The Structure and Geometry of Graphs. Graphs are ubiquitous mathematical structures that model relational information such as information flows, transportation networks, and biochemical pathways. It is often desirable to have a geometric representation of a graph. For example, a programmer will better understand a computer program if the flow of information within the program is represented by a visually appealing drawing. The focus of the project will be the interplay between graph structure th ....The Structure and Geometry of Graphs. Graphs are ubiquitous mathematical structures that model relational information such as information flows, transportation networks, and biochemical pathways. It is often desirable to have a geometric representation of a graph. For example, a programmer will better understand a computer program if the flow of information within the program is represented by a visually appealing drawing. The focus of the project will be the interplay between graph structure theory and geometric properties of graphs. Moreover, the project will have significant applications to other area of mathematics and computer science, including computational complexity, analysis of data structures, and three-dimensional information visualisation.Read moreRead less
Monopoles, instantons and metrics. This Project is pure basic research in the general area of differential geometry or the study of manifolds. Manifolds are higher dimensional analogues of surfaces such as the surface of the sphere or the surface of a doughnut. This Project studies monopoles and instantons which are solutions of partial differential equations arising in physics. These solutions and the so-called moduli spaces of all solutions have been used in the last two decades by the worlds ....Monopoles, instantons and metrics. This Project is pure basic research in the general area of differential geometry or the study of manifolds. Manifolds are higher dimensional analogues of surfaces such as the surface of the sphere or the surface of a doughnut. This Project studies monopoles and instantons which are solutions of partial differential equations arising in physics. These solutions and the so-called moduli spaces of all solutions have been used in the last two decades by the worlds leading mathematicians to revolutionize the study of three and four dimensional manifolds.Read moreRead less
Characterizing and classifying ovoids, flocks and generalized quadrangles. This project lies within the framework of the classification and characterization of fundamental structures in finite geometry. This research area is the site of much international activity, in which the proposed research team plays a central role. The aim of the project is to pursue twin goals: the classification of ovoids in three dimensional projective space, a famous long-standing problem; and the classification of ce ....Characterizing and classifying ovoids, flocks and generalized quadrangles. This project lies within the framework of the classification and characterization of fundamental structures in finite geometry. This research area is the site of much international activity, in which the proposed research team plays a central role. The aim of the project is to pursue twin goals: the classification of ovoids in three dimensional projective space, a famous long-standing problem; and the classification of certain generalized quadrangles. Our approach is novel as it utilises recently discovered links between these areas. The expected outcomes are significant progress towards these goals, as well as the development of new techniques in finite geometry.Read moreRead less
Geometry of manifolds of non-negative scalar curvature. This proposal will benefit Australia in several ways: Its outcomes will make Australia a world leader in research on scalar curvature, and consequently help Australia secure its position among world leaders in research on differential geometry and differential equations; Overseas world-class experts will be attracted to Australia by the impact of this research to develop further collaboration; More and more talented Australian students will ....Geometry of manifolds of non-negative scalar curvature. This proposal will benefit Australia in several ways: Its outcomes will make Australia a world leader in research on scalar curvature, and consequently help Australia secure its position among world leaders in research on differential geometry and differential equations; Overseas world-class experts will be attracted to Australia by the impact of this research to develop further collaboration; More and more talented Australian students will be motivated to pursue science-based and mathematics based studies, thereby improving the mathematical skills of the Australian workforce.Read moreRead less
Moduli spaces. This project will offer a great opportunity for Australian researchers and students to engage in internationally competitive research in mathematics. Moduli spaces are fundamental to our understanding of mathematics and modern mathematical physics. It is crucial that Australian scientists and students take active part in these developments. The training of Honours and PhD students in various aspects of moduli spaces, and in the mathematics and mathematical physics that it address ....Moduli spaces. This project will offer a great opportunity for Australian researchers and students to engage in internationally competitive research in mathematics. Moduli spaces are fundamental to our understanding of mathematics and modern mathematical physics. It is crucial that Australian scientists and students take active part in these developments. The training of Honours and PhD students in various aspects of moduli spaces, and in the mathematics and mathematical physics that it addresses, is an integral part of this application.Read moreRead less
