Singularities and surgery in geometric evolution equations. The analysis of geometric evolution equations is a very active area of mathematical research internationally. The applications of such systems to physical problems such as crystal growth and flame propagation are also of great interest in the broader scientific community. The proposed research addresses questions central to the understanding of curvature flows. The project will yield internationally significant results in theoretical ....Singularities and surgery in geometric evolution equations. The analysis of geometric evolution equations is a very active area of mathematical research internationally. The applications of such systems to physical problems such as crystal growth and flame propagation are also of great interest in the broader scientific community. The proposed research addresses questions central to the understanding of curvature flows. The project will yield internationally significant results in theoretical mathematics, with applications in physics, engineering and image processing. These results will enhance Australia's reputation for high quality theoretical mathematical research with real world applications.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 evolution equations and global effects of curvature. This project aims to approach several important problems in global differential geometry, by inventing new processes to deform geometric objects to simpler ones. The deformations are described by carefully constructed geometric evolution equations, designed to exhibit behaviour suited to the given problem. The project proposes methods for building such equations, and new techniques for their analysis. The research is expected to yi ....Geometric evolution equations and global effects of curvature. This project aims to approach several important problems in global differential geometry, by inventing new processes to deform geometric objects to simpler ones. The deformations are described by carefully constructed geometric evolution equations, designed to exhibit behaviour suited to the given problem. The project proposes methods for building such equations, and new techniques for their analysis. The research is expected to yield significant new results, both in differential geometry and in nonlinear heat equations, and should provide substantial progress towards resolving several important long-standing conjectures.
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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.
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Energy, Cosmic Censorship and Black Hole Stability. Human progress is achieved by confronting fundamental questions, at the leading edge of knowledge. This project will lead to better understanding of space-time physics, and of the properties of singular solutions of non-linear hyperbolic equations. Such equations govern a wide range of physical phenomena, including fluid flow, weather and electromagnetic fields.
Global properties of evolution on manifolds. The aim is to analyze global properties of solutions of parabolic equations on manifolds and in particular the equations associated with a family of Hormander fields.
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