Spaces and Operators

Research Area B

Bartels, Bellissard (until 2021), Böhm, Courtney (since 2022), Cuntz, Ebert, Echterhoff, Gardella (until 2021), Hein (since 2020), Holzegel (since 2020), Joachim, Kerr (since 2021), Kramer, de Laat, Lohkamp, Löwe, Ludwig (since 2022), Santoro (since 2022), Siffert (since 2020), Weber (since 2022), Weiss, Wiemeler (since 2020), Wilking, Winter, Wulkenhaar, Zeidler.

In Area B we will study spaces and in particular manifolds, dynamical systems, geometric structures on manifolds, symmetries and automorphisms of manifolds. A central topic is the classification theory of separable nuclear C*-algebras. We will apply and advance a wide range of methods and tools, such as the Ricci flow, heat flow methods, Gromov-Hausdorff convergence of sequences of manifolds, singular foliations, topological and algebraic K-theory, surgery theory, cobordism categories, index theory, notions of dimension and regularity for dynamical systems and C*-algebras, notions of positive and negative curvature, and the isomorphism conjectures of Baum-Connes and Farrell-Jones. We will establish a differentiable diameter sphere theorem and study manifolds of positive curvature and torus actions on them. We will study the topology of spaces of positive scalar curvature metrics on manifolds and the topology of groups of homeomorphisms and diffeomorphisms of manifolds. For nuclear C*-algebras we will attack the UCT problem and quasi-diagonality.

• B1. Smooth, singular and rigid spaces in geometry.

Many interesting classes of Riemannian manifolds are precompact in the Gromov–Hausdorff topology. The closure of such a class usually contains singular metric spaces. Understanding the phenomena that occur when passing from the smooth to the singular object is often a first step to prove structure and finiteness results. In some instances one knows or expects to define a smooth Ricci flow coming out of the singular objects. If one were to establish uniqueness of the flow, the differentiable stability conjecture would follow. If a dimension drop occurs from the smooth to the singular object, one often knows or expects that the collapse happens along singular Riemannian foliations or orbits of isometric group actions.

Rigidity aspects of isometric group actions and singular foliations are another focus in this project. For example, we plan to establish rigidity of quasi-isometries of CAT(0) spaces, as well as rigidity of limits of Type III Ricci flow solutions and of positively curved manifolds with low-dimensional torus actions. We will also investigate area-minimising hypersurfaces by means of a canonical conformal completion of the hypersurface away from its singular set.

• B2. Topology.

We will analyse geometric structures from a topological point of view. In particular, we will study manifolds, their diffeomorphisms and embeddings, and positive scalar curvature metrics on them. Via surgery theory and index theory many of the resulting questions are related to topological K-theory of group C*-algebras and to algebraic K-theory and L-theory of group rings.

Index theory provides a map from the space of positive scalar curvature metrics to the K-theory of the reduced C*-algebra of the fundamental group. We will develop new tools such as parameterised coarse index theory and combine them with cobordism categories and parameterised surgery theory to study this map. In particular, we aim at rationally realising all K-theory classes by families of positive scalar curvature metrics. Important topics and tools are the isomorphism conjectures of Farrell–Jones and Baum–Connes about the structure of the K-groups that appear. We will extend the scope of these conjectures as well as the available techniques, for example from geometric group theory and controlled topology. We are interested in the K-theory of Hecke algebras of reductive p-adic Lie groups and in the topological K-theory of rapid decay completions of complex group rings. Via dimension conditions and index theory techniques we will exploit connections to operator algebras and coarse geometry. Via assembly maps in algebraic K- and L-theory we will develop index-theoretic tools to understand tangential structures of manifolds. We will use these tools to analyse the smooth structure space of manifolds and study diffeomorphism groups. Configuration categories will be used to study spaces of embeddings of manifolds.

• B3. Operator algebras & mathematical physics.

The development of operator algebras was largely motivated by physics since they provide the right mathematical framework for quantum mechanics. Since then, operator algebras have turned into a subject of their own. We will pursue the many fascinating connections to (functional) analysis, algebra, topology, group theory and logic, and eventually connect back to mathematical physics via random matrices and non-commutative geometry.

