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.

Gabriel Angelini-Knoll: Algebraic K-theory of minimal skew-fields in homotopy theory

In algebra, the minimal fields are determined by their characteristic and their algebraic K-theory is of fundamental importance. For example, the algebraic K-theory of the rational numbers recovers special values of the Riemann zeta function. In homotopy theory, there are additional minimal skew fields known as Morava K-theory, which depend on a height n and a prime p. In my talk, I will report on joint work with J. Hahn and D. Wilson on the algebraic K-theory of Morava K-theory. Our approach uses syntomic cohomology after Bhatt-Morrow-Scholze and Hahn-Raksit-Wilson. As consequences, we prove that the telescope conjecture, Lichtenbaum--Quillen property, and redshift hold for the algebraic K-theory of Morava K-theory at arbitrary heights and primes. The computation also exhibits a higher height analogue of Poutou-Tate duality.

Amine Koubaa: Purity in the tame cohomology with p-adic coefficients

Let X be a scheme which is smooth over a field k. Assume that char(k)=p>0. I will sketch a proof of cohomological purity for tame cohomology of X, defined by Hübner and Schmidt, with p-adic coefficients. We prove that, under the assumption of resolution of singularities, the following purity isomorphism of sheaves in the tame topology of X

Ri^! \nu_m(n)\cong \nu_m(n-r)[-r]
 
exists, where \nu_m(n) are the logarithmic de Rham-Witt sheaves, and i:Z\to X is a smooth closed immersion of pure codimension r.

Claudius Heyer: A 6-functor formalism for smooth mod p representations

The formalism of six operations was introduced by Grothendieck to show that many phenomena in the étale cohomology of schemes can be formally deduced from a small set of axioms. Since then these six operations have been constructed in many of other contexts like D-modules, motives and rigid-analytic geometry. But only recently has there been a formal definition of a 6-functor formalism, mainly due to Liu–Zheng and then further simplified by Mann in his PhD thesis. Also in Fargue–Scholze's geometrization of the local Langlands correspondence the six operations are a guiding theme.

In this talk I will report on joint work with Lucas Mann where we construct a full 6-functor formalism in the setting of smooth representations of p-adic Lie groups with mod p coefficients (in fact, we allow arbitrary discrete coefficient rings). As an application we use the formalism to construct a canonical anti-involution on derived Hecke algebras generalizing earlier work by Schneider–Sorensen.

• Gästeseminare vergangener Semester