# Zahlen- und Gruppentheorie

### Forschungsschwerpunkt A

Bays, Cuntz, Deninger, Hartl, Hellmann, Hille, Hils, Jahnke, Kwiatkoswka, Nikolaus, Scherotzke (bis 2020), R. Schindler, Schlutzenberg, Schneider, Scholbach, Schürmann, Tent.

Die Forschung betrifft hauptsächlich die Themenfelder Arithmetische Geometrie und Darstellungstheorie sowie Gruppen- und Modelltheorie. Wir nennen einige der großen Ziele: Es soll in möglichst großer Allgemeinheit eine p-adische lokale Langlands Korrespondenz konstruiert werden, welche endlich dimensionale p-adische Darstellungen der Galoisgruppe p-adischer Körper mit unendlich dimensionalen Darstellungen reduktiver p-adischer Gruppen in Beziehung setzt. In diesem Kontext und in vielen weiteren spielen Modulräume p-adischer Darstellungen eine wichtige Rolle. Weiterhin sollen neue Kohomologietheorien in Algebra und Arithmetischer Geometrie entwickelt werden, die unter anderem zu einem tieferen Verständnis von Hasse Weil Zetafunktionen führen sollen.

In der Gruppentheorie sollen neuartige Zugänge zu Burnside Gruppen und dem fundamentalen Burnside Problem weiter entwickelt werden. In der Modelltheorie wird der Fokus auf dem Erforschen bis hin zur Charakterisierung von NIP Gruppen und Körpern sowie dem Klassifizieren und der Konstruktion üppiger streng minimaler Strukturen liegen.

## Weitere Forschungsprojekte von Mitgliedern des Forschungsschwerpunkts A

$\bullet$ Christopher Deninger, Thomas Nikolaus: CRC 1442: Geometry: Deformation and Rigidity - A04: New cohomology theories for arithmetic schemes (2020-2024)
We develop, study and compute certain global cohomology theories for schemes. These cohomology theories may be viewed as deformations of ‘classical’ cohomology theories over mixed characteristic or over the sphere spectrum. For example, de Rham–Witt cohomology is a deformation of de Rham cohomology over mixed characteristic. Topological Hochschild homology is a deformation of ordinary Hochschild homology over the sphere spectrum. The motivation for the project is to use these cohomology theories to attack deep problems in algebra, topology and arithmetic geometry. Our most ambitious application concerns zeta functions.

$\bullet$ Eugen Hellmann, Peter Schneider: CRC 1442: Geometry: Deformation and Rigidity - A01: Automorphic forms and the p-adic Langlands programme (2020-2024)
The p-adic Langlands programme aims to establish a relation between p-adic representations of p-adic reductive groups and p-adic representations of Galois groups of p-adic local fields. We plan an in depth study of the smooth mod p representation theory of reductive groups on the level of derived categories. In first relevant test cases we want to construct functors from representations of reductive groups to sheaves on deformation spaces of Galois representations.

$\bullet$ Eugen Hellmann, Urs Hartl, Peter Schneider: CRC 1442: Geometry: Deformation and Rigidity - A02: Moduli spaces of p-adic Galois representations (2020-2024)
p-adic Galois representations in finite Zp-modules are equivalent to (phi,Gamma)-modules for Qp. In this project, we develop the theory of (phi,Gamma)-modules further in the direction of finite extensions of Qp and their function field analogues. We will also use (phi,Gamma)-modules to construct moduli spaces of p-adic Galois representations. We aim to decompose special fibres on these moduli spaces into cycles in a way that mirrors multiplicity formulas in representation theory.

$\bullet$ Urs Hartl: CRC 1442: Geometry: Deformation and Rigidity - A03: Special cycles on moduli spaces of G-shtukas (2020-2024)
Moduli stacks of global G-shtukas are the function field analogue of Shimura varieties. Their geometry was investigated by the PI and his collaborators. The long term goal of the project is to investigate arithmetic properties of these moduli stacks, special cycles on them and the intersection numbers of the latter. In particular, we want to develop a function field analogue of the Kudla programme for Shimura varieties and prove arithmetic fundamental lemmas for function fields.

$\bullet$ Katrin Tent: CRC 1442: Geometry: Deformation and Rigidity - C04: Hyperbolic groups acting sharply 2- or 3-transitively and the Burnside problem (2020-2024)
Until recent 'free' constructions of the PI and her collaborators, the only known sharply 2- and 3-transitive permutation groups were those arising from linear transformations of the affine or projective line. We will investigate the limitations of this class of groups. The quest for sharply 2- and 3-transitive groups in positive characteristic not arising from linear transformations leads us to the Burnside problem, for which we propose a new approach yielding much lower bounds on the exponent for infinite such groups.

$\bullet$ Aleksandra Kwiatkowska: CRC 1442: Geometry: Deformation and Rigidity - C05: Rigidity of group topologies and universal minimal flows (2020-2024)
We will study automatic continuity and universal minimal flows for topological groups acting on geometric objects. First, we focus on rigidity of group topologies, motivated by a question due to Rosendal on the existence of a locally compact infinite group with the automatic continuity property. We want to study this question for Burger–Mozes groups. Furthermore, motivated by a question of Evans–Hubička–Nešetřil, we want to understand when universal minimal flows of kaleidoscopic groups are metrisable. Our methods will involve interactions between group theory, topological dynamics and Ramsey theory.

$\bullet$ Thomas Nikolaus: CRC 1442: Geometry: Deformation and Rigidity - C02: Homological algebra for stable ∞-categories (2020-2024)
The goal of this project is to study the emerging area of homological algebra for stable infinity-categories. Concretely the major objective of this project is to study non-commutative motives as introduced by Blumberg–Gepner–Tabuada and the homotopy theory of chain complexes of stable infinity-categories that will be developed as part of the project, following up on pioneering work of Dyckerhoff. Moreover, we will explore the notion of a stable (infinity,infinity)-category and the corresponding higher version of spectra, which will besides its general importance also be relevant in setting up a rigid higher version of Quinn’s Ad-theories.

$\bullet$ Martin Bays, Katrin Tent, Franziska Jahnke, Martin Hils: Geometric and Combinatorial Configurations in Model Theory (2020-2022)
Model theory studies structures from the point of view of first-order logic. It isolates combinatorial properties of definable sets and uses these to obtain algebraic consequences. A key example is the group configuration theorem, a powerful tool in geometric stability used, e.g., to prove the trichotomy for Zariski geometries and in recent applications to combinatorics. Valued fields are an example of the confluence of stability theory and algebraic model theory. While Robinson studied algebraically closed valued fields already in 1959, the tools from geometric stability were only made available in this context in work of Haskell-Hrushovski-Macpherson, brought to bear in Hrushovski-Loeser's approach to non-archimedean geometry. In the project, we aim to strengthen the recent relations between model theory and combinatorics, develop the model theory of valued fields using tools from geometric stability and carry out an abstract study of the configurations which are a fundamental tool in these two areas.