Comparing Einstein to Newton: a mathematical foundation for the Newtonian limit and post-Newtonian expansions. This proposal will benefit the nation in the following ways: (i) to make Australia a world leader in post-Newtonian research, (ii) to contribute to Australia's existing commitment to the search for gravitational waves by providing theoretical tools that will aid in the analysis of gravitational wave data, (iii) to train the next generation of Australian gravitational researchers in a fi ....Comparing Einstein to Newton: a mathematical foundation for the Newtonian limit and post-Newtonian expansions. This proposal will benefit the nation in the following ways: (i) to make Australia a world leader in post-Newtonian research, (ii) to contribute to Australia's existing commitment to the search for gravitational waves by providing theoretical tools that will aid in the analysis of gravitational wave data, (iii) to train the next generation of Australian gravitational researchers in a field whose importance will only grow as the field of gravitational wave astronomy matures, and (iv) to facilitate visits by my collaborators to Australia who will bring world class expertise for the benefit of both Australian students and experts in general relativity.Read moreRead less
Quantum Spectra. Fundamental quantum processes will play a key role in emerging technologies in the twenty-first century across diverse industries including quantum information technology, quantum computers and electronics, quantum optics, nanoscale quantum microscopes and superconductor technology. Australia has a strong base of expertise in the underpinning quantum disciplines. This project in strategic basic research within mathematical physics will develop a comprehensive and consistent math ....Quantum Spectra. Fundamental quantum processes will play a key role in emerging technologies in the twenty-first century across diverse industries including quantum information technology, quantum computers and electronics, quantum optics, nanoscale quantum microscopes and superconductor technology. Australia has a strong base of expertise in the underpinning quantum disciplines. This project in strategic basic research within mathematical physics will develop a comprehensive and consistent mathematical description of quantum processes. This research will lead to a deeper understanding of quantum processes, keep Australia at the leading edge of international developments and increase Australia's capacity to develop and implement these new technologies.Read moreRead less
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.
The Sakai scheme-Askey table correspondence, analogues of isomonodromy and determinantal point processes. The Australian mathematical sciences enjoys two research groups with active interests on Painleve equations in applied mathematics which are able to address difficult problems. Such a problem is to give a formulation of Sakai's 2001 classification of the Painleve equations in a form most suitable for applications. For this links will be made with a seemingly distinct area of mathematics - t ....The Sakai scheme-Askey table correspondence, analogues of isomonodromy and determinantal point processes. The Australian mathematical sciences enjoys two research groups with active interests on Painleve equations in applied mathematics which are able to address difficult problems. Such a problem is to give a formulation of Sakai's 2001 classification of the Painleve equations in a form most suitable for applications. For this links will be made with a seemingly distinct area of mathematics - the Askey table from the theory of hypergeometric orthogonal polynomials. A number of tractable PhD projects are suggested by this proposal.Read moreRead less
New Geometric and Entropy Techniques for Differential Equations. The three main practical outcomes of this mathematical research will be better predictability of salt movement responsible for land degradation, better predictability of surface evolution of microelectronic components in nanoscale technology and an open source computer package that harnesses new and powerful geometrical techniques to solve differential equations. The project will train the next generation of researchers in the math ....New Geometric and Entropy Techniques for Differential Equations. The three main practical outcomes of this mathematical research will be better predictability of salt movement responsible for land degradation, better predictability of surface evolution of microelectronic components in nanoscale technology and an open source computer package that harnesses new and powerful geometrical techniques to solve differential equations. The project will train the next generation of researchers in the mathematical modelling of critical physical processes and it will bring international experts to Australia to work on these vital problems.Read moreRead less
GEOMETRIC NUMERICAL INTEGRATION. Many scientific phenomena in physics, astronomy, and chemistry, are modelled by ordinary differential equations (ODEs). Often these equations have no solution in closed form, and one relies on numerical integration. Traditionally this is done using Runge-Kutta methods or linear multistep methods. In the last decade, however, we (and others) have discovered novel classes of so-called "geometric" numerical integration methods that preserve qualititative featur ....GEOMETRIC NUMERICAL INTEGRATION. Many scientific phenomena in physics, astronomy, and chemistry, are modelled by ordinary differential equations (ODEs). Often these equations have no solution in closed form, and one relies on numerical integration. Traditionally this is done using Runge-Kutta methods or linear multistep methods. In the last decade, however, we (and others) have discovered novel classes of so-called "geometric" numerical integration methods that preserve qualititative features of certain ODE's exactly (in contrast to traditional methods), leading to crucial stability improvements. Extending concepts from dynamical systems theory and traditional numerical ODEs, this project will improve, extend and systematize this new field of geometric integration.Read moreRead less
