Research Group — Zero-Dimensional Symmetry


January 2021

Ph.D. scholarship available

A Ph.D. position is available in 2021 to work with Dr Stephan Tornier and other members of the Zero-Dimensional Symmetry Research Group on the ARC DECRA project "Effective classification of closed vertex-transitive groups acting on trees".

Candidates with a strong interest and demonstrated skills in at least one of the following areas are encouraged to apply: (topological) group theory, computational algebra (e.g. with GAP), graph theory, combinatorics.
See the advertisement for more details.




December 2020

Young Mathematicians Program

The Young Mathematicians Program allows students in Years 7-11 to explore new ideas and be challenged in a supportive and interactive environment. One of the five zoom sessions lead by Ian Renner explored the theme of symmetry under the guidance of George Willis and Stephan Tornier.




November 2020

DECRA success

Stephan Tornier has been awarded a Discovery Early Career Researcher Award for his project "Effective classification of closed vertex-transitive groups acting on trees". The award funds his position for three years, a Ph.D. scholarship and research cost.




September 2020

Symmetry - A video in the NSW Department of Education's SISP program

The NSW Department of Education's STEM Industry School Partnerships (SISP) program aims to provide an educational model that engages students, inspires them to study STEM and prepares them for STEM careers. Below is a contribution from Stephan Tornier on the topic of Symmetry.




September 2020

Summer Projects 2020/21

The following student research projects are related to the Australian Research Council project on 0-Dimensional Symmetry. While they are independent, each one gives a different view on the overall project. Another, more technical, description of this research is that it concerns totally disconnected, locally compact groups. An overview which attempts to explain the broader research project to non-experts may be seen here. (MFO, CC BY-NC-SA 3.0 )

Isolated points in colourings of vertex transitive graphs

(Michal Ferov)

The aim of this project is to describe infinite symmetric graphs that admit a rigid colouring. In the case of locally finite graph, the space of all minimal colourings can be naturally seen as a topological space. This project is concerted with the question of to what extent does the topology of the space of colourings correspond to the geometry and symmetries of the graphs. First natural step is to study how far is the space of colourings away from being perfect, i.e. studying the isolated points. In the space of colourings, isolated points correspond to colourings that are rigid, meaning that if any other colouring agrees with it on a large enough finite subgraph, then they have to be the same. The goal of the project is to give some combinatorial of graphs that graphs that admit rigid colourings, or the converse - a combinatorial description of graphs that do not admit rigid colourings. The project is part of a program of research on 0-dimensional groups, which includes symmetries of infinite graphs. Students working on this project will further their knowledge of combinatorics and graph theory. Knowledge of point-set topology and group theory is advantageous but not necessary.
Local actions of algebraic groups acting on trees

(Stephan Tornier)

Symmetry is a fundamental organising principle in mathematics, science and and the arts. It is formalised in the algebraic notion of a 'group'. The symmetry groups of infinite networks, or graphs, constitute a current research frontier. It has proven fruitful to study these groups by analysing their 'local actions', i. e. the permutation groups that the fixator of a vertex in the graph induces on spheres of varying radii around that vertex. The primary aim of this project is to make the local actions of several theoretically derived symmetry groups of graphs tractable by implementing them on a computer using computational group theory tools, such as GAP. A second step would be to study the resulting permutation groups and their interdependence using a mix of theoretical and computational tools. A student who takes this project will extend his/her knowledge of algebra and learn how to use computer algebra systems designed for computations in group theory, including coding skills.
Geometric representations of self-replicating groups

(George Willis)

This project aims to find geometries that have so-called self-replicating, or fractal, groups as their symmetries. The self-replicating nature of these groups is described by representing them as symmetries of rooted trees but that way of thinking about them hides other patterns that are of interest. The purpose of the project is to see the groups geometrically. Computer algebra software will be used to analyse how the groups act on pairs, triples, etc. of vertices of the trees and then study the polyhedra in which these are edges, faces, etc. Observed patterns in the geometries may be extrapolated to produce new families of self-replicating groups. The project is part of a program of research on symmetry groups of infinite networks, known as 0-dimensional groups. Self-replicating 0-dimensional groups are analogous to eigenvalues and eigenvectors in linear algebra and its applications. It is not necessary to understand this bigger picture in order to do the project however. Students taking this project will extend their knowledge of algebra, graph theory and mathematical software. 'Group' is an algebraic notion and 'rooted tree' is a combinatorial one which arises in the study of data structures.
Symmetry of Graphs

