The contraction subgroup for $x$ in the locally compact group, $G$, $\mathrm{con}(x) = \left\{ g\in G \mid x^ngx^{-n} \to 1\text{ as }n\to\infty \right\}$, and the Levi subgroup is $\mathrm{lev}(x) = \left\{ g\in G \mid \{x^ngx^{-n}\}_{n\in\mathbb{Z}} \text{ has compact closure}\right\}$. The following will be shown. Let $G$ be a totally disconnected, locally compact group and $x\in G$. Let $y\in{\sf lev}(x)$. Then there are $x'\in G$ and a compact subgroup, $K\leq G$ such that:
-$K$ is normalised by $x'$ and $y$,
-$\mathrm{con}(x') = \mathrm{con}(x)$ and $\mathrm{lev}(x') = \mathrm{lev}(x)$ and
-the group $\langle x',y,K\rangle$ is abelian modulo $K$, and hence flat.
If no compact open subgroup of $G$ normalised by $x$ and no compact open subgroup of $\mathrm{lev}(x)$ normalised by $y$, then the flat-rank of $\langle x',y,K\rangle$ is equal to $2$.

It will be seen in this talk that certain geometrical theorems may be proved rigorously by checking only three cases. The idea is what Doron Zeilberger calls an 'Ansatz' -- that once we know the general form of a solution we can find the exact solution by checking a few cases. He gives examples where formulae usually established by induction can in fact be proved by checking a small number of cases. We shall do the same for Napoleon's Theorem and also for geometric theorems which don't seem to have been known either to the ancients or to Napoleon Bonaparte.

High-school mathematics only will be assumed. Themes such as computer algebra and notions of proof will be touched on, as will the historical context of ideas such as calculus and complex numbers seen in first-year university mathematics courses.

This project aims to investigate algebraic objects known as 0-dimensional groups, which are a mathematical tool for analysing the symmetry of infinite networks. Group theory has been used to classify possible types of symmetry in various contexts for nearly two centuries now, and 0-dimensional groups are the current frontier of knowledge. The expected outcome of the project is that the understanding of the abstract groups will be substantially advanced, and that this understanding will shed light on structures possessing 0-dimensional symmetry. In addition to being cultural achievements in their own right, advances in group theory such as this also often have significant translational benefits. This will provide benefits such as the creation of tools relevant to information science and researchers trained in the use of these tools.

The project aims to develop novel techniques to investigate Geometric analysis on infinite dimensional bundles, as well as Geometric analysis of pathological spaces with Cantor set as fibre, that arise in models for the fractional quantum Hall effect and topological matter, areas recognised with the 1998 and 2016 Nobel Prizes. Building on the applicant's expertise in the area, the project will involve postgraduate and postdoctoral training in order to enhance Australia's position at the forefront of international research in Geometric Analysis. Ultimately, the project will enhance Australia's leading position in the area of Index Theory by developing novel techniques to solve challenging conjectures, and mentoring HDR students and ECRs.

This project aims to solve hard, outstanding problems which have impeded our ability to progress in the area of quantum or noncommutative calculus. Calculus has provided an invaluable tool to science, enabling scientific and technological revolutions throughout the past two centuries. The project will initiate a program of collaboration among top mathematical researchers from around the world and bring together two separate mathematical areas into a powerful new set of tools. The outcomes from the project will impact research at the forefront of mathematical physics and other sciences and enhance Australia's reputation and standing.

(joint work with R. Grigorchuk ad D. Horadam) The tree representation theorem represents a certain group associated with the scale of an automorphism of a t.d.l.c. group as acting by symmetries of a regular (unrooted) tree. It shows that groups acting on regular trees are a fundamental part of the theory of t.d.l.c. groups.
There is also an extensive theory of self-similar and self-replicating groups of symmetries of rooted trees which has developed from the discovery (or creation) of examples such as the Grigorchuk groups.
It will be seen in this talk that these two branches of research are studying essentially the same groups.

See here for an abstract.

This talk will highlight links between topics studied in undergraduate mathematics on one hand and frontiers of current research in analysis and symmetry on the other. The approach will be semi-historical and will aim to give an impression of what the research is about. 



