GrDyAp by EU Research

Studying the fundamentals of infinite symmetry Groups are of fundamental importance throughout mathematics, and new analytical methods are being developed to gain deeper insights into unresolved questions in the field. Professor Andreas Thom tells us about the work of the GrDyAp project in connecting group theory, functional analysis, and the theory of dynamical systems, which could open up new perspectives on long-standing mathematical problems. The study of symmetry is at the heart of pure mathematics, with researchers building on the existing foundations of group theory to approach long-standing problems in the field. Groups arise as symmetries of objects and are thus of fundamental importance across different branches of mathematics, now researchers in the ERC-funded GrDyAp project are looking deeper into the subject. “The project tries to connect group theory, functional analysis, and the theory of dynamical systems,” says Andreas Thom, Professor of Mathematics at Technische Universität Dresden, the project’s Principal Investigator. This work also touches on several other areas within the pure mathematics field. “My research is solely in pure mathematics, but it brings together different branches of pure mathematics,” continues Professor Thom. A group itself can be understood as a type of algebraic structure consisting of symmetries which can act on an object, like a geometric figure or something more complicated. For example, a cube has 24 symmetries which act on the cube; this may on the surface seem difficult to fully understand, but things get clearer when one looks at the cube in more detail. “Each symmetry of the cube has a concrete meaning. An abstract group cannot be understood directly, precisely because a natural object is missing, on which it acts by

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symmetries (whose symmetries it is). Now, in geometric group theory, such an object is constructed — the Cayley graph,” explains Professor Thom. Symmetries can explain complex behaviour of phenomena, ranging from material sciences to quantum physics to everyday objects, says Professor Thom. “Why is a dice fair? Because it has so many symmetries that make it obvious that each side has equal probability,” he points out.

Infinite symmetry groups The major goal in the project now is to study infinite symmetry groups, with Professor Thom and his colleagues looking to develop

novel methods to approach some of the major challenges in infinite group theory. Groups can be found throughout mathematics, and while significant progress has been made in classifying some types of groups, notably finite symmetry groups, Professor Thom says that infinite symmetry groups are particularly challenging. “Infinite groups are harder to study because one cannot write down a complete multiplication table. One has to understand them by other means, for example via the geometric object whose symmetries they describe,” he outlines. “One such object is the Cayley graph, but frequently there are also even more geometric objects, such as

methods to address these types of problems, including approximation theorems. “My focus recently has been more on approximation and the stability of groups,” he outlines. Permutations are symmetries of finite sets. The Gromov conjecture, which states that all groups are approximable by permutations, is a problem of particular interest to Professor Thom. “Stability says that any approximation can be made a true equality. It is very surprising that stable groups exist, and together with my PhD student Marcus de Chiffre - who is also in the ERC group - we have found the first non-trivial examples of stable groups,” he says.

Mathematical concepts The wider goal of mathematics in general, according to Bill Thurston, is to reduce any confusion which still remains around core mathematical concepts and to build a deeper understanding of the interplay between basic notions. While this research is largely fundamental in nature, the hope is that the insights gained will prove useful in other mathematical contexts. “We are always looking for generalizations,” stresses Professor Thom. These novel methods could in future be

applied on other mathematical problems, for example the Bergeron-Venkatesh conjecture. “It might be that at some point in future I will realize that a particular method I previously developed is also useful to understand this conjecture. This has happened with other problems in the past, and I am aware of a lot of maths problems in neighboring areas,” says Professor Thom. Further investigation in these areas would require deep understanding and sophisticated methods, underlining the importance of interdisciplinary research. Professor Thom believes it is important to share expertise and learn about the methods that are being applied in other fields. “It is always fun to work in an interdisciplinary way and to apply methods from one field in another,” he stresses. This could then open up new avenues of research and spark further investigation; one longerterm goal for Professor Thom is to use these novel methods to address the growth of torsion in sequences of lattices in hyperbolic groups, while he says there are also many other unresolved questions. “Group theory and functional analysis are fascinating topics, there are still so many interesting open problems,” he says.

