Group algorithms: Complexity, Theory and Practice. The symmetry of a mathematical or physical system is often best described by an abstract structure called a group, and groups are commonly represented as groups of permutations or matrices. In this project we shall design and analyse a general algorithmic framework for computing with finite groups. In the context of permutation groups and matrix groups we will produce prototype implementations. The proposed framework has the potential to revolut ....Group algorithms: Complexity, Theory and Practice. The symmetry of a mathematical or physical system is often best described by an abstract structure called a group, and groups are commonly represented as groups of permutations or matrices. In this project we shall design and analyse a general algorithmic framework for computing with finite groups. In the context of permutation groups and matrix groups we will produce prototype implementations. The proposed framework has the potential to revolutionise algorithmic group theory as it draws together theoretical and computational models of groups.Read moreRead less
Computing with large groups: probability distributions and fast randomised algorithms. Fast algorithms produced by the project will impact on the practical management of symmetry in large scale searches, which have important industrial applications. Hence the project addresses the Priority Goals Breakthrough Science and Smart Information Use. The project will enhance Australia's leading position in Computational Algebra. Implementations of our algorithms will be incorporated in the Computer Alge ....Computing with large groups: probability distributions and fast randomised algorithms. Fast algorithms produced by the project will impact on the practical management of symmetry in large scale searches, which have important industrial applications. Hence the project addresses the Priority Goals Breakthrough Science and Smart Information Use. The project will enhance Australia's leading position in Computational Algebra. Implementations of our algorithms will be incorporated in the Computer Algebra system Magma, based at the University of Sydney, distributed world-wide, and used intensively in research and teaching. The project will attract international and Australian graduate students and postdoctoral researchers, and strengthen research activities in Australia by enhancing already strong international collaborations. Read moreRead less
Applications of Group Theory to Finite Geometry. Group theory and geometry have influenced one another for over a century. The most important structures in geometry are the symmetric ones and the most important groups act on geometries. Recent developments in finite geometry, although informed by symmetry, have used a minimum of group theory. The project aims to redress this, by applying results from a broad range of finite group theory to the presently hot topics in finite geometry. Our aim is ....Applications of Group Theory to Finite Geometry. Group theory and geometry have influenced one another for over a century. The most important structures in geometry are the symmetric ones and the most important groups act on geometries. Recent developments in finite geometry, although informed by symmetry, have used a minimum of group theory. The project aims to redress this, by applying results from a broad range of finite group theory to the presently hot topics in finite geometry. Our aim is to achieve a paradigm shift, by finding substantively different structures than those presently known. Should it succeed, much activity in geometry would follow, seeking geometric interpretation of these group theoretic results. Our focus is necessitated by the lack of a result characterising the underlying groups of symmetric generalised quadrangles.Read moreRead less
Permutation groups and their interplay with symmetry in finite geometry and graph theory. A strong mathematical community in Australia provides the foundations for future discoveries in technology, science and business. The use of group theory to characterise symmetric generalised quadrangles, partial quadrangles, and strongly regular graphs, and the construction of new examples of such objects, will enhance Australia's leading position in Group Theory, Algebraic Graph Theory and Finite Geometry ....Permutation groups and their interplay with symmetry in finite geometry and graph theory. A strong mathematical community in Australia provides the foundations for future discoveries in technology, science and business. The use of group theory to characterise symmetric generalised quadrangles, partial quadrangles, and strongly regular graphs, and the construction of new examples of such objects, will enhance Australia's leading position in Group Theory, Algebraic Graph Theory and Finite Geometry. This project will also strengthen the collaboration between Australian, Belgian and Italian Universities and support young researchers, developing expertise in a world-leading research group, to drive Australia's future in mathematics.Read moreRead less
Symmetrical graphs, generalized polygons and expanders. This project proposes to study a class of highly symmetrical graphs -- locally s-arc-transitive graphs. Studying the class of graphs has been one of the central topics in algebraic graph theory for over 50 years. This class of graphs has been effectively used in computer science, communication network, group theory, geometry, and other areas. This project will develop new methods to solve several fundamental problems regarding locally s-arc ....Symmetrical graphs, generalized polygons and expanders. This project proposes to study a class of highly symmetrical graphs -- locally s-arc-transitive graphs. Studying the class of graphs has been one of the central topics in algebraic graph theory for over 50 years. This class of graphs has been effectively used in computer science, communication network, group theory, geometry, and other areas. This project will develop new methods to solve several fundamental problems regarding locally s-arc-transitive graphs, and apply the outcomes to solve important problems in communication networks, graph theory, group theory, and geometry.Read moreRead less
