A new generation of fractals: theory, computation, and applications particularly to digital imaging. The project develops the mathematical and algorithmic foundations of superfractals and applies these results to a number of different areas, including in particular, digital imaging. For example, the ``third generation'' of mobile communications (3G), combines wireless mobile technology with high data transmission capacities. Currently the requirement for extensive bandwidth is a problem for e ....A new generation of fractals: theory, computation, and applications particularly to digital imaging. The project develops the mathematical and algorithmic foundations of superfractals and applies these results to a number of different areas, including in particular, digital imaging. For example, the ``third generation'' of mobile communications (3G), combines wireless mobile technology with high data transmission capacities. Currently the requirement for extensive bandwidth is a problem for efficient use. Superfractals and the associated colouring algorithm could be used to develop a new system to produce synthetic content for wireless devices that would require only low bandwidth.Read moreRead less
3D Image segmentation and shape characterisation driven by topological persistence. Tomographic imaging is emerging as a new tool to help tackle a remarkable array of scientific challenges. What distinguishes healthy bone from that of osteoporosis sufferers? How does groundwater contamination spread? Why is a macadamia nut so hard to crack? What causes the iridescence in a butterfly wing? These are just a few of the questions being answered at tomographic facilities in Australia alone. By co ....3D Image segmentation and shape characterisation driven by topological persistence. Tomographic imaging is emerging as a new tool to help tackle a remarkable array of scientific challenges. What distinguishes healthy bone from that of osteoporosis sufferers? How does groundwater contamination spread? Why is a macadamia nut so hard to crack? What causes the iridescence in a butterfly wing? These are just a few of the questions being answered at tomographic facilities in Australia alone. By combining sophisticated mathematics with cutting edge image-processing algorithms, this project will yield a new class of topology driven image analysis techniques that will improve the accuracy and reliability of predictions made from tomographic images.Read moreRead less
HARMONIC ANALYSIS AND BOUNDARY VALUE PROBLEMS FOR ELLIPTIC SYSTEMS. It is of the utmost necessity for Australia to develop the theoretical
expertise needed in the current era. The type of mathematics under
investigation here is closely allied to that needed in much of the
current boom in communication technology and medical research. The
training which would be provided to the research associates is
considerable, and would flow on to produce the expertise needed to
keep the coming gen ....HARMONIC ANALYSIS AND BOUNDARY VALUE PROBLEMS FOR ELLIPTIC SYSTEMS. It is of the utmost necessity for Australia to develop the theoretical
expertise needed in the current era. The type of mathematics under
investigation here is closely allied to that needed in much of the
current boom in communication technology and medical research. The
training which would be provided to the research associates is
considerable, and would flow on to produce the expertise needed to
keep the coming generation involved in modern technological development. I will maintain my large collaborative effort with
leading mathematicians from the US, France and other countries, thus
helping to keep Australia at the forefront of a significant field of
research.Read moreRead less
HARMONIC ANALYSIS OF ELLIPTIC SYSTEMS ON RIEMANNIAN MANIFOLDS. It is of the utmost necessity for Australia to develop the theoretical expertise needed in the current era. The type of mathematics under investigation here is closely allied to that needed in much of the current boom in communication technology and medical research. The training which would be provided to the research associates is considerable, and would flow on to produce the expertise needed to keep the coming generation invol ....HARMONIC ANALYSIS OF ELLIPTIC SYSTEMS ON RIEMANNIAN MANIFOLDS. It is of the utmost necessity for Australia to develop the theoretical expertise needed in the current era. The type of mathematics under investigation here is closely allied to that needed in much of the current boom in communication technology and medical research. The training which would be provided to the research associates is considerable, and would flow on to produce the expertise needed to keep the coming generation involved in modern technological development. I will maintain my active collaborative effort with leading mathematicians from the US, France and other countries, thus helping to keep Australia at the forefront of a significant field of research.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
