Geometric methods in quantum theory. Quantum theory is the fundamental language of physics, it describes the small scale structure of matter and possibly space-time. The advent of sophisticated models, particularly of quarks has emphasised geometric structure as a basic component of the theory. The issues thrown up by quantum theory are similar to problems encountered in the geometry of manifolds so that tools from the latter have been successfully employed in the former and vice-versa. ....Geometric methods in quantum theory. Quantum theory is the fundamental language of physics, it describes the small scale structure of matter and possibly space-time. The advent of sophisticated models, particularly of quarks has emphasised geometric structure as a basic component of the theory. The issues thrown up by quantum theory are similar to problems encountered in the geometry of manifolds so that tools from the latter have been successfully employed in the former and vice-versa. Previous work of the Chief Investigators has shown the importance of geometric structures known as gerbes which this Project will extend and apply in novel ways.Read moreRead less
Geometric problems from quantum theory. This Proposal is fundamental, basic research at the forefront of the application of mathematics to physical theories. The problems that will be worked on are central to much of the research activity which is presently occuring in leading centres and institutes internationally. By being a part of that research we ensure that not only is Australia involved in todays mathematical and physical advances but that we also have Australian mathematicians trained ....Geometric problems from quantum theory. This Proposal is fundamental, basic research at the forefront of the application of mathematics to physical theories. The problems that will be worked on are central to much of the research activity which is presently occuring in leading centres and institutes internationally. By being a part of that research we ensure that not only is Australia involved in todays mathematical and physical advances but that we also have Australian mathematicians trained to take advantage of the benefits those advances will bring in the future.Read moreRead less
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
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
Noncommutative geometry: new frontiers. This project is at the leading edge of fundamental mathematics and will result in important scientific advances. As a result Australian science will be seen to be at the forefront internationally. This area of mathematics is having a high impact at the moment so that research training is an important aspect. There will be PhD students trained as part of the project and honours students exposed to the latest advances. Australians would normally need to go ....Noncommutative geometry: new frontiers. This project is at the leading edge of fundamental mathematics and will result in important scientific advances. As a result Australian science will be seen to be at the forefront internationally. This area of mathematics is having a high impact at the moment so that research training is an important aspect. There will be PhD students trained as part of the project and honours students exposed to the latest advances. Australians would normally need to go to leading international centres such as Paris to partake in projects of this nature. That high profile research of this kind can be done in Australia will enhance our capacity to retain scientific talent.Read moreRead less
Noncommutative geometry and applications. Noncommutative geometry is a new field of mathematical research having its origins in quantum mechanics. The main feature of this theory is that it deals with geoemtric and topological aspects of objects for which the order in which we make measurements or perform operations or observations is relevant to the outcome. This happens with microscopic systems especially those at the atomic or subatomic length scale. The methods have been motivated by and fou ....Noncommutative geometry and applications. Noncommutative geometry is a new field of mathematical research having its origins in quantum mechanics. The main feature of this theory is that it deals with geoemtric and topological aspects of objects for which the order in which we make measurements or perform operations or observations is relevant to the outcome. This happens with microscopic systems especially those at the atomic or subatomic length scale. The methods have been motivated by and found application in condensed matter physics, string theory, random media, algebraic structures and the geometry and topology of manifoldsRead 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
Geometric evolution equations and global effects of curvature. This project aims to approach several important problems in global differential geometry, by inventing new processes to deform geometric objects to simpler ones. The deformations are described by carefully constructed geometric evolution equations, designed to exhibit behaviour suited to the given problem. The project proposes methods for building such equations, and new techniques for their analysis. The research is expected to yi ....Geometric evolution equations and global effects of curvature. This project aims to approach several important problems in global differential geometry, by inventing new processes to deform geometric objects to simpler ones. The deformations are described by carefully constructed geometric evolution equations, designed to exhibit behaviour suited to the given problem. The project proposes methods for building such equations, and new techniques for their analysis. The research is expected to yield significant new results, both in differential geometry and in nonlinear heat equations, and should provide substantial progress towards resolving several important long-standing conjectures.
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Triangulated categories and their applications. This project is at the cutting edge of modern, international research in mathematics. Having work of this calibre done in Australia raises our international prestige, and makes Australia a more attractive place for top-notch hi-tech companies. Furthermore, training our young people to such a high standard will have the long-term effect of raising our profile.
Energy, Cosmic Censorship and Black Hole Stability. Human progress is achieved by confronting fundamental questions, at the leading edge of knowledge. This project will lead to better understanding of space-time physics, and of the properties of singular solutions of non-linear hyperbolic equations. Such equations govern a wide range of physical phenomena, including fluid flow, weather and electromagnetic fields.