Further research projects of Research Area B members

$\bullet$ Christoph Böhm, Arnulf Jentzen: Dynamical systems and irregular gradient flows (2021-2025)
The central goal of this project is to study asymptotic properties for gradient flows (GFs) and related dynamical systems. In particular, we intend to establish a priori bounds and related regularity properties for solutions of GFs, we intend to study the behaviour of GFs near unstable critical regions, we intend to derive lower and upper bounds for attracting regions, and we intend to establish convergence speeds towards global attrators. Special attention will be given to GFs with irregularities (discontinuities) in the gradient and for such GFs we also intend to reveal sufficient conditions for existence, uniqueness, and flow properties in dependence of the given potential. We intend to accomplish the above goals by extending techniques and concepts from differential geometry to describe and study attracting and critical regions, by using tools from convex analysis such as subdifferentials and other generalized derivatives, as well as by employing concepts from real algebraic geometry to describe domains of attraction. In particular, we intend to generalize the center-stable manifold theorem from the theory of dynamical systems to the considered non-smooth setting. Beside finite dimensional GFs, we also study GFs in their associated infinite dimensional limits. The considered irregular GFs and related dynamical systems naturally arise, for example, in the context of molecular dynamics (to model the configuration of atoms along temporal evoluation) and machine learning (to model the training process of artificial neural networks).

$\bullet$ Bianca Santoro, Hans-Joachim Hein, Johan Klemmensen: CRC 1442: Geometry: Deformation and Rigidity - B05: Scalar curvature in Kähler geometry (2021-2024)
In this project we propose to study the degeneration of Kähler manifolds with constant or bounded scalar curvature under a non-collapsing assumption. For Riemannian manifolds of bounded sectional curvature, this is the content of the classical Cheeger-Gromov convergence theory from the 1970s. For Riemannian manifolds of bounded Ricci curvature, definitive results were obtained by Cheeger-Colding-Naber in the past 10-20 years, with spectacular applications to the Kähler-Einstein problem on Fano manifolds. Very little is currently known under only a scalar curvature bound even in the Kähler case. We propose to make progress in two different directions: (I) Gather examples of weak convergence phenomena related to the stability of the Positive Mass Theorem for Kähler metrics and to Taubes' virtually infinite connected sum construction for ASD 4-manifolds. (II) Study uniqueness and existence of constant scalar curvature Kähler metrics on non-compact or singular spaces by using direct PDE methods.

$\bullet$ Rudolf Zeidler: SPP 2026: Geometry at Infinity - Subproject: Duality and the coarse assembly map (2021-2023)
The coarse co-assembly map was introduced by Emerson and Meyer as a dual to the coarse assembly map. These two maps are mutually adjoint with respect to canonical pairings. The main objective of our project is to further develop this coarse duality theory by studying analogues of multiplicative structures known from algebraic topology such as cup and cap products as well as external and slant products. Many of these multiplicative structures have direct interpretations in terms of Dirac operators and vector bundles leading to new applications in index theory and geometric topology. In addition, we will develop coarse versions of well-known results from algebraic topology involving multiplicative structures.Most notably, we want to investigate a coarse version of Poincaré duality.Furthermore, the question of whether coarse assembly and co-assembly are isomorphisms will also be examined in certain cases. We have a special focus on spaces which admit nice coarse compactifications constructed from coarse geometric versions of contractions or, more generally, deformation retractions. Such coarse deformation retractions are also key to the construction of secondary cup and cap products, establishing a close link between the two main aspects of our project.