GrDyAp Groups, Dynamics, and Approximation

Project Objectives

The study of inﬁnite symmetry groups is a particularly challenging part of group theory, as most of the tools from the sophisticated theory of ﬁnite groups break down, so new global methods of study have to be found. The interaction of group theory and the study of group rings with methods from ring theory, probability, Riemannian geometry, functional analyis, and the theory of dynamical systems has been extremely fruitful in a variety of situations. In the GrDyAp project, Professor Thom and his colleagues aim to extend this line of approach and introduce novel approaches to long standing and fundamental problems.

Project Funding

ERC - Consolidator Grant

Contact Details

Project Coordinator, Professor Andreas Thom Institute of Geometry TU Dresden 01069 Dresden T: +49 351 463-43074 E: andreas.thom@tu-dresden.de W: https://tu-dresden.de/mn/math/ geometrie/thom/forschung/erc-projekte

Professor Andreas Thom

manifolds, or more analytic objects, such as Banach spaces.” A lot of attention in research is now focused on developing new methods to study these infinite symmetry groups, work which brings together elements of different branches of mathematics. For his part, Professor Thom is applying randomization and algebraic approximation techniques in this area. “I have been using randomization techniques to construct laws of groups, special equations that all elements of the group satisfy. In a sense, these are generalizations of the identity x+y=y+x for the integers, but they are much more complicated in form and valid in much more complicated groups,” he outlines. “Algebraic approximation on the other hand is a special interplay between ring theory and functional analysis that I have used in joint work with Guillermo Cortinas (Buenos Aires) to prove Rosenberg’s Conjecture.” This particular conjecture relates to a special question about algebraic K-theory of rings of continuous functions, while Professor Thom and his colleagues are also investigating other interesting problems in the field. One is Dixmier’s Conjecture, put forward by the French mathematician Jacques Dixmier,

which centres on the representations of a group on Hilbert space, the most symmetrical object imaginable. “Dixmier’s conjecture is now almost 70 years old and has inspired a lot of research over time. Together with Maria Gerasimova, a PhD student in my ERC group, we have made some progress relating this problem with properties of the Cayley graph of a group, in fact all the Cayley graphs of the group,” says Professor Thom.

are also used to study discrete objects. In fact they are very useful,” outlines Professor Thom. Amenability is an analytic property of the action of a group on a Cayley graph, first described by John von Neumann in the late ‘20s. “Dixmier’s conjecture is about the understanding of amenable groups. Dixmier asked whether all unitarisable groups are amenable,” continues Professor Thom. The converse had already been proved to

Andreas Thom is Professor of Mathematics at Technische Universität Dresden, a position he has held since October 2016. is a member of several mathematical societies, including the Max-Planck Institute, ‘Mathematics in the Sciences’ and is also a member of the editorial board of ‘Mathematische Annalen’.

My research is solely in pure mathematics, but it brings together different branches of pure mathematics, such as group theory, the theory of dynamical systems, algebraic topology, and functional analysis. The Dixmier conjecture itself is about the understanding of amenable groups, which define an important class of discrete groups, in a sense close to familiar groups like the group of integers or finite groups. Given that discrete groups are more combinatorial in nature, it might be expected that mathematicians would not be keen to use analytical methods like derivatives and real valued functions in this area, yet this is in fact not the case. “In modern mathematics analytical methods

Marcus de Chiffre

be true, namely that amenable groups can be unitarised. It follows that Dixmier’s questions would really imply that a group is amenable exactly when it is unitarisable, hence unitarisability would be a characterization of amenability, a topic of great interest to Professor Thom. “This would indeed be very striking and I would really like to know an answer to this question,” he says. Alongside group theory, Professor Thom is also investigating the possibility of using other

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Maria Gerasimova

Tiling of the hyperbolic plane (Image source Wikimedia Commons).