CARMA Research Group
Symmetry
About us
Leader: George Willis
Symmetry is accounted for mathematically through the algebraic concept of a 'group'. Our research focusses on '0dimensional', aka 'totally disconnected, locally compact', groups, which arise as symmetries of graphs (in the sense of networks). We aim to analyse the structure of these groups and thus to understand the types of symmetries that graphs may possess. Whereas connected locally compact groups are well understood, much remains to be done in the totally disconnected case and we are filling significant gaps in knowledge. The research also has significant links with harmonic analysis, number theory and geometry — ideas from these fields are used in our research and our results feed back to influence these other fields. Another part of our research aims to develop computational tools for working with the groups and for visualising the corresponding symmetries.
We collaborate with researchers in Europe, Asia and North and South America in this work, which is being supported by Australian Research Council funds of $2.8 million in the period 201822.
More information about us and our research can be at https://zerodimensional.group. Feel free to contact us at contact@zerodimensional.group.
People
Members of this research group:
 George Willis
 Michal Ferov
 Alejandra Garrido
 Colin Reid
 David Robertson
 Stephan Tornier
Activities

"What is a selfsimilar group?"
Dr Alejandra Garrido
Abstract:
Selfsimilarity (when part of an object is a scaled version of the whole) is one of the most basic forms of symmetry. While known and used since ancient times, its use and investigation took off in the 1980s thanks to the advent of fractals, whose infinite selfsimilar structure has captured the imagination of mathematicians and lay people alike.
Selfsimilar fractals are highly symmetrical, so much so that even their symmetry groups exhibit selfsimilarity. In this talk, I will introduce and discuss groups which are selfsimilar, or fractal, in an algebraic sense; their connections to fractals, symbolic dynamics and automata theory; how they produce fascinating new examples in group theory, and some research questions in this lively new area. 
Symmetry in Newcastle
12:00 pm, Friday, 1st Nov 2019
V109, Mathematics Building
Schedule:
121: Anthony Dooley
12: Lunch
23: Colin Reid
33.30: Tea
3.304.30: Michael Barnsley"Classification of nonsingular systems and critical dimension"
— Prof Anthony Dooley
Abstract:
A nonsingular measurable dynamical system is a measure space $X$ whose measure $\mu$ has the property that $\mu $ and $\mu \circ T$ are equivalent measures (in the sense that they have the same sets of measure zero). Here $T$ is a bimeasurable invertible transformation of $X$. The basic building blocks are the \emph{ergodic} measures. Von Neumann proposed a classification of nonsingular ergodic dynamical systems, and this has been elaborated subsequently by Krieger, Connes and others. This work has deep connections with C*algebras. I will describe some work of myself, collaborators and students which explore the classification of dynamical systems from the point of view of measure theory. In particular, we have recently been exploring the notion of critical dimension, a study of the rate of growth of sums of RadonNikodym derivatives $\Sigma_{k=1}^n \frac{d\mu \circ T^k}{d\mu}$. Recently, we have been replacing the single transformation $T$ with a group acting on the space $X$.
"Piecewise powers of a minimal homeomorphism of the Cantor set"
— Dr Colin Reid
Abstract:
Let $X$ be the Cantor set and let $g$ be a minimal homeomorphism of $X$ (that is, every orbit is dense). Then the topological full group $\tau[g]$ of $g$ consists of all homeomorphisms $h$ of $X$ that act 'piecewise' as powers of $g$, in other words, $X$ can be partitioned into finitely many clopen pieces $X_1,...,X_n$ such that for each $i$, $h$ acts on $X_i$ as a constant power of $g$. Such groups have attracted considerable interest in dynamical systems and group theory, for instance they characterize the homeomorphism up to flip conjugacy (GiordanoPutnamSkau) and they provided the first known examples of infinite finitely generated simple amenable groups (JuschenkoMonod). My talk is motivated by the following question: given $h\in\tau[g]$ for some minimal homeomorphism $g$, what can the closures of orbits of $h$ look like? Certainly $h\in\tau[g]$ is not minimal in general, but it turns out to be quite close to being minimal, in the following sense: there is a decomposition of $X$ into finitely many clopen invariant pieces, such that on each piece $h$ acts a homeomorphism that is either minimal or of finite order. Moreover, on each of the minimal parts of $h$, then either $h$ or $h^{1}$ has a 'positive drift' with respect to the orbits of $g$; in fact, it can be written in a canonical way as a conjugate of a product of induced transformations (aka first return maps) of $g$. No background knowledge of topological full groups is required; I will introduce all the necessary concepts in the talk.
"Dynamics on Fractals"
— Prof Michael Barnsley
Abstract:
I will outline a new theory of fractal tilings. The approach uses graph iterated function systems (IFS) and centers on underlying symbolic shift spaces. These provide a zero dimensional representation of the intricate relationship between shift dynamics on fractals and renormalization dynamics on spaces of tilings. The ideas I will describe unify, simplify, and substantially extend key concepts in foundational papers by Solomyak, Anderson and Putnam, and others. In effect, IFS theory on the one hand, and selfsimilar tiling theory on the other, are unified.
The work presented is largely new and has not yet been submitted for publication. It is joint work with Andrew Vince (UFL) and Louisa Barnsley. The presentation will include links to detailed notes. The figures illustrate 2d fractal tilings.
By way of recommended background reading I mention the following awardwinning paper: M. F. Barnsley, A. Vince, Selfsimilar polygonal tilings, Amer. Math. Monthly 124 (1017) 905921.