Equations of Monge-Ampere type and applications. Many fundamental problems in geometry, physics and applied sciences are related to equations of Monge-Ampere type. In recent years there have been rapid developments in the study of these equations with major breakthroughs made by the proposers. This project aims at new discoveries and findings in theory and applications by resolving outstanding open problems, and enhance Australian leadership, expertise, and training in key areas of mathematics a ....Equations of Monge-Ampere type and applications. Many fundamental problems in geometry, physics and applied sciences are related to equations of Monge-Ampere type. In recent years there have been rapid developments in the study of these equations with major breakthroughs made by the proposers. This project aims at new discoveries and findings in theory and applications by resolving outstanding open problems, and enhance Australian leadership, expertise, and training in key areas of mathematics and its applications.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
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 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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Stability conditions on triangulated categories and related aspects of homological mirror symmetry. The proposed research studies one of the deepest questions in nature through superstring theory and mathematics with leading experts around the world. So, the proposed project maintains the Australia's profile in science. Also, the proposed project fits within the the Research Priority: Frontier Technologies for Building and Transforming Australian Industries. We will have exciting mathematical di ....Stability conditions on triangulated categories and related aspects of homological mirror symmetry. The proposed research studies one of the deepest questions in nature through superstring theory and mathematics with leading experts around the world. So, the proposed project maintains the Australia's profile in science. Also, the proposed project fits within the the Research Priority: Frontier Technologies for Building and Transforming Australian Industries. We will have exciting mathematical discussions which stimulate Australian students. They will be able to take advantage of such experience, especially when they need innovation. Thus, it is an investment for future of Australian industries.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
Generalized Geometries and their Applications. Geometry is one of the pillars of both ancient and modern mathematics. It also plays a vital role in many scientific applications, in particular in physics. Progress on the mathematical aspects and the applications have often gone hand in hand, as for example with differential geometry and general relativity. Geometry is a very fruitful area for interdisciplinary research.
Australia has a long tradition and a recognized research strength in Mat ....Generalized Geometries and their Applications. Geometry is one of the pillars of both ancient and modern mathematics. It also plays a vital role in many scientific applications, in particular in physics. Progress on the mathematical aspects and the applications have often gone hand in hand, as for example with differential geometry and general relativity. Geometry is a very fruitful area for interdisciplinary research.
Australia has a long tradition and a recognized research strength in Mathematical Physics, and this project will contribute to maintaining that status. An integral part of this proposal is student involvement and postgraduate research training, for which the topic lends itself particularly well.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
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
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.