## Further research projects of Research Area A members

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**CRC 1442: Geometry: Deformation and Rigidity - A04: New cohomology theories for arithmetic schemes ** The goal of this project is to study cohomology theories for schemes in order to attack important open problems in arithmetic. Among these theories are topological periodic homology (TP), topological cyclic homology (TC), rational de Rham–Witt cohomology, prismatic cohomology, K-theory, L-Theory and leafwise cohomology of associated dynamical systems. We will prove structural results about those theories as well as make further calculations of specific cases.

onlineProject members:

**Christopher Deninger**,

**Thomas Nikolaus**• **CRC 1442: Geometry: Deformation and Rigidity - D03: Integrability **

We investigate blobbed topological recursion for the general Kontsevich matrix model, as well as the behaviour of Baker–Akhiezer spinor kernels for deformations of the spectral curve and for the quartic Kontsevich model. We study relations between spin structures and square roots of Strebel differentials, respectively between topological recursion and free probability. We examine factorisation super-line bundles on infinite-dimensional Grassmannians and motivic characteristic classes for intersection cohomology sheaves of Schubert varieties.

onlineProject members:

**Jörg Schürmann**,

**Raimar Wulkenhaar**,

**Yifei Zhao**• **CRC 1442: Geometry: Deformation and Rigidity - A01: Automorphic forms and the p-adic Langlands programme **

The past years have seen tremendous progress in the development of a categorical approach to the arithmetic of the Langlands programme. In the context of the p-adic Langlands programme the main features of this approach are the study of derived categories of p-adic representations of p-adic Lie groups, the study of (coherent) sheaves on moduli of Galois representations associated to such representations, and the development of a more geometric approach to such representations. The project addresses all these three aspects of the programme.

onlineProject members:

**Peter Schneider**,

**Eugen Hellmann**• **CRC 1442: Geometry: Deformation and Rigidity - A02: Moduli spaces of p-adic Galois representations **

Representations of the absolute Galois group of a p-adic local field with p-adic coefficients are studied most fruitfully in terms of semi-linear algebra objects called (phi,Gamma)-modules. In part of the project we will advance the study of (phi,Gamma)-modules. In another part we use (phi,Gamma)-modules to construct and study moduli spaces of Galois representation that occur in the context of the p-adic Langlands programme.

onlineProject members:

**Peter Schneider**,

**Eugen Hellmann**• **CRC 1442: Geometry: Deformation and Rigidity - C04: Group theoretic aspects of negative curvature **

We will continue to investigate iterated small cancellation groups. We recently gave a relatively short proof of the infinity of the free Burnside groups of sufficiently large odd exponent which is based on a few important principles well-studied in hyperbolic groups. In a next step we will develop a meta theorem based on these principles which will be applicable to a wider range of problems, in particular in the context of sharply multiply transitive groups. We will also study complexes associated to hyperbolic groups from a model theoretic perspective.

onlineProject members:

**Katrin Tent**• **CRC 1442: Geometry: Deformation and Rigidity - Z01: Central Task of the Collaborative Research Centre ** online

Project members: **Eugen Hellmann**

• **CRC 1442: Geometry: Deformation and Rigidity **

From its historic roots, geometry has evolved into a cornerstone in modern mathematics, both as a tool and as a subject in its own right. On the one hand many of the most important open questions in mathematics are of geometric origin, asking for example to what extent an object is determined by geometric properties. On the other hand, abstract mathematical problems can often be solved by associating them to more geometric objects that can then be investigated using geometric tools. A geometric point of view on an abstract mathematical problem quite often opens a path to its solution.Deformations and rigidity are two antagonistic geometric concepts which can be applied in many abstract situations making transfer of methods particularly fruitful. Deformations of mathematical objects can be viewed as continuous families of such objects, like for instance evolutions of a shape or a system with time. The collection of all possible deformations of a mathematical object can often be considered as a deformation space (or moduli space), thus becoming a geometric object in its own right. The geometric properties of this space in turn shed light on the deeper structure of the given mathematical objects. We think of properties or of quantities associated with mathematical objects as rigid if they are preserved under all (reasonable) deformations.A rigidity phenomenon refers to a situation where essentially no deformations are possible.Rigidity then implies that objects which are approximately the same must in fact be equal, making such results important for classifications.The overall objective of our research programme can be summarised as follows:Develop geometry as a subject and as a powerful tool in theoretical mathematics focusing on the dichotomy of deformations versus rigidity. Use this unifying perspective to transfer deep methods and insights between different mathematical subjects to obtain scientific breakthroughs, for example concerning the Langlands programme, positive curvature manifolds, K-theory, group theory, and C*-algebras.

onlineProject members:

**Eugen Hellmann**• **CRC 1442: Geometry: Deformation and Rigidity - A05: Moduli spaces of local shtukas in mixed characteristic **

We study the geometry and cohomology of moduli spaces of local G-shtukas, a class of moduli spaces that plays a central role in the geometrisation of Langlands correspondences. More precisely, we are interested in the geometry of the image of the period maps, want to investigate étale sheaves on the moduli spaces and aim at the local Langlands correspondence for covering groups via a metaplectic geometric Satake equivalence.

onlineProject members:

**Yifei Zhao**,

**Eva Viehmann**• **CRC 1442: Geometry: Deformation and Rigidity - C02: Homological algebra for stable ∞-categories **

The general goal of the project is to study the homological algebra of stable infinity-categories and Poincaré infinity-categories. This is done through the theory of non-commutative motives and Efimov K-Theory. Concrete goals are to give a new approach to controlled algebra (thereby attacking open problems and conjectures in geometric topology) and obtain new structural results about the category of motives. The latter thus yields new results about K-Theory and TC, e.g. we try to resolve the long-standing open question about a universal property for TC.

onlineProject members:

**Thomas Nikolaus**• **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.

onlineProject 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**