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
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
Quantum chaos and scattering theory. The project will involve mathematical research of the highest international standard, as well as research training of postgraduate students and postdoctoral researchers, in a very active and far-reaching field. Progress in this field will have implications in areas ranging from engineering (e.g. nanotechnology, quantum computing) and mathematical analysis (e.g. theory of partial differential equations) through to number theory.
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).
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).
Geometric Spectral and Scattering Theory. Spectral and scattering theory is the mathematical study of natural frequencies (eigenvalues) and modes of vibration (eigenfunctions) of systems arising in geometry, physics, and engineering. As such, it has important applications in numerous areas including medical imaging, geological surveying and the transmission of information along optical fibres. In this project I will solve a variety of problems involving high-frequency asymptotics of eigenvalues, ....Geometric Spectral and Scattering Theory. Spectral and scattering theory is the mathematical study of natural frequencies (eigenvalues) and modes of vibration (eigenfunctions) of systems arising in geometry, physics, and engineering. As such, it has important applications in numerous areas including medical imaging, geological surveying and the transmission of information along optical fibres. In this project I will solve a variety of problems involving high-frequency asymptotics of eigenvalues, quantum chaos, eigenfunction concentration and nonlinear wave propagation.Read moreRead less
Hardy spaces of differential forms and applications. Hardy spaces on Euclidean spaces were developed in the 1970's following the fundamental work of Stein, Weiss and Fefferman. These spaces play an important role in harmonic analysis, as they are the natural spaces on which to consider singular integral operators. They arise in many contexts, such as when using Jacobians in non-linear partial differential equations. Recently the French participants and the Australian participants have have obt ....Hardy spaces of differential forms and applications. Hardy spaces on Euclidean spaces were developed in the 1970's following the fundamental work of Stein, Weiss and Fefferman. These spaces play an important role in harmonic analysis, as they are the natural spaces on which to consider singular integral operators. They arise in many contexts, such as when using Jacobians in non-linear partial differential equations. Recently the French participants and the Australian participants have have obtained different but related results concerning Hardy spaces of exact differential forms. The time is now ripe to construct a unified theory.
Read moreRead less
Nonlinear Partial Differential Equations: Singularities, Potential Theory, and Geometric Applications. The main objective of the project is to study properties of solutions to fully nonlinear, elliptic partial differential equations. Rather than studying more traditional existence-uniqueness problems the main task will be to investigate qualitative questions. These concern the behaviour of solutions to the equations, the description of possible pathologies and singularities the solutions can hav ....Nonlinear Partial Differential Equations: Singularities, Potential Theory, and Geometric Applications. The main objective of the project is to study properties of solutions to fully nonlinear, elliptic partial differential equations. Rather than studying more traditional existence-uniqueness problems the main task will be to investigate qualitative questions. These concern the behaviour of solutions to the equations, the description of possible pathologies and singularities the solutions can have, and conditions for the absence of singularities. Understanding of the singular behaviour of solutions is very important for applications in geometry, physics, elasticity, and mechanics. From this point of view, probably the most important problem is to find explicit information about singularities of solutions.Read moreRead less