(George Willis)

A graph (a network of vertices and edges) is \emph{vertex-transitive} if it 'looks' the same at all vertices. In algebraic terms, this means that, for any pair of vertices, there is an automorphism of the graph which maps the first vertex to the second. A necessary, but not sufficient, condition for a graph to be vertex-transitive is that all vertices should have the same valency. Analysing graph symmetry therefore involves a deeper study of the relationship between graphs and their automorphism groups than vertex valencies. Vertex-transitive graphs need not be edge-transitive, for example, the horizontal edges of a triangular pyramid lie on 3-cycles in the graph but the vertical edges do not. The particular question investigated in this project is how its symmetry group changes as edges are added to, or removed from, a graph. For finite graphs, a classical theorem of Burnside is relevant when the graph has prime order and, for infinite graphs, the goal is to reduce the vertex valency since this number controls important features of the symmetry group. Students taking this project will extend their knowledge of combinatorics and algebra, and how these two topics interact. The project will be jointly supervised by Brian Alspach.
Visualising $0$-dimensional symmetry

(George Willis)

The word 'symmetry' brings to mind visual images and geometry. It has a broader meaning in mathematics, where we think of regularly repeating patterns and invariance under transformations as displaying symmetry, and where the language of algebra is used to describe symmetry. Visualising the patterns or the dynamics of the transformations remains an effective tool for understanding the algebra however. This project aims to develop software for visualising various aspects of $0$-dimensional symmetry, which is the symmetry of infinite networks and arises in number theory and other parts of algebra as well. The aim is to produce software which may be used by researchers and which will be made available on web-pages of the $0$-Dimensional Symmetry project. Students taking this project will extend their knowledge of algebra, analysis, mathematical software and coding skills. 'Totally disconnected' and 'locally compact' are topological notions; 'group' is an algebraic one; and other concepts will be met in the course of the project.



August 2020

Simon Marais Mathematics Competition

Do you enjoy solving mathematical problems? Join the UoN team now, either as in individual or as a pair, to compete against students from all over the Asia-Pacific region and earn your share of the $100,000 prize pool. See the flyer for registration information and the website for further information.




August 2020

Research Assistant Positions

We are looking to employ research assistants to help with projects to do with graph isomorphisms and groups. The descriptions are given below. These projects would be suitable for mathematics or computer science students. One of the projects requires knowledge of group theory and the other does not.

1. Computations with finite graphs

Description: A research assistant is required to work on a project that computes, for small finite graphs, whether the graph can be the 1-sphere of a vertex-transitive graph. The assistant will work under the supervision of Stephan Tornier and George Willis to implement algorithms that check whether a given graph satisfies compatibility conditions that allow it to be extended to a vertex-transitive graph. We aim to build a catalogue of all such graphs up to some size.

Essential: Basic knowledge of graph theory and programming experience.
Desirable: LaTeX

2. Computations with self-replicating groups

Description: A research assistant is required to work on a project that searches for automorphism group of finite rooted trees that are quotients of self-replicating groups acting on an infinite tree. The assistant will work under the supervision of Stephan Tornier and George Willis to implement algorithms that find such groups incrementally on trees of increasing depth. At first, this work will be verifying previous calculations but we also aim to extend the range of trees for which these groups are known.

Essential: Group theory (Algebra / MATH3120) and programming experience.
Desirable: LaTeX




July 2020

Label Refinement for Graphs

A seminar given by George Willis on "Label Refinement for Graphs"




July 2020

Children's University Newcastle

As part of their 2020 CU On Campus Discovery Days, held online due to COVID-19, the Children's University Newcastle is showing videos about researchers at The University of Newcastle and their work. Below is a contribution from George Willis on the topic of Symmetry.