Physics of Risk: new tools to survey the Australian market and beyond. The lives of most Australians depend on the dynamics of financial markets that affects investments, savings, business, employment, growth, wealth and -ultimately- the daily functioning of our society. Understanding, monitoring and managing the dynamics of financial markets is of crucial importance to policy-makers, financial institutions and businesses that are increasingly faced with managing risk, planning strategies and ta ....Physics of Risk: new tools to survey the Australian market and beyond. The lives of most Australians depend on the dynamics of financial markets that affects investments, savings, business, employment, growth, wealth and -ultimately- the daily functioning of our society. Understanding, monitoring and managing the dynamics of financial markets is of crucial importance to policy-makers, financial institutions and businesses that are increasingly faced with managing risk, planning strategies and taking decisions in an increasingly complex market-place. The project is also of importance to the continued evolution of physics in this country contributing to the emergence of a strong new area of statistical physics concerned with the ?real world? in a manner hitherto unknown.Read moreRead less
HARMONIC ANALYSIS, BOUNDARY VALUE PROBLEMS, AND MAXWELL'S EQUATIONS IN LIPSCHITZ DOMAINS. Boundary value problems for partial differential equations arise naturally when physical problems are expressed in mathematical terms. This project concerns the systematic development of the harmonic analysis of partial differential operators, and of the corresponding boundary integrals in order to solve such problems on irregular regions. Particular emphasis is given to studying the behaviour of electrom ....HARMONIC ANALYSIS, BOUNDARY VALUE PROBLEMS, AND MAXWELL'S EQUATIONS IN LIPSCHITZ DOMAINS. Boundary value problems for partial differential equations arise naturally when physical problems are expressed in mathematical terms. This project concerns the systematic development of the harmonic analysis of partial differential operators, and of the corresponding boundary integrals in order to solve such problems on irregular regions. Particular emphasis is given to studying the behaviour of electromagnetic waves both inside and outside irregularly shaped surfaces, and their propagation through it.Read moreRead less
Novel geometric invariants. Quantum theory is the language of fundamental physics, it describes the small scale structure of matter and possibly space-time. Sophisticated models in condensed matter physics and string theory have exposed geometric and topological structure as basic building blocks of the theory. Issues thrown up by quantum theory are very similar to, and have provided techniques to solve, problems in the geometry of three and four dimensional manifolds. Exciting two way exchanges ....Novel geometric invariants. Quantum theory is the language of fundamental physics, it describes the small scale structure of matter and possibly space-time. Sophisticated models in condensed matter physics and string theory have exposed geometric and topological structure as basic building blocks of the theory. Issues thrown up by quantum theory are very similar to, and have provided techniques to solve, problems in the geometry of three and four dimensional manifolds. Exciting two way exchanges of methods, problems and solutions have emerged. This project aims to settle fundamental questions in the interaction between these two fields.Read moreRead less
Asymptotic Geometric Analysis and Learning Theory. Learning Theory is used in various real-world applications in diverse research areas, ranging from Biology (e.g. DNA sequencing) to Information Sciences. Therefore, having a deep understanding of fundamental questions in Learning Theory, and in particular, pin-pointing the parameters that make a learning problem hard would have a significant practical impact. This projects aims to achieve this goal, and in addition, we expect it would have a hig ....Asymptotic Geometric Analysis and Learning Theory. Learning Theory is used in various real-world applications in diverse research areas, ranging from Biology (e.g. DNA sequencing) to Information Sciences. Therefore, having a deep understanding of fundamental questions in Learning Theory, and in particular, pin-pointing the parameters that make a learning problem hard would have a significant practical impact. This projects aims to achieve this goal, and in addition, we expect it would have a high theoretical value, as the questions we shall address are of independent interest to pure mathematicians.Read moreRead less
Parallel and Distributed Machine Learning - Smart Data Analysis in the Multicore Era. In large data centres our research will lead to reduced energy consumption by using graphics cards which have a much better computation to power ratio than traditional processors. On desktop computers, it will make machine learning practical by enabling efficient algorithms for spam filtering and content analysis. On networked systems it will lead to distributed inference, caching and collaborative filtering ap ....Parallel and Distributed Machine Learning - Smart Data Analysis in the Multicore Era. In large data centres our research will lead to reduced energy consumption by using graphics cards which have a much better computation to power ratio than traditional processors. On desktop computers, it will make machine learning practical by enabling efficient algorithms for spam filtering and content analysis. On networked systems it will lead to distributed inference, caching and collaborative filtering applications which will both reduced the bandwidth required and make the internet safer for users. Finally, it will enable rapid deployment of sensor networks for monitoring and detection, such as for environmental monitoring and safeguarding Australia's borders.Read moreRead less