Automated texture selection and classification methods for detection of osteoarthritis in knee radiographs. In Australia there are 1-2 million OA sufferers, a condition that costs approximately $9 billion annually. This project will address an important problem of early detection and monitoring of OA and this remains in line with the National Research Priority 2. Potential outcomes of the project will result in better diagnosis and treatment of OA, reduced discomfort to the individual and saving ....Automated texture selection and classification methods for detection of osteoarthritis in knee radiographs. In Australia there are 1-2 million OA sufferers, a condition that costs approximately $9 billion annually. This project will address an important problem of early detection and monitoring of OA and this remains in line with the National Research Priority 2. Potential outcomes of the project will result in better diagnosis and treatment of OA, reduced discomfort to the individual and saving to the national economy. This project will improve existing activity and rehabilitation programs such as exercise of lower limbs and it will help in developing diets for healthy people and OA sufferers.Read moreRead less
Symmetries of finite digraphs. Highly symmetrical graphs are well-studied and, in many respects, the theory for dealing with them is well-established. By comparison, our understanding of symmetrical digraphs is much poorer. There are some rather basic questions about these about which we know shamefully little. The aim of this project is to remedy this shortage of knowledge by extending many important results and theories about symmetrical graphs to digraphs.
Finite linearly representable geometries and symmetry. Finite geometry has profound mathematical connections to the theory of symmetry. Advances in finite geometry and in symmetry have historically led to advances in diverse areas such as algebra, computing, and theoretical physics. The project aims to characterise basic geometric objects called "projective planes'' and "generalised polygons'' using their symmetry properties. To achieve these aims, conceptual links between certain elements in cl ....Finite linearly representable geometries and symmetry. Finite geometry has profound mathematical connections to the theory of symmetry. Advances in finite geometry and in symmetry have historically led to advances in diverse areas such as algebra, computing, and theoretical physics. The project aims to characterise basic geometric objects called "projective planes'' and "generalised polygons'' using their symmetry properties. To achieve these aims, conceptual links between certain elements in classical symmetry groups and geometric planes and polygons must be developed. The density of these certain elements has important applications to probabilistic geometric algorithms.Read moreRead less
Exceptionally symmetric combinatorial designs. Advances in digital technologies are underpinned by powerful mathematics; use of symmetry greatly simplifies complex problems. This project aims to exploit the mathematical theory of groups to advance our understanding of combinatorial designs with exceptional symmetry. New designs have become prominent through links with networks and error correcting codes. The project expects to generate constructions and classifications in these areas by utilisin ....Exceptionally symmetric combinatorial designs. Advances in digital technologies are underpinned by powerful mathematics; use of symmetry greatly simplifies complex problems. This project aims to exploit the mathematical theory of groups to advance our understanding of combinatorial designs with exceptional symmetry. New designs have become prominent through links with networks and error correcting codes. The project expects to generate constructions and classifications in these areas by utilising powerful group theory. As well as innovative methods for studying designs with symmetry based on group actions, expected outcomes include enhanced international collaboration, and highly trained combinatorial mathematicians to strengthen Australia’s research standing in fundamental science. Read moreRead less
The synchronisation hierarchy of permutation groups. This project aims to make significant advances in understanding finite primitive permutation groups, which are the basic building blocks of the mathematical study of symmetry. A recently-developed perspective, inspired by the notion of a synchronising automaton, has revealed that these groups fall into a natural hierarchy. While the outline of this synchronisation hierarchy is known, many questions remain about exactly which primitive groups l ....The synchronisation hierarchy of permutation groups. This project aims to make significant advances in understanding finite primitive permutation groups, which are the basic building blocks of the mathematical study of symmetry. A recently-developed perspective, inspired by the notion of a synchronising automaton, has revealed that these groups fall into a natural hierarchy. While the outline of this synchronisation hierarchy is known, many questions remain about exactly which primitive groups lie in which layers. Answering these questions using techniques from group theory, graph theory and finite geometry will substantially deepen our understanding. The benefits of this include new knowledge and enhanced insight into this fundamental class of groups and new tools for their analysis.Read moreRead less