Quantization of polyhedral surfaces. Recent developments in the theory of discrete surfaces have revealed their fascinating links to many other areas of mathematics including integrable systems and quantum geometry. Rapid progress in this field is motivated by applications in pure mathematics, mathematical physics, computer graphics and engineering. Australian researchers are world recognized experts in integrable systems and this project will link them together with German experts in discrete d ....Quantization of polyhedral surfaces. Recent developments in the theory of discrete surfaces have revealed their fascinating links to many other areas of mathematics including integrable systems and quantum geometry. Rapid progress in this field is motivated by applications in pure mathematics, mathematical physics, computer graphics and engineering. Australian researchers are world recognized experts in integrable systems and this project will link them together with German experts in discrete differential geometry. The project will advance our knowledge base in fundamental and applied sciences and offer a unique research training opportunity for students in contemporary areas of pure and applied mathematics.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
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
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
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
Minimal surfaces. Recent stunning progress in topology, in particular a possible solution to one of the Clay Institute million dollar problems, using techniques from partial differential equations and minimal surfaces has made this area a hot topic. To attract researchers in this field to visit Australia and to train students in this area is a major part of this project.