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 shape of chaos: geometric advances in partially hyperbolic dynamics. This project aims to use recent advances in geometry and topology to discover new forms of chaotic dynamical systems and further classify the forms of chaos which are possible. Many systems in nature exhibit chaotic dynamics as they change in time. Not all systems are chaotic in the same way, and identifying the form of chaos and its qualitative properties is crucial to truly understanding the system. Chaotic dynamics aris ....The shape of chaos: geometric advances in partially hyperbolic dynamics. This project aims to use recent advances in geometry and topology to discover new forms of chaotic dynamical systems and further classify the forms of chaos which are possible. Many systems in nature exhibit chaotic dynamics as they change in time. Not all systems are chaotic in the same way, and identifying the form of chaos and its qualitative properties is crucial to truly understanding the system. Chaotic dynamics arise in chemical reactions, celestial mechanics, industrial mixing processes, fusion reactors, and many other processes. This project will aid in predicting the possible long-term behaviours of these systems.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
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
Measure theoretic frameworks for limsup sets. This project aims to develop new powerful measure theoretic techniques in mathematics that will be used in establishing some indispensable results in analytical number theory (Diophantine approximation) and dynamical systems. The plan is to construct new techniques and to use them in situations where existing techniques are not applicable. As a consequence of the proposed frameworks, not only we aim to resolve a few long-standing problems such as the ....Measure theoretic frameworks for limsup sets. This project aims to develop new powerful measure theoretic techniques in mathematics that will be used in establishing some indispensable results in analytical number theory (Diophantine approximation) and dynamical systems. The plan is to construct new techniques and to use them in situations where existing techniques are not applicable. As a consequence of the proposed frameworks, not only we aim to resolve a few long-standing problems such as the Generalised Baker-Schmidt Problem (1970) but also envisage that the proposed frameworks will have far-reaching applications beyond the confines of Diophantine approximation and dynamical systems, for example, geometric measure theory, geometric probability and stochastic geometry etc. Read moreRead less
Computer-aided proofs for non-hyperbolic dynamics and blenders . This project aims to develop methods to rigorously detect certain geometric structures in systems that are known to imply chaos and are robust under perturbation. Such structures include blenders and robust heterodimensional cycles and homoclinic tangencies.
This project expects to generate new knowledge in the area of non hyperbolic dynamics utilising a novel combination of recent developments in Dynamical Systems and techniques ....Computer-aided proofs for non-hyperbolic dynamics and blenders . This project aims to develop methods to rigorously detect certain geometric structures in systems that are known to imply chaos and are robust under perturbation. Such structures include blenders and robust heterodimensional cycles and homoclinic tangencies.
This project expects to generate new knowledge in the area of non hyperbolic dynamics utilising a novel combination of recent developments in Dynamical Systems and techniques from rigorous numerics.
Expected outcomes of this project include an efficient computation platform aimed at detecting and verifying chaos-inducing objects in complex dynamical systems.
This should provide significant benefits, such as an increased understanding of non-hyperbolic dynamical systems. Read moreRead less