Arithmetic and Groups

© MM

Research Area A

Bays (until 2023), Cuntz, Deninger, Gardam (2022-2023), Hartl, Hellmann, Hille, Hils, Jahnke, Kwiatkoswka, Lourenço (since 2024), Nikolaus, Scherotzke (until 2020), R. Schindler, Schlutzenberg (until 2023), Schneider, Scholbach (until 2022), Schürmann, Tent, Viehmann (since 2022), Zhao (since 2024).

Research Area A will focus on problems of algebraic nature with methods ranging from representation and cohomology theory to model theory and mathematical logic. The p-adic and characteristic p Langlands correspondences that relate arithmetic invariants with automorphic representations are major research topics. We also aim at developing new powerful cohomology theories in algebra and arithmetic geometry. One of the long-term goals is to understand Hasse-Weil zeta functions cohomologically.

In group theory, we plan to develop new techniques for small cancellation theory in order to construct sharply 2-transitive groups in sufficiently large finite characteristic. On the model theoretic side, the focus will be on understanding and classifying NIP groups and fields as well as on classifying and constructing ample strongly minimal structures. In set theory and descriptive set theory, we will be studying Varsovian models as well as Polish group actions.

  • A1. Arithmetic, geometry and representations.

    The Langlands programme relates representations of (the adele valued points of) reductive groups G over Q - so-called automorphic representations - with certain representations of the absolute Galois group of Q. This programme includes the study of these objects over general global fields (finite extension of Q or Fp (t)) and local fields as well. In its local form the classical programme only considered l-adic Galois representations of p-adic fields for unequal primes l neq p. In order to allow for a p-adic variation of the objects, it is absolutely crucial to extend it to the case l = p. In the global situation, the automorphic representations in question can often be realised in (or studied via) the cohomology of a tower of Shimura varieties (or related moduli spaces) attached to the group G.

    We will focus on the following directions within this programme:
    The p-adic and mod p Langlands programme asks for an extension of such a correspondence involving certain continuous representations with p-adic respectively mod p coefficients. Broadening the perspective to p-adic automorphic forms should, for example, enable us to capture all Galois representations, not just those having a particular Hodge theoretic behaviour at primes dividing p. This extended programme requires the introduction of derived categories. We will study differential graded Hecke algebras and their derived categories on the reductive group side. On the Galois side, we hope to use derived versions of the moduli spaces of p-adic Galois representations introduced by Emerton and Gee.

    The geometric Langlands programme is a categorification of the Langlands programme. We plan to unify the different approaches using motivic methods.
    In another direction, we study the geometry and arithmetic of moduli stacks of global G-shtukas over function fields. Their cohomology has been the crucial tool to establish large parts of the local and global Langlands programme over function fields. Variants of G-shtukas are also used to construct and investigate families of p-adic Galois representations.

    Cohomology theories are a universal tool pervading large parts of algebraic and arithmetic geometry. We will develop and study cohomology theories, especially in mixed characteristic, that generalise and unify étale cohomology, crystalline cohomology and de Rham cohomology as well as Hochschild cohomology in the non-commutative setting. Developing (topological) cyclic homology in new contexts is an important aim. A main goal is to construct a cohomology theory that can serve the same purposes for arithmetic schemes as the l-adic or crystalline cohomology with their Frobenius actions for varieties over finite fields. Ideas from algebraic geometry, algebraic topology, operator algebras and analysis blend in these investigations.

  • A2. Groups, model theory and sets.

    Model theory and, more generally, mathematical logic as a whole has seen striking applications to arithmetic geometry, topological dynamics and group theory. Such applications as well as fundamental questions in geometric group theory, on the foundations of model theory and in set theory are at the focus of our work.
    The research in our group reaches from group theoretic questions to the model theory of groups and valued fields as well as set theory. The study of automorphism groups of first order structures as topological groups, for examples, uses tools from descriptive set theory leading to pure set theoretic questions.

     

Further research projects of Research Area A members

• 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.
online
Project members: Martin Hils

• Geometry and Arithmetic of Uniformized Structures online
Project members: Eva Viehmann

• 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: Katrin Tent, Martin Hils