Dynamics of eigenvalue/eigenspace algorithms with applications to signal processing. Many problems in signal and systems lead naturally to an eigenvalue/eigenspace determination and tracking problem; for example (acoustic) echo-cancellation, crosstalk suppression in ADSL modems, direction of arrival determination with an array of sensors, linear system identification etc. Exploiting methods from global analysis and dynamical systems theory we will study the available algorithms for eigenspace de ....Dynamics of eigenvalue/eigenspace algorithms with applications to signal processing. Many problems in signal and systems lead naturally to an eigenvalue/eigenspace determination and tracking problem; for example (acoustic) echo-cancellation, crosstalk suppression in ADSL modems, direction of arrival determination with an array of sensors, linear system identification etc. Exploiting methods from global analysis and dynamical systems theory we will study the available algorithms for eigenspace determination to characterise their computational efficiency, accuracy and effectiveness in various data scenarios. The analysis will lead to improved designs for eigenvalue/eigenspace algorithms, as well as design tools to engineer algorithms to specific situations.Read moreRead less
A semiclassical approach to spectral theory. Spectral theory is the branch of mathematics dealing with natural frequencies (eigenvalues) and modes of vibration (eigenfunctions) of systems arising in geometry, quantum physics and engineering. As such, they have important applications in seismic and medical imaging, nanotechnology, and optical communications. This project aims to use recently developed mathematical tools to advance our understanding of high energy eigenvalues and eigenfunctions, a ....A semiclassical approach to spectral theory. Spectral theory is the branch of mathematics dealing with natural frequencies (eigenvalues) and modes of vibration (eigenfunctions) of systems arising in geometry, quantum physics and engineering. As such, they have important applications in seismic and medical imaging, nanotechnology, and optical communications. This project aims to use recently developed mathematical tools to advance our understanding of high energy eigenvalues and eigenfunctions, as well as new algorithms for numerically computing them.Read moreRead less
Mathematics for breaking limits of speed and density in magnetic memories. The aim of this project is to develop a mathematical theory and numerical models of stochastic partial differential equations for magnetic nano-structures. Such materials will yield next-generation magnetic memories with up to three orders of magnitude faster switching speeds and dramatically increased data storage density. New mathematical theories will help understand their sensitivity to small random fluctuations that ....Mathematics for breaking limits of speed and density in magnetic memories. The aim of this project is to develop a mathematical theory and numerical models of stochastic partial differential equations for magnetic nano-structures. Such materials will yield next-generation magnetic memories with up to three orders of magnitude faster switching speeds and dramatically increased data storage density. New mathematical theories will help understand their sensitivity to small random fluctuations that can destroy stored information. This project aims to revolutionise mathematical modelling of magnetic memories and put Australia at the forefront of international research. Technological advances to create much smaller and faster memory devices are expected to enable groundbreaking ways of managing and mining big data.Read moreRead less
Group orbits in garmonic analysis and ergodic theory. Researchers from many areas need a type of mathematical analysis which involves the behaviour of a system - which may be a set of data points - under repeated application of some operation or group of operations. The structures arising from this kind of process are known as group orbits. The project gives information about their nature. Two major types of orbits are considered, coming from actions of discrete groups on measure spaces, and fro ....Group orbits in garmonic analysis and ergodic theory. Researchers from many areas need a type of mathematical analysis which involves the behaviour of a system - which may be a set of data points - under repeated application of some operation or group of operations. The structures arising from this kind of process are known as group orbits. The project gives information about their nature. Two major types of orbits are considered, coming from actions of discrete groups on measure spaces, and from smooth actions of Lie groups on manifolds, where powerful geometric methods are available. The project will yield new understandings of entropy, and new approaches to Fourier analysis.Read moreRead less
Ergodic theory and number theory. Recent advances in the theory of measured dynamical systems investigated by the proponents include new versions of entropy, and the study of spectral theory for non-singular systems. These will be further developed in this joint project with the French CNRS. The results are expected to have interesting applications in physics and number theory.