Geometric Integration. This project gives an important boost to Australia's strength in the niche area of geometric numerical integration,in the face of strong international competition. It gathers 7 world experts from 5 countries to create new computer programs to improve calculations in dynamics, with applications ranging from astronomy, physics, chemistry, biology, and meteorology to finance. It strengthens Australia's links with the mathematical software industry, and will lead to world-clas ....Geometric Integration. This project gives an important boost to Australia's strength in the niche area of geometric numerical integration,in the face of strong international competition. It gathers 7 world experts from 5 countries to create new computer programs to improve calculations in dynamics, with applications ranging from astronomy, physics, chemistry, biology, and meteorology to finance. It strengthens Australia's links with the mathematical software industry, and will lead to world-class graduates and research training.Read moreRead less
Geometric numerical integration of differential equations. Differential equations (DEs) play a central role in modelling scientific phenomena in physics, biology, chemistry, astronomy, meteorology, and geoscience. We have developed new ways of solving DEs, using geometric integration, which have significant advantages over traditional methods because of the crucial nonlinear stability they provide.
This project, combining 7 world experts from 6 countries and 1 early career researcher, will pl ....Geometric numerical integration of differential equations. Differential equations (DEs) play a central role in modelling scientific phenomena in physics, biology, chemistry, astronomy, meteorology, and geoscience. We have developed new ways of solving DEs, using geometric integration, which have significant advantages over traditional methods because of the crucial nonlinear stability they provide.
This project, combining 7 world experts from 6 countries and 1 early career researcher, will place Australia at the forefront of this intensive international activity.
It will significantly strengthen Australia's links with the mathematical software industry (e.g. Wolfram Research, Inc), and will lead to world class graduates and research training.
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Green functions, correlation functions and differential equations. Classical and quantum exact solutions are established cornerstones in Australian applied mathematical research. In this project, we will:- 1). Address long standing open problems, whose resolution will add to mathematical knowledge and enhance Australia's reputation as a leading contributor to these topics; 2). List concrete and tractable sub-projects that will engage young scientists, whose training we are particularly keen on, ....Green functions, correlation functions and differential equations. Classical and quantum exact solutions are established cornerstones in Australian applied mathematical research. In this project, we will:- 1). Address long standing open problems, whose resolution will add to mathematical knowledge and enhance Australia's reputation as a leading contributor to these topics; 2). List concrete and tractable sub-projects that will engage young scientists, whose training we are particularly keen on, in vigorous and internationally competitive research; 3). Facilitate collaborations between various Australian research groups, all of whom are very well positioned to contribute to it; 4). Bring leading scientists to visit Australia to the benefit of the entire Australian mathematical community.Read moreRead less
Integrable structures in models of complex systems. The CI is in the happy circumstance of having almost completed (now in the proof reading stage) a large monograph on random matrices commissioned by Princeton University Press. This gives great international profile to the CI, and more generally Australian mathematical sciences in the subject matter of the proposal. To build on this base it is essential that significant new results, impacting on the work of others, continue to be obtained by t ....Integrable structures in models of complex systems. The CI is in the happy circumstance of having almost completed (now in the proof reading stage) a large monograph on random matrices commissioned by Princeton University Press. This gives great international profile to the CI, and more generally Australian mathematical sciences in the subject matter of the proposal. To build on this base it is essential that significant new results, impacting on the work of others, continue to be obtained by the CI. All indications are that the new ideas relating integrable structures and random matrices underpinning this proposal will fulfil this goal. For the postdoctral researcher involved the stimulating atmosphere of discovery should provide ideal training in mathematical research.Read moreRead less