March 2020

Ph.D. scholarship available

A PhD scholarship opportunity is available for students to investigate totally disconnected, locally compact groups under the supervision of ARC Laureate Professor George Willis. The students will join a team seeking to bring our understanding of these groups to a level comparable to that of finite.




February 2020

Visiting Researcher

Dr Waltraud Lederle from UC Louvain (Belgium) is visiting the School of Mathematical and Physical Sciences, specifically the Zero-Dimensional Symmetry research group, from February 10 to April 7. Her visit is supported by the Sydney Mathematical Research Institute and the Australian Research Council.




October 2019

First-Year Summer Projects

We offer six summer project scholarships around the topic "Puzzles, Codes and Groups" for first year students to be conducted in January and February 2020. Apply to juliane.turner@newcastle.edu.au with your student number. A flyer for the project can be found here.

In this summer project you will explore the mathematical formalisation of the everyday notion of symmetry, which is the algebraic concept of a 'group'. Using groups, we are able to state with certainty whether a given puzzle can be solved (the 15-puzzle can not!) and, if so, compute how many steps are needed. As an example of the far-reaching applicability of this concept, we will look into error-correcting codes, such as the Golay code, which are critical in any digital communication, including NASA's Voyager program. Apart from theoretical studies in the area of groups, possible subprojects include the analysis of a given puzzle, the creation of new ones and the formulation of solution algorithms, as well as the design, analysis and implementation of codes. A team of up to six students will work on this project.




September 2019

Summer Projects 2019/20

The following student research projects are related to the Australian Research Council project on 0-Dimensional Symmetry. While they are independent, each one gives a different view on the overall project. Another, more technical, description of this research is that it concerns totally disconnected, locally compact groups. An overview which attempts to explain the broader research project to non-experts may be seen here. (MFO, CC BY-NC-SA 3.0 )

Symmetry of Graphs

A graph (a network of vertices and edges) is \emph{vertex-transitive} if it 'looks' the same at all vertices. In algebraic terms, this means that, for any pair of vertices, there is an automorphism of the graph which maps the first vertex to the second. A necessary, but not sufficient, condition for a graph to be vertex-transitive is that all vertices should have the same valency. Analysing graph symmetry therefore involves a deeper study of the relationship between graphs and their automorphism groups than vertex valencies. Vertex-transitive graphs need not be edge-transitive, for example, the horizontal edges of a triangular pyramid lie on 3-cycles in the graph but the vertical edges do not. The particular question investigated in this project is how its symmetry group changes as edges are added to, or removed from, a graph. For finite graphs, a classical theorem of Burnside is relevant when the graph has prime order and, for infinite graphs, the goal is to reduce the vertex valency since this number controls important features of the symmetry group. Students taking this project will extend their knowledge of combinatorics and algebra, and how these two topics interact. The project will be jointly supervised by Brian Alspach.
Solvable and nilpotent groups

The notion of a solvable group originated with the work of É. Galois (1832), who showed that a polynomial equation has a solution by radicals if and only if its group of symmetries is solvable. For example, the formula $x=-b\pm\sqrt{b^{2}-4ac}/2a$ is the solution of a quadratic equation by radicals and symmetries of the equation swap the $+$ and $-$ signs. (A group that is not solvable has some factors which are simple.) Groups of upper triangular $n\times n$ real matrices are solvable and also have the topological property of being connected. It may be shown that these are essentially all the connected solvable groups. This project investigates solvable totally disconnected, locally compact groups. Our starting point is groups of upper triangular matrices having integer entries. These groups have the property of being nilpotent, which is stronger than solvability. Students taking this project will extend their knowledge of algebra, analysis and number theory. 'Totally disconnected' and 'locally compact' are topological notions; 'group', 'solvable and 'nilpotent' are algebraic ones; and the integer matrices embed into matrices over the real numbers as well as over other number fields.
Symmetries of Rooted Trees