Operator Integrals and Derivatives. The project is a contribution to the study of non-commutative differential and integral calculus. The novelty of the present project lies in the study of smoothness properties of functions whose domains and ranges are spaces of unbounded, non-commuting operators on some Hilbert space. Our general approach will be based on a detailed investigation of properties of double operator integrals, which permit smoothness estimates of operator-functions. It can be expe ....Operator Integrals and Derivatives. The project is a contribution to the study of non-commutative differential and integral calculus. The novelty of the present project lies in the study of smoothness properties of functions whose domains and ranges are spaces of unbounded, non-commuting operators on some Hilbert space. Our general approach will be based on a detailed investigation of properties of double operator integrals, which permit smoothness estimates of operator-functions. It can be expected that the new techniques generated will find further application in areas of mathematical physics and non-commutative geometry related to quantized calculus.
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Entropy and maximal entropy in Markov systems. Entropy is a measure of how well-ordered a system is: chaotic systems have high entropy. Two approaches to entropy are available, via the limiting behaviour of the orbits of points, which yields topological entropy, and via the behaviour of the distributions of measures of partitions, yielding measure-theoretic entropy. The topological entropy is the least upper bound of entropies of all possible measures. We study when there is a measure which real ....Entropy and maximal entropy in Markov systems. Entropy is a measure of how well-ordered a system is: chaotic systems have high entropy. Two approaches to entropy are available, via the limiting behaviour of the orbits of points, which yields topological entropy, and via the behaviour of the distributions of measures of partitions, yielding measure-theoretic entropy. The topological entropy is the least upper bound of entropies of all possible measures. We study when there is a measure which realises this bound, describing the structure of such systems via Markov and Bratteli diagrams. Our methods will be applied to new versions of entropy for non-singular systems. This will assist in the description of chaotic behaviour.Read moreRead less
Symmetries in analysis. Technical research is like an iceberg. The 10% you see in applications is supported by 90% hidden, long-term, sometimes abstruse or theoretical-sounding work. The area of mathematical analysis has, for over 200 years, proved its worth as part of the unseen 90%, giving us such important tools as Fourier analysis, statistical mechanics and quantum mechanics. Australia is known as a world leader in mathematical analysis, and it is important for the country to maintain that e ....Symmetries in analysis. Technical research is like an iceberg. The 10% you see in applications is supported by 90% hidden, long-term, sometimes abstruse or theoretical-sounding work. The area of mathematical analysis has, for over 200 years, proved its worth as part of the unseen 90%, giving us such important tools as Fourier analysis, statistical mechanics and quantum mechanics. Australia is known as a world leader in mathematical analysis, and it is important for the country to maintain that edge in a number of key disciplines, so we can continue to participate in global technological advance. The project has an international focus which will enable that to happen. It will also provide training for the next generation of mathematicians. Read moreRead less
Dynamical systems: theory and practice. Mathematical science has proven a crucial platform for science and technology: it may have a long lead-time to application but its impacts are more profound than glamorous technical developments. Australia has an economic imperative to maintain investment in fundamental mathematics. Dynamical systems underpin a wide range of applications in physics, engineering, information science, finance and economics. This project will improve our capacity to model sy ....Dynamical systems: theory and practice. Mathematical science has proven a crucial platform for science and technology: it may have a long lead-time to application but its impacts are more profound than glamorous technical developments. Australia has an economic imperative to maintain investment in fundamental mathematics. Dynamical systems underpin a wide range of applications in physics, engineering, information science, finance and economics. This project will improve our capacity to model systems and to study their evolution, giving us better predictive power. It will keep Australia in the forefront of international research, providing a basis of expertise not otherwise available to Australian researchers and industry. Read moreRead less
Non-commutative analysis and differential calculus. This project is in an area of central mathematical importance and will lead to important scientific advances that will keep Australia at the forefront internationally in this field of research. There is an emphasis on international networking and we will collaborate with leading researchers in USA and France.