Zahlen- und Gruppentheorie

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Forschungsschwerpunkt A

Bays, Cuntz, Deninger, Gardam (seit 2022), Hartl, Hellmann, Hille, Hils, Jahnke, Kwiatkoswka, Nikolaus, Scherotzke (bis 2020), R. Schindler, Schlutzenberg, Schneider, Scholbach (bis 2022), Schürmann, Tent, Viehmann (seit 2022).

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

Model Theory of Valued Fields with Endomorphism
We propose a model-theoretic investigation of valued fields with non-surjective endomorphism. The model theory of valued fields with automorphism, in particular the Witt Frobenius case treated by Bélair, Macintyre and Scanlon, was extensively developed over the last 15 years, e.g., obtaining Ax-Kochen-Ershov principles for various classes of σ-henselian valued difference fields. We plan to generalize these results to the non-surjective context. The most natural example of a non-surjective endomorphism is the Frobenius map on an imperfect field. In analogy to the Witt Frobenius, the main examples for our study are Cohen fields over imperfect residue fields endowed with a lift of the Frobenius. The model theory of Cohen fields was recently developed in the work of Anscombe and Jahnke. In order to obtain Ax-Kochen-Ershov type results in our setting, it will be necessary to firstunderstand the equicharacteristic 0 case. This case is interesting in its own right, as it encompasses the asymptotic theory of Cohen fields with Frobenius lift. We are particularly interested in obtaining relative completeness and transfer of model theoretic tameness notions from value group and residue field to the valued difference field, as well as in identifying model companions for various subclasses. In all these cases, the endomorphism is an isometry of the valued field.
Another natural example of a σ-henselian valued difference fields is given by ultraproducts ofseparably closed valued fields with Frobenius. Here, the endomorphism is no longer an isometry, and the induced automorphism of the value group is ω-increasing. By work of Chatzidakis and Hrushovski the residue difference field of such an ultraproduct is existentially closed as a field with distinguished endomorphism. We aim to show the analogous result for the valued difference field, namely that it is existentially closed in a natural language, and infer existence and an axiomatization of the model-companion from this.
Project members: Franziska Jahnke, Martin Hils

Long extenders, Varsovian models, combinatorics We are investigating some questions in inner model theory. online
Project members: Jan Kruschewski, Farmer Schlutzenberg

CRC 1442: Geometry: Deformation and Rigidity - A04: New cohomology theories for arithmetic schemes 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. online
Project members: Christopher Deninger, Thomas Nikolaus

CRC 1442: Geometry: Deformation and Rigidity - A01: Automorphic forms and the p-adic Langlands programme 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. online
Project members: Eugen Hellmann, Peter Schneider

CRC 1442: Geometry: Deformation and Rigidity - A02: Moduli spaces of p-adic Galois representations 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. online
Project members: Eugen Hellmann, Urs Hartl, Peter Schneider

CRC 1442: Geometry: Deformation and Rigidity - A03: Special cycles on moduli spaces of G-shtukas 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. online
Project members: Urs Hartl

CRC 1442: Geometry: Deformation and Rigidity - C04: Hyperbolic groups acting sharply 2- or 3-transitively and the Burnside problem 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. online
Project members: Katrin Tent

CRC 1442: Geometry: Deformation and Rigidity - C05: Rigidity of group topologies and universal minimal flows 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. online
Project members: Aleksandra Kwiatkowska

CRC 1442: Geometry: Deformation and Rigidity - C02: Homological algebra for stable ∞-categories 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. online
Project members: Thomas Nikolaus

Geometric and Combinatorial Configurations in Model Theory 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. online
Project members: Martin Bays, Katrin Tent, Franziska Jahnke, Martin Hils