Arithmetic

The theme of Arithmetic geometry is the understanding of algebraic varieties over the rational numbers Q or arithmetic schemes over the integers Z. Our research in this direction can be divided into projects within the framework of the Langlands programme, and into projects that concern the construction of new cohomology theories for algebraic varieties. A common motivation for both directions is the aim to understand properties of the Hasse–Weil zeta-function of algebraic varieties, a vast generalisation of the Riemann zeta-function.

The Langlands programme lies at the crossroads of different mathematical disciplines, involving number theory, algebraic geometry, representation theory and harmonic analysis. It aims to establish and exploit a deep structural relation between algebraic varieties over Q and objects in representation theory, so-called automorphic representations. Using such a correspondence, questions in number theory and algebraic geometry can be transformed into questions about automorphic representations and vice versa. For example the Langlands programme conjecturally describes the l-adic cohomology of Shimura varieties in terms of automorphic representations and hence computes their zeta-function as a product of automorphic L-functions.

In order to improve the cohomological understanding of more general varieties or arithmetic schemes (as opposed to the special case of Shimura varieties) it will be crucial to develop stronger cohomology theories that replace the l-adic and crystalline cohomologies in the case of varieties over finite fields.

Projects involved: A1,A2,A3 and A4

K-theory

K-theory arises as an important invariant in different areas of mathematics, from arithmetic geometry to geometric topology to operator algebras. It associates algebraic invariants to mathematical structures in various situations, for example to rings and more generally to schemes, C∗-algebras, stable ∞-categories and topological spaces. By its very definition, algebraic K- theory has its origins in algebraic geometry. The functor K0 was invented to formulate the Grothendieck–Riemann–Roch theorem. Higher K-theory is used to study intersection theory on varieties and special values of L-functions. Questions in geometric topology, such as Borel’s conjecture about the topological rigidity of aspherical manifolds and Novikov’s conjecture about the rigidity of higher signatures, are, via surgery theory, equivalent to statements about the algebraic K-theory and about the related L-theory of group rings. K-theory of ring spectra makes an appearance in the parametrised h-cobordism theorem and is crucial more generally in studying moduli spaces of manifolds. Topological K-theory is used to detect essential variations in families of metrics with positive curvature. The isomorphism conjectures of Baum–Connes and Farrell–Jones are examples of prominent conjectures regarding K-theory. Here geometry is both a subject and a tool. Questions of geometric origin are translated into questions in K-theory; these in turn can sometimes be answered using geometric methods. For example deformations of K-theory classes along flows are used in connection with Borel’s conjecture. Topological Hochschild homology and topological cyclic homology, which are deformations of ordinary Hochschild Homology, are good approximations of K-theory and are used in many of the known calculations of algebraic K-theory. These invariants are best understood from the point of view of higher algebra and ∞-categories. Topological K-theory is essential to the classification of (sufficiently regular) nuclear C∗-algebras.

Projects involved: A1, A4, B3, C2, C3, C4, D1, D3, D4 and D5

Moduli spaces

Moduli spaces are objects of fundamental importance in several areas within pure mathematics. Roughly, a moduli space parametrises families of mathematical objects with certain prescribed properties. On one hand their local geometry allows to obtain additional information about the individual parametrised objects. On the other hand, global invariants help to understand for example rigidity properties and are interesting in their own right, but also often yield interesting applications beyond the parametrised family.

Within the CRC, we are interested in moduli spaces arising in Arithmetic Geometry (parametrising for example Galois representations or G-shtukas), whose study aims at applications within the Langlands programme. Further we study diffeomorphism groups in Topology and moduli of stable complex curves, motivated by questions from Quantum Field Theory and Mathematical Physics. Despite these diverse origins, the overall questions that we consider are closely related. A central topic has been and is to study cycles, cohomology classes and intersection numbers on moduli spaces. This geometric task is complemented by investigating suitable stratifications of the moduli spaces.

Projects involved: A2, A4, B3, B6 and D3

Curvature

The class of manifolds admitting metrics with positive (non-negative) sectional curvature is poorly understood. For example there is not a single obstruction known that distinguishes the class of simply connected closed manifolds admitting positively curved metrics from the corresponding class of manifolds admitting nonnegatively curved metrics. On the other hand in dimension above 24 all known simply connected positively curved manifolds are given by deformations of rank 1 symmetric spaces. Under fairly mild symmetry assumptions one can expect to find a conceptual explanation of the rigidity of the topology of positively curved manifolds. Moreover, one looks at classes of positively curved Riemannian manifold as a whole. Here the global topological properties of the moduli spaces are a key issue. Taking the closure of a moduli space in the Gromov–Hausdorff topology, one often finds singular objects in the limit. Understanding the phenomena that occur when passing from the smooth to the singular object often result in finiteness theorems for certain classes of manifolds. Ricci flow frequently plays a role since it is usually the best bet for finding a canonical smoothing of a singular object.

Projects involved: B1, B2, B3, B4, B5 and B6

Groups and Dynamics

The study of symmetry and space through the medium of groups and their actions has long been a central theme in modern mathematics, indeed one that cuts across a wide spectrum of research within the entire CRC. In the projects of this research objective we focus on those aspects of group and dynamics that are related to measure and topology in their barest abstract sense. We will also view infinite discrete groups and other locally compact groups as geometric or combinatorial objects. This direction of research draws on tools from functional analysis, probability, combinatorics, and graph theory. It branches out into many parts of mathematics, from the rigidity theory of Lie groups and their lattices to the study of manifolds and geometry through invariants like K-theory.

Projects involved: C3, C4, C5, D1, D4 and D5

Singularities

Singularities naturally occur in the study of geometric objects and of differential equations. In two of our well-established research areas and from a more algebro-geometric point of view, singularities are playing an increasingly important role. The study of singularities of moduli spaces of Galois representations and characteristic classes for singular varieties as well as perverse sheaves and mixed Hodge modules. A more analytic point of view on singularities centres around singularities of Kähler metrics with bounded scalar curvature and singularities of the PDEs of General Relativity, black hole spacetimes. The example of Hilbert modular surfaces shows very well how seemingly disparate fields and methods are unified through the lens of singularity theory: Hilbert modular surfaces are moduli spaces of abelian surfaces and hence arise in Arithmetic Geometry; but they are also equipped with Kähler–Einstein metrics of negative Ricci curvature, whose behaviour at the cusps is symptomatic of the singularities of Kähler–Einstein metrics more generally.

Projects involved: A2, B2, B5, B6 and D3