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
Representations of dynamical systems, amenability, and proper actions. Mathematicians study abstract objects by representing them in terms of well-understood concrete models, and need to know when a representation is faithful, in the sense that the model contains complete information. Dynamical systems are an abstraction of physical systems suitable for studying time evolution and symmetries. The project aims to determine when important representations of dynamical systems are faithful, or, in ....Representations of dynamical systems, amenability, and proper actions. Mathematicians study abstract objects by representing them in terms of well-understood concrete models, and need to know when a representation is faithful, in the sense that the model contains complete information. Dynamical systems are an abstraction of physical systems suitable for studying time evolution and symmetries. The project aims to determine when important representations of dynamical systems are faithful, or, in mathematical language, when the dynamical system is amenable. The proposed strategy involves extending Rieffel's notion of proper actions; the construction should be of wide applicability apart from the intended applications to amenability.Read moreRead less
The structure of quantum groups. We propose to study the structure of mathematical objects used in describing symmetries of micro-scale phenomena. The project will significantly develop already well established Australian-Korean cooperation in this exciting and rapidly growing area of research. The results will be immediately applicable to related fields of mathematics, most notably to noncommutative geometry. In the long run, the outcomes will help in better understanding of fundamental problem ....The structure of quantum groups. We propose to study the structure of mathematical objects used in describing symmetries of micro-scale phenomena. The project will significantly develop already well established Australian-Korean cooperation in this exciting and rapidly growing area of research. The results will be immediately applicable to related fields of mathematics, most notably to noncommutative geometry. In the long run, the outcomes will help in better understanding of fundamental problems of modern quantum physics.Read moreRead less
Lifting the curse of dimensionality - bringing together the quasi Monte Carlo and sparse grid methods. This project is expected to lead to improved methods for handling high-dimensional problems (i.e. problems with many variables) that arise in finance, statistics, commerce, physics, and many other fields. In turn this could lead to significant economic benefit, especially to high-value service industries such as the finance industry. By strengthening international collaboration, it will also ....Lifting the curse of dimensionality - bringing together the quasi Monte Carlo and sparse grid methods. This project is expected to lead to improved methods for handling high-dimensional problems (i.e. problems with many variables) that arise in finance, statistics, commerce, physics, and many other fields. In turn this could lead to significant economic benefit, especially to high-value service industries such as the finance industry. By strengthening international collaboration, it will also help to maintain Australia's strong position in international research in the mathematical sciences.Read moreRead less
Computational Schemes for Initial-Boundary Value Problems. Many physical phenomena can be modelled as initial-boundary value problems described by partial differential equations. Simulations of such models require efficient and robust computational algorithms. The main aim of this project is to propose numerical algorithms for two dimensional spatial problems and three dimensional time-space models. A major focus of the project is to investigate methods that require about half the computation ....Computational Schemes for Initial-Boundary Value Problems. Many physical phenomena can be modelled as initial-boundary value problems described by partial differential equations. Simulations of such models require efficient and robust computational algorithms. The main aim of this project is to propose numerical algorithms for two dimensional spatial problems and three dimensional time-space models. A major focus of the project is to investigate methods that require about half the computational resources over celebrated schemes for solving boundary value problems.Read moreRead less
The development of a two-colour flow cytometric assay for the detection of whole cell biosensors in environmental samples. Macquarie University and the University of Copenhagen have expertise in fluorescence detection and whole cell biosensors respectively. The project will take advantage of these skills and develop a sensitive assay for monitoring biosensor bacteria in soil. The technology will be significant as it will enable real time analysis of antibiotic production in situ through the de ....The development of a two-colour flow cytometric assay for the detection of whole cell biosensors in environmental samples. Macquarie University and the University of Copenhagen have expertise in fluorescence detection and whole cell biosensors respectively. The project will take advantage of these skills and develop a sensitive assay for monitoring biosensor bacteria in soil. The technology will be significant as it will enable real time analysis of antibiotic production in situ through the detection of GFP expression. This work will then be used to isolate new antibiotic produces and will be extended to research into the bioavailability of toxic compounds and stress. An existing collaboration between the two institutions will be extended enabling the transfer and application of biosensor technology to Australia.Read moreRead less
Hecke Algebras in Algebra and Analysis. The aim of this program is to adapt techniques from harmonic analysis and operator-algebraic representation theory to study Hecke algebras arising in algebraic and geometric settings. The relevant analytic structures are C*-algebras and the fundamental question is then "Which Hecke algebras have a faithful enveloping C*-algebra?" We investigate this question, first by developing an appropriate theory of crossed products by semigroups and, second, by using ....Hecke Algebras in Algebra and Analysis. The aim of this program is to adapt techniques from harmonic analysis and operator-algebraic representation theory to study Hecke algebras arising in algebraic and geometric settings. The relevant analytic structures are C*-algebras and the fundamental question is then "Which Hecke algebras have a faithful enveloping C*-algebra?" We investigate this question, first by developing an appropriate theory of crossed products by semigroups and, second, by using the notion of topologization which enables the Hecke algebra to be studied in the context of topological groups.Read moreRead less
Increasing the effectiveness of quantitative verification. The ability to analyse the performance of complex systems and protocols is a vital part of the development of large-scale computer applications. Methods that improve the effectiveness of the analysis task would increase the competitiveness of the software industry, and would attract future development work (in complex systems) to Australia. The results of this project will have a direct influence on currently available design tools; the ....Increasing the effectiveness of quantitative verification. The ability to analyse the performance of complex systems and protocols is a vital part of the development of large-scale computer applications. Methods that improve the effectiveness of the analysis task would increase the competitiveness of the software industry, and would attract future development work (in complex systems) to Australia. The results of this project will have a direct influence on currently available design tools; the fact that Australian institutions will be (in part) responsible for key theoretical results in this growing field will strengthen Australia's position worldwide as an international centre for computer science.Read moreRead less
Metabolic engineering of Zymomonas mobilis for higher value fermentation products. This project will provide an opportunity to revolutionise the Australian chemical and sugar industries. Unlike the traditional methods of the petrochemical industry, bioconversion of carbohydrates to chemicals such as succinic acid via fermentation is cosiderably environmental friendly. For the sugar industry this project will provide an opportunity to produce not only conventional sugar products but also high val ....Metabolic engineering of Zymomonas mobilis for higher value fermentation products. This project will provide an opportunity to revolutionise the Australian chemical and sugar industries. Unlike the traditional methods of the petrochemical industry, bioconversion of carbohydrates to chemicals such as succinic acid via fermentation is cosiderably environmental friendly. For the sugar industry this project will provide an opportunity to produce not only conventional sugar products but also high value commodities via the process integration of succinic acid production using agricultural residues such as bagasse. This will contribute to significant levels of job creation in Australia, and further benefits will be that such products will be important both for import replacement and export potential.Read moreRead less