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
Propagation of singularities for the Schrodinger equation. The time-dependent Schrodinger equation governs the evolution of quantum particles. In this project we aim to use new techniques from mathematical scattering theory to analyse solutions of the Schrodinger equation and obtain sharp bounds on their singularities. Controlling such singularities will allow us to deduce quantitative bounds on the number of eigenvalues in certain situations, and provide new techniques for studying nonlinear Sc ....Propagation of singularities for the Schrodinger equation. The time-dependent Schrodinger equation governs the evolution of quantum particles. In this project we aim to use new techniques from mathematical scattering theory to analyse solutions of the Schrodinger equation and obtain sharp bounds on their singularities. Controlling such singularities will allow us to deduce quantitative bounds on the number of eigenvalues in certain situations, and provide new techniques for studying nonlinear Schrodinger equations. Read moreRead less
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
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
Stochastic modelling and analysis of spatio-temporal processes with fractal characteristics. Interest has grown in recent years on the derivation of fractal models to represent certain physical phenomena such as diffusion and transport in porous media, oceanic and atmospheric turbulence, climatology, etc. This project focuses on the phenomenon of diffusion on domains with multifractal geometry. Recent advances in harmonic analysis on fractals and our own development of fractional generalized ran ....Stochastic modelling and analysis of spatio-temporal processes with fractal characteristics. Interest has grown in recent years on the derivation of fractal models to represent certain physical phenomena such as diffusion and transport in porous media, oceanic and atmospheric turbulence, climatology, etc. This project focuses on the phenomenon of diffusion on domains with multifractal geometry. Recent advances in harmonic analysis on fractals and our own development of fractional generalized random fields allow us to formulate a comprehensive program to tackle some key problems including modeling, processing and statistical estimation of fractional diffusion. Advances made in this program will in turn benefit the developments in related scientific fields.Read moreRead less
Insect host/vector genetic responses to rhabdovirus infection. Rhabdoviruses cause important diseases in humans, animals and plants. These viruses are transmitted by insect vectors in which they persist and propagate, an intimate and specific association. Insect-rhabdovirus interactions will be studied at the molecular level using a planthopper-maize rhabdovirus model. Insect genes specifically induced by rhabdovirus infection as well as the viral genes themselves will be identified and characte ....Insect host/vector genetic responses to rhabdovirus infection. Rhabdoviruses cause important diseases in humans, animals and plants. These viruses are transmitted by insect vectors in which they persist and propagate, an intimate and specific association. Insect-rhabdovirus interactions will be studied at the molecular level using a planthopper-maize rhabdovirus model. Insect genes specifically induced by rhabdovirus infection as well as the viral genes themselves will be identified and characterized using genomics and bioinformatics tools. We will extend the same approaches to a comparative analysis of these planthopper genes with other insect-rhabdovirus systems under investigation in our institutes. Identification of viral genes expressed in insects, the insect genes that respond to virus infection, and interpretation of the roles of these genes in insects, may indicate new opportunities to control serious plant and animal diseases through control of virus transmission.Read moreRead less
Using integrated frontier and smart technologies to identify new drug targets for parasites causing major diseases in humans and animals. The national/community benefits are (1) enhanced focus on animal and human health biotechnology through the development of anti-parasite compounds and/or vaccines; (2) improved and sustainable control of important parasites with decreased risk of induction of drug resistance; (3) increased profitability of agricultural animal production; (4) consolidation of a ....Using integrated frontier and smart technologies to identify new drug targets for parasites causing major diseases in humans and animals. The national/community benefits are (1) enhanced focus on animal and human health biotechnology through the development of anti-parasite compounds and/or vaccines; (2) improved and sustainable control of important parasites with decreased risk of induction of drug resistance; (3) increased profitability of agricultural animal production; (4) consolidation of a technology platform for further applications in genomics and post-genomics of pathogens of global significance and construction of a pipeline for the validation of drug targets; (5) capturing the benefits from fundamental research and strengthening links between fundamental and applied research; (6) increasing the quality and quantity of scientifically skilled people in biotechnology.Read moreRead less
Automated, smart genomic data integration for the exploration of developmentally regulated molecules in parasites of major socioeconomic importance. The national/community benefits are (1) enhanced focus on animal and human health biotechnology through the development of safe anti-parasite compounds/vaccines; (2) improved and sustainable control of key parasites with decreased risk of induction of drug resistance; (3) increased profitability of agricultural animal production; (4) consolidation o ....Automated, smart genomic data integration for the exploration of developmentally regulated molecules in parasites of major socioeconomic importance. The national/community benefits are (1) enhanced focus on animal and human health biotechnology through the development of safe anti-parasite compounds/vaccines; (2) improved and sustainable control of key parasites with decreased risk of induction of drug resistance; (3) increased profitability of agricultural animal production; (4) consolidation of a technology platform for further applications in genomics and post-genomics of pathogens of global significance and construction of a pipeline for the validation of drug targets; (5) capturing the benefits from fundamental research and strengthening links between fundamental and applied research; (6) increasing the quality and quantity of scientifically skilled people in biotechnology.Read moreRead less