for a mathematical audience and I will try to highlight the connections with modern branches of mathematics.

Devarshi Mukherjee: Derived analytic geometry and localising invariants

We introduce a version of K-theory and other localising invariants for derived analytic spaces using Efimov's continuous K-theory. Our model of derived analytic geometry uses complete bornological algebras as building blocks. The main results include descent of K-theory for various topologies that arise in derived analytic geometry, and a version of the Grothendieck-Riemann-Roch Theorem. This is joint work with Jack Kelly.

Alberto Merici: Tame and logarithmic motivic homotopy

It is well known that integral p-adic étale cohomology theories do not fit well with motivic homotopy theory because of the Artin—Schreier sequence. In recent years, two approaches to this problem emerged: motives for log schemes of Binda-Park-Østvær and the tame cohomology of Hübner-Schmidt. In this talk, I will prove a comparison theorem between the two theories, which implies some motivic properties of tame cohomology, as conjectured by Hübner-Schmidt.

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