 SYMMETRY IN NEWCASTLE
 Location: Room V109, Mathematics Building (Callaghan Campus) The University of Newcastle
 Dates: Fri, 1^{st} Nov 2019  Fri, 1^{st} Nov 2019

Schedule:
121: Anthony Dooley
12: Lunch
23: Colin Reid
33.30: Tea
3.304.30: Michael Barnsley
 Speaker: Prof Anthony Dooley, University of Technology Sydney
 Title: Classification of nonsingular systems and critical dimension
 Abstract for Classification of nonsingular systems and critical dimension:
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 $X$.
 Speaker: Dr Colin Reid, CARMA, The University of Newcastle
 Title: Piecewise powers of a minimal homeomorphism of the Cantor set
 Abstract for Piecewise powers of a minimal homeomorphism of the Cantor set:
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.
 Speaker: Prof Michael Barnsley, Mathematical Sciences Institute, Australian National University
 Title: Dynamics on Fractals
 Abstract for Dynamics on Fractals:
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.
 Abstract for Classification of nonsingular systems and critical dimension:
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 $X$.
 Abstract for Piecewise powers of a minimal homeomorphism of the Cantor set:
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.
 Abstract for Dynamics on Fractals:
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.
 [Permanent link]
 ZERODIMENSIONAL SYMMETRY SEMINAR
 Speaker: Dr Colin Reid, CARMA, The University of Newcastle
 Title: Endomorphisms of profinite groups
 Location: Room MC102, McMullin (Callaghan Campus) The University of Newcastle
 Time and Date: 2:00 pm, Mon, 10^{th} Sep 2018
 Abstract:
Given a profinite group $G$, we can consider the semigroup $\mathrm{End}(G)$ of continuous homomorphisms from $G$ to itself. In general $\lambda \in\mathrm{End}(G)$ can be
injective but not surjective, or vice versa: consider for instance the case when $G$ is the group $F_p[[t]$ of formal power series over a finite field, $n$ is an integer, and
$\lambda_n$ is the continuous endomorphism that sends $t^k$ to $t^{k+n}$ if $k+n \ge 0$ and $0$ otherwise. However, when $G$ has only finitely many open subgroups of each
index (for instance, if $G$ is finitely generated), the structure of endomorphisms is much more restricted: given $\lambda \in\mathrm{End}(G)$, then $G$ can be written as a
semidirect product $N \rtimes H$ of closed subgroups, where $\lambda$ acts as an automorphism on $H$ and a contracting endomorphism on $N$. When $\lambda$ is open and
injective, the structure of $N$ can be restricted further using results of Glöckner and Willis (including the very recent progress that George told us about a few weeks ago). This
puts some restrictions on the profinite groups that can appear as a '$V_+$' group for an automorphism of a t.d.l.c. group.
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 CARMA COLLOQUIUM
 Speaker: Dr Colin Reid, CARMA, The University of Newcastle
 Title: Totally disconnected, locally compact groups
 Location: Room V205, Mathematics Building (Callaghan Campus) The University of Newcastle
 Time and Date: 4:00 pm, Tue, 28^{th} Mar 2017
 Abstract:
Totally disconnected, locally compact (t.d.l.c.) groups are a large class of topological groups that arise from a few different sources, for instance as automorphism groups of a range of algebraic and combinatorial structures, or from the study of isomorphisms between finite index subgroups of a given group. A general theory has begun to emerge in recent years, based on the interaction between smallscale and largescale structure in t.d.l.c. groups. I will give a survey of some ways in which these groups arise and some of the tools that have been developed for understanding them.
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 CARMA GROUP THEORY RHD MEETING
 Speaker: Dr Colin Reid, CARMA, The University of Newcastle
 Title: Proof of the plocalisation theorem
 Location: Room V205, Mathematics Building (Callaghan Campus) The University of Newcastle
 Time and Date: 11:00 am, Thu, 13^{th} Oct 2016
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 CARMA GROUP THEORY RHD MEETING
 Speaker: Dr Colin Reid, CARMA, The University of Newcastle
 Title: Introduction to plocalisations
 Location: Room V205, Mathematics Building (Callaghan Campus) The University of Newcastle
 Time and Date: 11:00 am, Thu, 1^{st} Sep 2016
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 CARMA COLLOQUIUM
 Speaker: Dr Colin Reid, CARMA, The University of Newcastle
 Title: Locally normal subgroups of totally disconnected groups
 Location: Room V129, Mathematics Building (Callaghan Campus) The University of Newcastle
 Time and Date: 4:00 pm, Thu, 1^{st} Nov 2012
 Abstract:
I will give an extended version of my talk at the AustMS meeting about some ongoing work with PierreEmmanuel Caprace and George Willis.
Given a locally compact topological group G, the connected component of the identity is a closed normal subgroup G_0 and the quotient group is totally disconnected. Connected locally compact groups can be approximated by Lie groups, and as such are relatively wellunderstood. By contrast, totally disconnected locally compact (t.d.l.c.) groups are a more difficult class of objects to understand. Unlike in the connected case, it is probably hopeless to classify the simple t.d.l.c. groups, because this would include for instance all simple groups (equipped with the discrete topology). Even classifying the finitely generated simple groups is widely regarded as impossible. However, we can prove some general results about broad classes of (topologically) simple t.d.l.c. groups that have a compact generating set.
Given a nondiscrete t.d.l.c. group, there is always an open compact subgroup. Compact totally disconnected groups are residually finite, so have many normal subgroups. Our approach is to analyse a t.d.l.c. group G (which may itself be simple) via normal subgroups of open compact subgroups. From these we obtain lattices and Cantor sets on which G acts, and we can use properties of these actions to demonstrate properties of G. For instance, we have made some progress on the question of whether a compactly generated topologically simple t.d.l.c. group is abstractly simple, and found some necessary conditions for G to be amenable.
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