# Spaces and Operators (Area B)

Bartels, Bellissard, Böhm, Cuntz, Ebert, Echterhoff, Gardella, Joachim, Kramer, de Laat, Lohkamp, Löwe, Weiss, 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 establishe 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.

# Research projects of Area B members

$\bullet$ 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.

$\bullet$ Eusebio Gardella: Amenability, structure and regularity of group actions on C*-algebras (2019-2021)
In mathematics, the study of symmetries and group actions is one of the most fundamental and prolific fields of research. In operator algebras, the classification of integer actions was instrumental in Connes' award-winning proof of uniqueness of amenable type III factors. Far reaching generalizations culminated in the classification of amenable group actions on amenable factors. Despite the efforts, the area of C*-dynamics is far less developed. However, recent breakthroughs have given new impetus to the field. This project will make exceptional progress in C*-dynamics, consolidating the area as one of the most active ones in C*-algebras. Along with the main goals of this project, we will also explore a number of problems related to dynamics both on C*-algebras and von Neumann algebras, whose solution will improve our understanding of the field. Some of these problems have tight connections to topological and measurable dynamics, and are thus of an interdisciplinary nature.

$\bullet$ Raimar Wulkenhaar: Nichtperturbative Gruppenfeldtheorie durch kombinatorische Dyson-Schwinger-Gleichungen und ihre algebraische Struktur (2019-2021)

$\bullet$ Raimar Wulkenhaar: RTG 2149 - Srong and Weak Interactions - from Hadrons to Dark Matter (2015-2020)
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.