Self-similar symmetry groups of rooted trees contribute to the study of totally disconnected, locally compact (or $0$-dimensional) groups in much the same way as eigenvalues and eigenvectors contribute to linear algebra. Whereas eigenvalues are complex numbers and we have a complete picture of what they all are, we are still at the stage of trying to describe self-similar symmetry groups of rooted trees. It is not necessary to understand the link with the theory of $0$-dimensional symmetry or with eigenvalues in order to describe these groups. This project aims to find an alternative geometric interpretation of some of the symmetry groups of trees. The idea being that describing these geometries might be a more natural approach to describing the groups. Computer algebra software will be used to analyse how the groups act on pairs, triples, etc. of vertices of the trees and then study the polyhedra in which these are edges, faces, etc. Observed patterns in the geometries may be able to be extrapolated to produce new families of groups. Students taking this project will extend their knowledge of algebra, graph theory and mathematical software. 'Group' is an algebraic notion and 'rooted tree' is a combinatorial one which arises in the study of data structures in computer science. 'Totally disconnected' and 'locally compact' are topological notions which, although relevant to the background, are not needed for this project.
Visualising $0$-dimensional symmetry

The word 'symmetry' brings to mind visual images and geometry. It has a broader meaning in mathematics, where we think of regularly repeating patterns and invariance under transformations as displaying symmetry, and where the language of algebra is used to describe symmetry. Visualising the patterns or the dynamics of the transformations remains an effective tool for understanding the algebra however. This project aims to develop software for visualising various aspects of $0$-dimensional symmetry, which is the symmetry of infinite networks and arises in number theory and other parts of algebra as well. The aim is to produce software which may be used by researchers and which will be made available on web-pages of the $0$-Dimensional Symmetry project. Students taking this project will extend their knowledge of algebra, analysis, mathematical software and coding skills. 'Totally disconnected' and 'locally compact' are topological notions; 'group' is an algebraic one; and other concepts will be met in the course of the project.
Groups Acting on Trees: Compatible Local Actions

The concept of symmetry is pervasive in mathematics and formalised in the algebraic notion of a 'group'. It is often natural to equip a group with a 'topology' - a generalisation of distance functions - which, in a sense, gives groups a shape. For example, the symmetry groups of infinite networks, or 'graphs', become zero-dimensional. A versatile and accessible class of these groups was defined by Burger-Mozes and refined by this project's supervisor: Picture an infinite graph in which every vertex has the same number of neighbours and consider only those symmetries of the graph which in a neighbourhood of every given vertex act like one of finitely many allowed 'local actions'. In order for the resulting family of symmetries to reflect these restrictions accurately, the local actions need to satisfy certain conditions.
The aim of this project is to find more general constructions of such local actions and use them to test the sharpness of an existing rigidity theorem. A student who takes this project will extend his/her knowledge of algebra and learn how to use computer algebra systems designed for computations in group theory, including coding skills.
Groups Acting on Trees without Involutive Inversions

The concept of symmetry is pervasive in mathematics and formalised in the algebraic notion of a 'group'. It is often natural to equip a group with a 'topology' - a generalisation of distance functions - which, in a sense, gives groups a shape. For example, the symmetry groups of infinite networks, or 'graphs', become zero-dimensional. A versatile and accessible class of these groups was defined by Burger-Mozes and refined by this project's supervisor: Picture an infinite graph in which every vertex has the same number of neighbours and consider only those symmetries of the graph which in a neighbourhood of every given vertex act like one of finitely many allowed 'local actions'. In order for the resulting family of symmetries to reflect these restrictions accurately, the local actions need to satisfy certain conditions.
The aim of this project is to define a new class of groups acting on said graph by restricting the local action on edge neighbourhoods rather than vertex neighbourhoods, and thereby gain a new perspective on existing examples of graph symmetry groups relating to the Weiss conjecture. A student who takes this project will extend his/her knowledge of algebra and topology, and develop proofs intertwining both.



April 2019

Special Semester at the Bernoulli Center

Our research group was recently awarded a special semester at the Bernoulli Center in Lausanne, Switzerland. The semester entitled "Locally compact groups acting on discrete structures" is to take place in the second half of 2020 and will comprise a summer school, several workshops as well as special lectures, bringing together researchers in the field across all academic levels.