$\bullet$ Wilhelm Winter: CRC 1442: Geometry: Deformation and Rigidity - D01: Amenable dynamics via C*-algebras (2020-2024)
Three regularity properties and their interplay have been at the heart of exciting recent developments in the structure and classification theory of nuclear C*-algebras: finite nuclear dimension, tensorial absorption of the Jiang–Su algebra, and strict comparison of positive elements. There are corresponding properties for group actions; we will study these dynamic regularity properties in order to gain new insights into amenable groups and their actions, and on rigidity properties of their associated C*-algebras. Taking a dual viewpoint, we will study - and to some extent classify - Cartan subalgebras of C*-algebras. These are maximal abelian subalgebras, the position of which encapsulates crucial information about the underlying dynamics of a C*-algebra.

$\bullet$ Siegfried Echterhoff: CRC 1442: Geometry: Deformation and Rigidity - D02: Exotic crossed products and the Baum–Connes conjecture (2020-2024)
The Baum–Connes conjecture on the K-theory of crossed products by group actions on C*-algebras is one of the central problems in noncommutative geometry. The conjecture holds for large classes of groups and has important applications in other areas of mathematics. However, there are groups for which the conjecture fails to be true and in this project we study a new formulation of the conjecture due to Baum, Guentner and Willett which avoids the known counterexamples for the classical one. This involves new exotic crossed product functors which differ from the classical maximal or reduced crossed products.

$\bullet$ Christoph Böhm, Burkhard Wilking: CRC 1442: Geometry: Deformation and Rigidity - Geometric evolution equations (2020-2024)
Hamilton’s Ricci flow is a geometric evolution equation on the space of Riemannian metrics of a smooth manifold. In a first subproject we would like to show a differentiable stability result for noncollapsed converging sequences of Riemannian manifolds with nonnegative sectional curvature, generalising Perelman’s topological stability. In a second subproject, next to classifying homogeneous Ricci solitons on non-compact homogeneous spaces, we would like to prove the dynamical Alekseevskii conjecture. Finally, in a third subproject we would like to find new Ricci flow invariant curvature conditions, a starting point for introducing a Ricci flow with surgery in higher dimensions.

$\bullet$ Michael Wiemeler, Burkhard Wilking: CRC 1442: Geometry: Deformation and Rigidity - B01: Curvature and Symmetry (2020-2024)
The question of how far geometric properties of a manifold determine its global topology is a classical problem in global differential geometry. In a first subproject we study the topology of positively curved manifolds with torus symmetry. We think that the methods used in this subproject can also be used to attack the Salamon conjecture for positive quaternionic Kähler manifolds. In a third subproject we study fundamental groups of non-negatively curved manifolds. Two other subprojects are concerned with the classification of manifolds all of whose geodesics are closed and the existence of closed geodesics on Riemannian orbifolds.

$\bullet$ Raimar Wulkenhaar: CRC 1442: Geometry: Deformation and Rigidity - D03: Integrability (2020-2024)
The project investigates a novel integrable system which arises from a quantum field theory on noncommutative geometry. It is characterised by a recursive system of equations with conjecturally rational solutions. The goal is to deduce their generating function and to relate the rational coefficients in the generating function to intersection numbers of tautological characteristic classes on some moduli space.

$\bullet$ Arthur Bartels: CRC 1442: Geometry: Deformation and Rigidity - C03: K-theory of group algebras (2020-2024)
The Farrell–Jones conjecture gives a homological formula for the K-theory of group rings. On the one hand this conjecture has concrete applications, e.g. to the rigidity of aspherical manifolds. On the other hand its underlying principle is very general, and applies for example also to Hecke algebras of p-adic Lie groups. We will extend the scope of positive results for this conjecture, develop new tools to study the conjecture and extend the framework where is can be formulated and applied.

$\bullet$ Arthur Bartels: CRC 1442: Geometry: Deformation and Rigidity - Z01: Central Task of the Collaborative Research Centre (2020-2024)

$\bullet$ Arthur Bartels: CRC 1442: Geometry: Deformation and Rigidity (2020-2024)
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.

$\bullet$ Michael Weiss, Johannes Ebert: CRC 1442: Geometry: Deformation and Rigidity - C01: Automorphisms and embeddings of manifolds (2020-2024)
This project concerns the homotopical properties of spaces of smooth and topological automorphisms of manifolds, their classifying spaces and spaces of smooth and topological embeddings of manifolds. Known characteristic classes for manifold bundles will play an important role. It is conceivable that new ones will be constructed. The action of automorphisms and embeddings on finite subsets of manifolds, more precisely on the configuration category of a manifold, will be exploited.

