# 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.