New Directions in Noncommutative Geometry. A. Connes' noncommutative geometry has recently become important in topology, geometry and physics. The central geometric objects in noncommutative geometry are called spectral triples. Spectral triples also provide the framework for studying some important classes of equations. This project will extend the definitions of spectral triples to cover additional important examples. This extension will provide the tools to study a broad class of boundary val ....New Directions in Noncommutative Geometry. A. Connes' noncommutative geometry has recently become important in topology, geometry and physics. The central geometric objects in noncommutative geometry are called spectral triples. Spectral triples also provide the framework for studying some important classes of equations. This project will extend the definitions of spectral triples to cover additional important examples. This extension will provide the tools to study a broad class of boundary value problems in the theory of equations. Such problems occur in several areas of modern physics. In addition, results obtained will be useful for studying the structure of the most important spectral triples, called noncommutative manifolds.Read moreRead less
Finite permutation groups and flag-transitive incidence structures. Mathematics is the enabling discipline for all the sciences and so a strong mathematical research community in Australia provides the foundations for future discoveries in science and technology. By developing new theory for permutation groups, producing a new paradigm for the study of Buekenhout geometries and classifying certain families of flag-transitive incidence structures, we will enhance Australia's leading position in P ....Finite permutation groups and flag-transitive incidence structures. Mathematics is the enabling discipline for all the sciences and so a strong mathematical research community in Australia provides the foundations for future discoveries in science and technology. By developing new theory for permutation groups, producing a new paradigm for the study of Buekenhout geometries and classifying certain families of flag-transitive incidence structures, we will enhance Australia's leading position in Permutation Group Theory, Algebraic Graph Theory and Finite Geometry. This will attract international and Australian postgraduate students and visitors, and strengthen the research activities of Australia by enhancing the collaboration between UWA and leading international universities.Read moreRead less
Noncommutative Algebraic Geometry. As algebra moves into the twenty-first century, we see a strong trend towards interactions with geometry. This project is right in the thick of this trend and will keep Australia abreast of some of the most interesting developments in algebra. The project seeks to start up a research group in noncommutative algebraic geometry which will foster a lively intellectual atmosphere. This will involve training postgraduate students, inviting international experts to g ....Noncommutative Algebraic Geometry. As algebra moves into the twenty-first century, we see a strong trend towards interactions with geometry. This project is right in the thick of this trend and will keep Australia abreast of some of the most interesting developments in algebra. The project seeks to start up a research group in noncommutative algebraic geometry which will foster a lively intellectual atmosphere. This will involve training postgraduate students, inviting international experts to give seminar talks and establishing relations with other Australian mathematicians in related areas.Read moreRead less
Three-dimensional geometry and topology. This project will carry out important fundamental research into the geometry and topology of 3-dimensional manifolds, an area of intense activity over the last 30 years.
The work has direct applications to physics, for example recent work in cosmology aimed at determining the global structure of our universe. Our work on knotting and symmetries of molecular graphs will also be of considerable interest in chemistry and biology.
The project will also ....Three-dimensional geometry and topology. This project will carry out important fundamental research into the geometry and topology of 3-dimensional manifolds, an area of intense activity over the last 30 years.
The work has direct applications to physics, for example recent work in cosmology aimed at determining the global structure of our universe. Our work on knotting and symmetries of molecular graphs will also be of considerable interest in chemistry and biology.
The project will also provide high quality training of undergraduate and graduate students in geometry and topology, and will increase international cooperation by developing closer links with colleagues and institutions overseas.Read moreRead less
Towards Mike Artin's conjecture. Non-commutative algebra and algebraic geometry are both classical branches of mathematics with much depth to them. As a result, the recent study of the interactions between the two disciplines has proven to be fertile ground for many important developments in mathematics. This project ensures that Australia remains a part of these developments.