$\bullet$ Johannes Ebert: CRC 1442: Geometry: Deformation and Rigidity - B03: Moduli spaces of metrics of positive curvature (2020-2024)
In this project, the space of Riemannian metrics of positive scalar curvature on closed manifolds will be studied. Central research questions concern the nontriviality of secondary index invariants, rigidity theorems for the homotopy type of those spaces and the action of the diffeomorphism group, and the comparison of two iterated loop space structures. We will use techniques from differential geometry, higher index theory, metric geometry, differential topology and homotopy theory.

$\bullet$ Raimar Wulkenhaar: GRK 2149 - Starke und schwache Wechselwirkung - von Hadronen zu Dunkler Materie (2020-2024)
The Research Training Group (Graduiertenkolleg) 2149 "Strong and Weak Interactions - from Hadrons to Dark Matter" funded by the Deutsche Forschungsgemeinschaft focuses on the close collaboration of theoretical and experimental nuclear, particle and astroparticle physicists further supported by a mathematician and a computer scientist. This explicit cooperation is of essence for the PhD topics of our Research Training Group.Scientifically this Research Training Group addresses questions at the forefront of our present knowledge of particle physics. In strong interactions we investigate questions of high complexity, such as the parton distributions in nuclear matter, the transition of the hot quark-gluon plasma into hadrons, or features of meson decays and spectroscopy. In weak interactions we pursue questions, which are by definition more speculative and which go beyond the Standard Model of particle physics, particularly with regard to the nature of dark matter. We will confront theoretical predictions with direct searches for cold dark matter particles or for heavy neutrinos as well as with new particle searches at the LHC.The pillars of our qualification programme are individual supervision and mentoring by one senior experimentalist and one senior theorist, topical lectures in physics and related fields (e.g. advanced computation), peer-to-peer training through active participation in two research groups, dedicated training in soft skills, and the promotion of research experience in the international community. We envisage early career steps through a transfer of responsibilities and international visibility with stays at external partner institutions. An important goal of this Research Training Group is to train a new generation of scientists, who are not only successful specialists in their fields, but who have a broader training both in theoretical and experimental nuclear, particle and astroparticle physics.

$\bullet$ Linus Kramer: Geometry and Topology of Artin Groups (2020-2023)

$\bullet$ Wilhelm Winter: Amenability, Approximation and Reconstruction (2019-2024)
Algebras of operators on Hilbert spaces were originally introduced as the right framework for the mathematical description of quantum mechanics. In modern mathematics the scope has much broadened due to the highly versatile nature of operator algebras. They are particularly useful in the analysis of groups and their actions. Amenability is a finiteness property which occurs in many different contexts and which can be characterised in many different ways. We will analyse amenability in terms of approximation properties, in the frameworks of abstract C*-algebras, of topological dynamical systems, and of discrete groups. Such approximation properties will serve as bridging devices between these setups, and they will be used to systematically recover geometric information about the underlying structures. When passing from groups, and more generally from dynamical systems, to operator algebras, one loses information, but one gains new tools to isolate and analyse pertinent properties of the underlying structure. We will mostly be interested in the topological setting, and in the associated C*-algebras. Amenability of groups or of dynamical systems then translates into the completely positive approximation property. Systems of completely positive approximations store all the essential data about a C*-algebra, and sometimes one can arrange the systems so that one can directly read of such information. For transformation group C*-algebras, one can achieve this by using approximation properties of the underlying dynamics. To some extent one can even go back, and extract dynamical approximation properties from completely positive approximations of the C*-algebra. This interplay between approximation properties in topological dynamics and in noncommutative topology carries a surprisingly rich structure. It connects directly to the heart of the classification problem for nuclear C*-algebras on the one hand, and to central open questions on amenable dynamics on the other.