Fundamental ideas in calculus, such as continuity, differentiation and integration, are first encountered in the setting of functions on the real line. In addition to topological properties of the line, the algebraic properties of being a group and a field, that the set of real numbers possesses, are also important. These properties express symmetries of the set of real numbers, and it turns out that this combination of calculus, algebra and symmetry extends to the setting of functions on locally compact groups, of which the group of rotations of a sphere and the group of automorphisms of a locally finite graph are examples. Not only do these groups frequently occur in applications, but theorems established prior to 1955 show that they are exactly the groups that support integration and differentiation. 



Integration and continuity of functions on the circle and the group of rotations of the circle are the basic ingredients for Fourier analysis, which deals with convolution function algebras supported on the circle. Since these basic ingredients extend to locally compact groups, so do the methods of Fourier analysis, and the study of convolution algebras on these groups is known as harmonic analysis. Indeed, there is such a close connection between harmonic analysis and locally compact groups that any locally compact group may be recovered from the convolution algebras that it carries. This fact has recently been exploited with the creation of a theory of `locally compact quantum groups' that axiomatises properties of the algebras appearing in harmonic analysis and does away with the underlying group. 


Locally compact groups have a rich structure theory in which significant advances are also currently being made. This theory divides into two cases: when the group is a connected topological space and when it is totally disconnected. The connected case has been well understood since the solution of Hilbert's Fifth Problem in the 1950's, which showed that they are essentially Lie groups. (Lie groups form the symmetries of smooth structures occurring in physics and underpinned, for example, the prediction of the existence of the Higgs boson.) For a long time it was thought that little could be said about totally disconnected groups in general, although important classes of such groups arising in number theory and as automorphism groups of graphs could be understood using techniques special to those classes. However, a complete general theory of these groups is now beginning to take shape following several breakthroughs in recent years. There is the exciting prospect that an understanding of totally disconnected groups matching that of the connected groups will be achieved in the next decade.

The topological and measure structures carried by locally compact groups make them precisely the class of groups to which the methods of harmonic analysis extend. These methods involve study of spaces of real- or complex-valued functions on the group and general theorems from topology guarantee that these spaces are sufficiently large. When analysing particular groups however, particular functions deriving from the structure of the group are at hand. The identity function in the cases of $(\mathbb{R},+)$ and $(\mathbb{Z},+)$ are the most obvious examples, and coordinate functions on matrix groups and growth functions on finitely generated discrete groups are only slightly less obvious.

In the case of totally disconnected groups, compact open subgroups are essential structural features that give rise to positive integer-valued functions on the group. The set of values of $p$ for which the reciprocals of these functions belong to $L^p$ is related to the structure of the group and, when they do, the $L^p$-norm is a type of $\zeta$-function of $p$. This is joint work with Thomas Weigel of Milan.

The restricted product over $X$ of copies of the $p$-adic numbers $\mathbb{Q}_p$, denoted $\mathbb{Q}_p(X)$, is self-dual and is the natural $p$-adic analogue of Hilbert space. The additive group of this space is locally compact and the continuous endomorphisms of the group are precisely the continuous linear operators on $\mathbb{Q}_p(X)$.

Attempts to develop a spectral theory for continuous linear operators on $\mathbb{Q}_p(X)$ will be described at an elementary level. The Berkovich spectral theory over non-Archimedean fields will be summarised and the spectrum of the linear operator $T$ compared with the scale of $T$ as an endomorphism of $(\mathbb{Q}_p(X),+)$.

The original motivation for this work, which is joint with Andreas Thom (Leipzig), will also be briefly discussed. A certain result that holds for representations of any group on a Hilbert space, proved by operator theoretic methods, can only be proved for representations of sofic groups on $\mathbb{Q}_p(X)$ and it is thought that the difficulty might lie with the lack of understanding of linear operators on $\mathbb{Q}_p(X)$ rather than with non-sofic groups.

Mathematics can often seen almost too good to be true. This sense that mathematics is marvellous enlivens learning and stimulates research but we tend to let remarkable things pass without remark after we become familiar with them. The miracles of Pythagorean triples and eigenvalues will be highlights of this talk.

The talk will include some ideas of what could be blending into our teaching program.