November 2018

PhD Scholarship

A PhD scholarship opportunity is available for students to investigate totally disconnected, locally compact groups under the supervision of ARC Laureate Professor George Willis. The students will join a team seeking to bring our understanding of these groups to a level comparable to that of finite.




October 2018

Summer Projects 2018/19

The following student research projects are related to the Australian Research Council project on 0-Dimensional Symmetry. While they are independent, each one gives a different view on the overall project. Another, more technical, description of this research is that it concerns totally disconnected, locally compact groups. An overview which attempts to explain the broader research project to non-experts may be seen here. (MFO, CC BY-NC-SA 3.0 )

Simple groups of infinite matrices

An essential step towards understanding $0$-dimensional symmetry is to describe the totally disconnected, locally compact (t.d.l.c.) groups which are simple. Simple groups are those which cannot be factored into smaller pieces and they are sometimes called the 'atoms of symmetry', or said to be analogues of the prime numbers in number theory. This project investigates t.d.l.c. groups of infinite matrices. It is suspected that these groups will be found to be simple and we will aim to show that by first studying corresponding groups of $n\times n$ matrices which are known to be simple. Students taking this project will extend their knowledge of algebra, analysis and number theory. 'Totally disconnected', 'locally compact' and '$0$-dimensional' are topological notions; 'group' and 'simple' are algebraic ones; and the matrix entries are numbers modulo a prime number $p$.
Solvable and nilpotent groups

The notion of a solvable group originated with the work of É. Galois (1832), who showed that a polynomial equation has a solution by radicals if and only if its group of symmetries is solvable. For example, the formula $x=-b\pm\sqrt{b^{2}-4ac}/2a$ is the solution of a quadratic equation by radicals and symmetries of the equation swap the $+$ and $-$ signs. (A group that is not solvable has some factors which are simple.) Groups of upper triangular $n\times n$ real matrices are solvable and also have the topological property of being connected. It may be shown that these are essentially all the connected solvable groups. This project investigates solvable totally disconnected, locally compact groups. Our starting point is groups of upper triangular matrices having integer entries. These groups have the property of being nilpotent, which is stronger than solvability. Students taking this project will extend their knowledge of algebra, analysis and number theory. 'Totally disconnected' and 'locally compact' are topological notions; 'group', 'solvable and 'nilpotent' are algebraic ones; and the integer matrices embed into matrices over the real numbers as well as over other number fields.
Free products of graphs

Symmetries of networks (or graphs) are '$0$-dimensional', and such symmetries are investigated through the algebraic technique of totally disconnected, locally compact groups. We are interested in highly symmetric, infinite graphs and one way to form such graphs is by gluing together infinitely many copies of finite graphs according to some regular instructions. This project investigates the symmetry groups of examples of graphs formed in this way and compares them with the symmetry groups of infinite regular trees, which are the most basic type of infinite regular graph. The aim is to determine whether the symmetry groups obtained in this way are simple and new. Students taking this project will extend their knowledge of algebra, analysis and combinatorics. 'Totally disconnected' and 'locally compact' are topological notions; 'group' and 'simple' are algebraic ones; and 'graphs' are a combinatorial concept.
Visualising $0$-dimensional symmetry

The word 'symmetry' brings to mind visual images and geometry. It has a broader meaning in mathematics, where we think of regularly repeating patterns and invariance under transformations as displaying symmetry, and where the language of algebra is used to describe symmetry. Visualising the patterns or the dynamics of the transformations remains an effective tool for understanding the algebra however. This project aims to develop software for visualising various aspects of $0$-dimensional symmetry, which is the symmetry of infinite networks and arises in number theory and other parts of algebra as well. The aim is to produce software which may be used by researchers and which will be made available on web-pages of the $0$-Dimensional Symmetry project. Students taking this project will extend their knowledge of algebra, analysis, mathematical software and coding skills. 'Totally disconnected' and 'locally compact' are topological notions; 'group' is an algebraic one; and other concepts will be met in the course of the project.

Contact us at contact[at]zerodimensional[.]group