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
The fundamental structure of combinatorial configurations. Combinatorial configurations are fundamental mathematical tools used to model physical problems in the information sciences. Combinatorial trades arise from the differences between combinatorial configurations. They uniquely determine the underlying structure of the configuration and are central to the determination of defining sets. With this proposal we shall study the existence, properties and applications of combinatorial trades and ....The fundamental structure of combinatorial configurations. Combinatorial configurations are fundamental mathematical tools used to model physical problems in the information sciences. Combinatorial trades arise from the differences between combinatorial configurations. They uniquely determine the underlying structure of the configuration and are central to the determination of defining sets. With this proposal we shall study the existence, properties and applications of combinatorial trades and the associated defining sets. Our results will have applications in the areas of biotechnology, information systems, information security and experimental design.Read moreRead less
Mathematics of Cryptography. The Australian economy and society requires fast, reliable, and secure communication. Current first-generation security solutions are not capable of supporting the efficiency and scalability requirements of mass-market adoption of wireless and embedded consumer applications. New security infrastructures are emerging and must be carefully, but rapidly, defined. Thus developing new mathematically solid tools in this area is an important and urgent tasks. In addition, t ....Mathematics of Cryptography. The Australian economy and society requires fast, reliable, and secure communication. Current first-generation security solutions are not capable of supporting the efficiency and scalability requirements of mass-market adoption of wireless and embedded consumer applications. New security infrastructures are emerging and must be carefully, but rapidly, defined. Thus developing new mathematically solid tools in this area is an important and urgent tasks. In addition, the intended work advances our knowledge of the theory and the quality of our culture. As such, it will promote the Australian science and will also have many practical applications in Cryptography, Computer Security and E-Commerce.Read moreRead less
Totally disconnected groups, representations and discrete mathematics. This project involves participation in programs at the Institute of Advanced Studies in Princeton and the nearby Center for Discrete Mathematics and Theoretical Computer Science that are designed to initiate collaborations across distinct mathematical research areas. These programs will set future research directions and could lead to innovations in computer science. Discoveries I have made in one of the research areas mean ....Totally disconnected groups, representations and discrete mathematics. This project involves participation in programs at the Institute of Advanced Studies in Princeton and the nearby Center for Discrete Mathematics and Theoretical Computer Science that are designed to initiate collaborations across distinct mathematical research areas. These programs will set future research directions and could lead to innovations in computer science. Discoveries I have made in one of the research areas mean that I may be able to make substantial contributions to these programs. Early involvement in influential programs such as these means that Australia is well placed to take advantage of developments that result and also enhances the reputation of Australian mathematics.Read moreRead less
Geometric representation of small-rank totally disconnected groups. Mathematics research creates and develops new concepts for understanding the world. Group theory is a branch of mathematics based on our innate sense of of symmetry. It was invented 200 hundred years ago and has grown into a language for analysing and classifying things ranging from wallpaper patterns to crystals, the fundamental particles of physics and Rubik's cube. The chief investigators have significant breakthroughs in the ....Geometric representation of small-rank totally disconnected groups. Mathematics research creates and develops new concepts for understanding the world. Group theory is a branch of mathematics based on our innate sense of of symmetry. It was invented 200 hundred years ago and has grown into a language for analysing and classifying things ranging from wallpaper patterns to crystals, the fundamental particles of physics and Rubik's cube. The chief investigators have significant breakthroughs in the study of symmetry groups of networks, giving Australia an international lead in this research. The project will develop the insights gained to make Australia a centre of expertise on these symmetry groups, which have applications to information and communication technology, among many others.Read moreRead less
Reflection groups. The study of symmetry in geometrical and abstract contexts is a central issue in such diverse areas as mathematical physics, singularity theory, algebraic geometry, quantum groups and the study of knots and braids. Group theory provides the mathematical framework for the analysis of symmetry. Reflection groups, simple examples of which are the symmetry groups of the five platonic solids, play a key role in all of the areas mentioned above. Thus an improved understanding of ref ....Reflection groups. The study of symmetry in geometrical and abstract contexts is a central issue in such diverse areas as mathematical physics, singularity theory, algebraic geometry, quantum groups and the study of knots and braids. Group theory provides the mathematical framework for the analysis of symmetry. Reflection groups, simple examples of which are the symmetry groups of the five platonic solids, play a key role in all of the areas mentioned above. Thus an improved understanding of reflection groups will significantly enhance the development of several important theories.
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Analysis of the structure of latin squares. The benefits to Australia of fundamental research in core disciplines such as mathematics are well documented. Discrete mathematics and combinatorics are boom disciplines of the computer age and this project seeks new knowledge concerning basic building blocks of combinatorial mathematics. The outcomes will be of interest to theoretical discrete mathematicians around the world, enhancing Australia's already high research profile in this important area ....Analysis of the structure of latin squares. The benefits to Australia of fundamental research in core disciplines such as mathematics are well documented. Discrete mathematics and combinatorics are boom disciplines of the computer age and this project seeks new knowledge concerning basic building blocks of combinatorial mathematics. The outcomes will be of interest to theoretical discrete mathematicians around the world, enhancing Australia's already high research profile in this important area of pure mathematical research. Importantly, the problems under investigation offer substantial opportunity for excellent postgraduate training, critical for the future of Australian research. Read moreRead less
Group actions: combinatorics, geometry and computation. Science today relies on digital technologies using quantised and digital information. Because of the discrete nature of digital information, much of the mathematics underpinning these advances comes from the core disciplines of algebra and combinatorics within which this proposal falls. All aspects of the proposal focus on strengthening theoretical understanding of algebraic and combinatorial structures, and increasing computational power f ....Group actions: combinatorics, geometry and computation. Science today relies on digital technologies using quantised and digital information. Because of the discrete nature of digital information, much of the mathematics underpinning these advances comes from the core disciplines of algebra and combinatorics within which this proposal falls. All aspects of the proposal focus on strengthening theoretical understanding of algebraic and combinatorial structures, and increasing computational power for working with them. The fundamental research outcomes, in terms of theorems, algorithms, and the training of young research mathematicians, will thus both enhance the high international standing of Australian mathematics, and strengthen Australia's capabilities in these important areas.Read moreRead less
Quantized representation theory. The representation theory of quantized algebras, or deformation algebras, is a rapidly expanding and exciting field. It has wide ranging applications from within mathematics, to knot theory and statistical mechanics. This project addresses several important open problems in the area with an emphasis on structural innovations and computing explicit numerical invariants.
Cycle decompositions of graphs. The benefits to Australia of fundamental research in core disciplines such as mathematics are well documented. This project aims to solve long-standing and significant open problems in the field of mathematics known as graph theory. Solving such problems will undoubtedly bring Australian research in this field to the fore, and help to enhance Australia's international research profile generally. The project offers substantial postgraduate training in the form of t ....Cycle decompositions of graphs. The benefits to Australia of fundamental research in core disciplines such as mathematics are well documented. This project aims to solve long-standing and significant open problems in the field of mathematics known as graph theory. Solving such problems will undoubtedly bring Australian research in this field to the fore, and help to enhance Australia's international research profile generally. The project offers substantial postgraduate training in the form of three excellent PhD projects in discrete mathematics. The computer age has ensured that this is a booming discipline and an increasing component of undergraduate syllabi around the world. It is thus a crucial area in which to be providing quality research training.Read moreRead less