I will review the creation and development of the concept of number and the role of visualisation in that development. The relationship between innate human capabilities on the one hand and mathematical research and education on the other will be discussed.

This colloquium will explain some of the background and significance of the concept of amenability. Arguments with finite groups frequently, without remark, count the number of elements in a subset or average a function over the group. It is usually important in these arguments that the result of the calculation is invariant under translation. Such calculations cannot be so readily made in infinite groups but the concepts of amenability and translation invariant measure on a group in some ways take their place. The talk will explain this and also say how random walks relate to these same ideas.

The link to the animation of the paradoxical decomposition is here.

You are invited to a celebration of the 21st anniversary of the Factoring Lemma. This lemma was the key to solving some long-standing open problems, and was the starting point of an investigation of totally disconnected, locally compact groups that has ensued over the last 20 years. In this talk, the life of the lemma will described from its conception through to a very recent strengthening of it. It will be described at a technical level, as well as viewed through its relationships with topology, geometry, combinatorics, algebra, linear algebra and research grants.

A birthday cake will be served afterwards.

Please make donations to the Mathematics Prize Fund in lieu of gifts.

George continues his series of talks on the embedding of Higman-Thompson groups into the groups of almost automorphisms of trees, structure theorems and scale calculations for these examples.

George continues his series of talks on the embedding of Higman-Thompson groups into the groups of almost automorphisms of trees, structure theorems and scale calculations for these examples.

George is going to continue his series of talks on the embedding of Higman-Thompson groups into the groups of almost automorphisms of trees, and over several weeks look at the structure theorems and scale calculations for these examples.

George is going to start giving some talks on the embedding of Higman-Thompson groups into the groups of almost automorphisms of trees, and over several weeks will look at the structure theorems and scale calculations for these examples.

Noncommutative geometry is based on fairly sophisticated methods: noncommutative C*-algebras are called noncommutative topological spaces, noncommutative von Neumann algebras are noncommutative measure spaces, and Hopf algebras and homological invariants describe the geometry. Standard topology, on the other hand, is based on naive intuitions about discontinuity: a continuous function is one whose graph does not have any gaps, and cutting and gluing are used to analyse and reconstruct geometrical objects. This intuition does not carry over to the noncommutative theory, and the dictum from quantum mechanics that it does not make sense any more to think about point particles perhaps explains a lack of expectation that it should. The talk will describe an attempt to make this transfer by computing the polar decompositions of certain operators in the group C*-algebras of free groups. The computation involves some identities and evaluations of integrals that might interest the audience, and the polar decomposition may be interpreted as a noncommutative version of the double angle formula familiar from high school geometry.

McCullough-Miller space X = X(W) is a topological model for the outer automorphism group of a free product of groups W. We will discuss the question of just how good a model it is. In particular, we consider circumstances under which Aut(X) is precisely Out(W).

The talk will explain joint work with Yehuda Shalom showing that the only homomorphisms from certain arithmetic groups to totally disconnected, locally compact groups are the obvious, or naturally occurring, ones. For these groups, this extends the supperrigidity theorem that G. Margulis proved for homomorphisms from high rank arithmetic groups to Lie groups. The theorems will be illustrated by referring to the groups $SL_3(\mathbb{Z})$, $SL_2(\mathbb{Z}[\sqrt{2}])$ and $SL_3(\mathbb{Q})$.

http://www.maths.usyd.edu.au/u/SemConf/InfGps/index.html

McCullough-Miller space X = X(W) is a topological model for the outer automorphism group of a free product of groups W. We will discuss the question of just how good a model it is. In particular, we consider circumstances under which Aut(X) is precisely Out(W).

The talk will explain joint work with Yehuda Shalom showing that the only homomorphisms from certain arithmetic groups to totally disconnected, locally compact groups are the obvious, or naturally occurring, ones. For these groups, this extends the supperrigidity theorem that G. Margulis proved for homomorphisms from high rank arithmetic groups to Lie groups. The theorems will be illustrated by referring to the groups $SL_3(\mathbb{Z})$, $SL_2(\mathbb{Z}[\sqrt{2}])$ and $SL_3(\mathbb{Q})$.

http://www.maths.usyd.edu.au/u/SemConf/InfGps/index.html