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.
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.
Functional and harmonic analysis of function spaces: synthesis, development and applications. Recent advances in mathematics are on the borderlines of its branches. This interdisciplinary project develops and binds the research areas attracting growing interest of prominent mathematicians during the last 30 years because of not only its theoretical value, but also its ties with the key equations describing a multitude of physical phenomena and the theoretical foundation of numerical methods. Th ....Functional and harmonic analysis of function spaces: synthesis, development and applications. Recent advances in mathematics are on the borderlines of its branches. This interdisciplinary project develops and binds the research areas attracting growing interest of prominent mathematicians during the last 30 years because of not only its theoretical value, but also its ties with the key equations describing a multitude of physical phenomena and the theoretical foundation of numerical methods. The Euler, Helmholtz, Lamb, Navier-Stokes and acoustic equations, studied in terms of function spaces, govern incompressible viscous fluid flows and wave propagations. Contributing to both pure mathematics and, particularly, Short-Term Tsunami Prediction, the project will enhance Australia's research reputation.Read moreRead less
Harmonic analysis on Lie groups via hypergroup convolution structures. This project studies convolution structures for conjugacy classes
of nilpotent and compact Lie groups and the connections with fusion rule algebras. The aims are to establish a suitable theory of almost periodic functions on a nilpotent Lie group to allow a
wrapping theorem to be formulated, to describe precisely the
class hypergroup of a compact Lie group, and to clarify the relations of the latter with fusion rule algebr ....Harmonic analysis on Lie groups via hypergroup convolution structures. This project studies convolution structures for conjugacy classes
of nilpotent and compact Lie groups and the connections with fusion rule algebras. The aims are to establish a suitable theory of almost periodic functions on a nilpotent Lie group to allow a
wrapping theorem to be formulated, to describe precisely the
class hypergroup of a compact Lie group, and to clarify the relations of the latter with fusion rule algebras. This will result in further understanding of the Kirillov orbit method and the have applications to conformal field theory.Read moreRead less
The Spectral Theory and Harmonic Analysis of Geometric Differential Operators. The project will involve mathematical research of the highest international standard in two very active and far-reaching field of mathematics: quantum chaos, and harmonic analysis. Progress in these fields will have implications in areas such as communications technology (e.g. image compression), quantum theory, and mathematical analysis (e.g. partial differential equations).
Boundedness of Singular integral operators and applications to Bochner-Riesz summability, Riesz transforms, and Hardy spaces. We aim to develop harmonic analysis methods to study singular integral operators and function spaces associated to these operators. We propose to study the long standing problem of convergence of Bochner-Riesz means in Fourier analysis, and investigate differential operators with non-smooth coefficients acting on rough domains, or acting on general spaces like manifolds. ....Boundedness of Singular integral operators and applications to Bochner-Riesz summability, Riesz transforms, and Hardy spaces. We aim to develop harmonic analysis methods to study singular integral operators and function spaces associated to these operators. We propose to study the long standing problem of convergence of Bochner-Riesz means in Fourier analysis, and investigate differential operators with non-smooth coefficients acting on rough domains, or acting on general spaces like manifolds. Expected outcomes are new techniques in harmonic analysis to be developed, with applications being solutions to a number of open problems in the theories of harmonic analysis, partial differential equations and function spaces.Read moreRead less