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
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
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.
Special Research Initiatives - Grant ID: SR0354466
Funder
Australian Research Council
Funding Amount
$20,000.00
Summary
Mathematics in Contemporary Science. The Mathematics in Contemporary Science Research Network brings contemporary methods of non-linear analysis and differential equations, geometric reasoning and relevant algebraic and topological ideas to enrich six application areas in modern science: Complex Systems, Computer Vision, Optimal Transportation, Nanotechnology, Physics and Shortest Networks. MiCS will develop both the mathematics and the application areas in parallel. It will focus on postgradu ....Mathematics in Contemporary Science. The Mathematics in Contemporary Science Research Network brings contemporary methods of non-linear analysis and differential equations, geometric reasoning and relevant algebraic and topological ideas to enrich six application areas in modern science: Complex Systems, Computer Vision, Optimal Transportation, Nanotechnology, Physics and Shortest Networks. MiCS will develop both the mathematics and the application areas in parallel. It will focus on postgraduate training through workshops, summer schools and web based resources and build long-term international collaborations with EU networks and NSERC, NSF and EPSRC institutes as well as bringing together academic and industry leaders.Read moreRead less
On the Geometry of Liquid Crystals and Biological Membranes. This project will provide fundamental insights via realistic mathematical models into two areas of technological importance in the development of certain advanced materials involving liquid crystals and biomembranes. The use of liquid crystal devices is ubiquitous in the design of optical display units. Biomembranes are of much current importance, in particular, in connection with sophisticated drug delivery systems. The design of adva ....On the Geometry of Liquid Crystals and Biological Membranes. This project will provide fundamental insights via realistic mathematical models into two areas of technological importance in the development of certain advanced materials involving liquid crystals and biomembranes. The use of liquid crystal devices is ubiquitous in the design of optical display units. Biomembranes are of much current importance, in particular, in connection with sophisticated drug delivery systems. The design of advanced `smart' materials which admit solitonic behaviour is an area at the forefront of materials science and as such is important to the continued development of an advanced technological base within Australia.Read moreRead less
Noncommutative geometry in representation theory and quantum physics. One of the most important problems in natural science is to understand the structure of spacetime at the Planck scale. Mathematical investigations in recent years have predicted that at this scale, spacetime becomes noncommutative. Taking this noncommutativity into account, the project brings together geometry, algebra and quantum mechanics to develop new mathematical theories required for addressing the problem. It promises ....Noncommutative geometry in representation theory and quantum physics. One of the most important problems in natural science is to understand the structure of spacetime at the Planck scale. Mathematical investigations in recent years have predicted that at this scale, spacetime becomes noncommutative. Taking this noncommutativity into account, the project brings together geometry, algebra and quantum mechanics to develop new mathematical theories required for addressing the problem. It promises to make fundamental contributions to both mathematics and theoretical physics. Read moreRead less
Singularities and surgery in geometric evolution equations. The analysis of geometric evolution equations is a very active area of mathematical research internationally. The applications of such systems to physical problems such as crystal growth and flame propagation are also of great interest in the broader scientific community. The proposed research addresses questions central to the understanding of curvature flows. The project will yield internationally significant results in theoretical ....Singularities and surgery in geometric evolution equations. The analysis of geometric evolution equations is a very active area of mathematical research internationally. The applications of such systems to physical problems such as crystal growth and flame propagation are also of great interest in the broader scientific community. The proposed research addresses questions central to the understanding of curvature flows. The project will yield internationally significant results in theoretical mathematics, with applications in physics, engineering and image processing. These results will enhance Australia's reputation for high quality theoretical mathematical research with real world applications.Read moreRead less
Noncommutative geometry of groups acting on buildings. Consider a tiling of the plane by triangles, where each triangle is labeled by an element of a finite alphabet. Suppose that only certain pairs of labels are allowed to be adjacent to each other in each direction. The tiled planes can be pasted together to form the abstract mathematical object known as a building. This building and its boundary, give rise to new families of C*-algebras and groups. The algebras have a rich structure which it ....Noncommutative geometry of groups acting on buildings. Consider a tiling of the plane by triangles, where each triangle is labeled by an element of a finite alphabet. Suppose that only certain pairs of labels are allowed to be adjacent to each other in each direction. The tiled planes can be pasted together to form the abstract mathematical object known as a building. This building and its boundary, give rise to new families of C*-algebras and groups. The algebras have a rich structure which it is proposed to investigate and link with geometric properties of the groups. New insights into geometry, dynamics and algebra are expected.Read moreRead less