Global wavefront propagation and non-elliptic Fredholm theory. Many significant phenomena in the natural world are described by partial differential equations that involve evolution in time. This project aims to develop new mathematical methods, involving recently discovered global wavefront set analysis and Fredholm theory, to solve such equations. These methods aim to extend the range of equations that can be solved as well as yield more information about solutions, in particular, their long-t ....Global wavefront propagation and non-elliptic Fredholm theory. Many significant phenomena in the natural world are described by partial differential equations that involve evolution in time. This project aims to develop new mathematical methods, involving recently discovered global wavefront set analysis and Fredholm theory, to solve such equations. These methods aim to extend the range of equations that can be solved as well as yield more information about solutions, in particular, their long-time asymptotics.Read moreRead less
Harmonic analysis and spectral analysis of differential operators. Harmonic analysis has had a profound influence in many areas of mathematics, including partial differential equations. This project is at the frontier of research, closely related to theoretical work of applied technology such as signal processing, financial modelling and it ensures that Australia maintains a high research profile in this field.
Geometric transforms and duality. This Proposal is fundamental, basic research at the forefront of modern differential geometry and its application to physics. It will ensure that Australia is involved in today's mathematical and physical advances and that we have Australian mathematicians trained to take advantage of the future benefits of these advances.
Indecomposable Structure in Representation Theory and Logarithmic Conformal Field Theory. Logarithmic conformal field theory describes non-local observables in statistical models of important physical systems (eg. polymers, percolation). This realisation has led to a recent explosion of activity among physicists and mathematicians. Mathematical physics in Australia is well-placed to capitalise on this activity, having several experts working in the area, and this project will significantly aug ....Indecomposable Structure in Representation Theory and Logarithmic Conformal Field Theory. Logarithmic conformal field theory describes non-local observables in statistical models of important physical systems (eg. polymers, percolation). This realisation has led to a recent explosion of activity among physicists and mathematicians. Mathematical physics in Australia is well-placed to capitalise on this activity, having several experts working in the area, and this project will significantly augment Australia's reputation within the international community by bringing (and developing) mathematical tools and insights which complement current research strengths. Such augmentations are vital to the well-being of mathematics and physics in Australia.Read moreRead less
Algebraic Structures in Mathematical Physics and Their Applications. Algebraic structures such as affine (super)algebras, quantised algebras and vertex operator algebras are among the most important discoveries in mathematics. They provide a universal common algebraic framework underlying applications in a wide range of physics (eg. statistical mechanics, string theory, condensed matter physics etc.) leading to a high level of research activity worldwide. The project harnessess the high level ....Algebraic Structures in Mathematical Physics and Their Applications. Algebraic structures such as affine (super)algebras, quantised algebras and vertex operator algebras are among the most important discoveries in mathematics. They provide a universal common algebraic framework underlying applications in a wide range of physics (eg. statistical mechanics, string theory, condensed matter physics etc.) leading to a high level of research activity worldwide. The project harnessess the high level of expertise in mathematical physics across Australia to focus on exciting new developments in the theory of these algebraic structures and their application to physics, thus ensuring Australia plays a leading role in this rapidly expanding field.Read moreRead less
Heat kernel and Riesz transform on non-compact metric measure spaces. This project will develop new techniques in heat kernel theory, with applications to such important topics as Schrodinger model for quantum mechanics. The proposed research is at the forefront of research in harmonic analysis and partial differential equations and will further enhance Australia's high international standing in these research fields.
Proper Group Actions in Complex Geometry. The results of the project will enhance Australia's performance in several key mathematical areas as well as mathematical applications to physics critical for expanding Australia's knowledge base and research capability. The project has strong international aspects, it will foster the international competitiveness of Australian research and establish long-term collaborations between Australian researchers and high profile world experts in the area of the ....Proper Group Actions in Complex Geometry. The results of the project will enhance Australia's performance in several key mathematical areas as well as mathematical applications to physics critical for expanding Australia's knowledge base and research capability. The project has strong international aspects, it will foster the international competitiveness of Australian research and establish long-term collaborations between Australian researchers and high profile world experts in the area of the proposal. It will create an opportunity for a Ph.D. graduate to be involved in top-class research as a Research Associate, and will attract Ph.D. and honours students thus enabling research training in a high-quality mathematical environment.Read moreRead less
Harmonic analysis of rough oscillations. This project intends to explore new perspectives in harmonic analysis. Harmonic analysis is a set of mathematical techniques used in many branches of science and engineering to analyse complex signals (functions). It is highly effective in modelling phenomena such as the propagation of electromagnetic waves, but it is currently limited to propagation occurring in a simple-enough medium. An intense international research effort in harmonic analysis is curr ....Harmonic analysis of rough oscillations. This project intends to explore new perspectives in harmonic analysis. Harmonic analysis is a set of mathematical techniques used in many branches of science and engineering to analyse complex signals (functions). It is highly effective in modelling phenomena such as the propagation of electromagnetic waves, but it is currently limited to propagation occurring in a simple-enough medium. An intense international research effort in harmonic analysis is currently under way to lift this limitation. This project is part of that effort, and aims to unite two of its fundamental directions of development: one focusing on the roughness of the medium; and one focusing on the interaction between highly oscillatory aspects of the function and the geometry of the medium.Read moreRead less
The Ricci curvature of homogeneous spaces. The geometry of homogeneous spaces is an area of research with applications in numerous fields, including topology, harmonic analysis, relativity and quantum theory. This project aims to resolve a fundamental problem in this area, known as the prescribed Ricci curvature problem for homogeneous metrics, and to settle the important and closely related question of Ricci iteration existence and convergence. Moreover, the project aims to exploit the interpla ....The Ricci curvature of homogeneous spaces. The geometry of homogeneous spaces is an area of research with applications in numerous fields, including topology, harmonic analysis, relativity and quantum theory. This project aims to resolve a fundamental problem in this area, known as the prescribed Ricci curvature problem for homogeneous metrics, and to settle the important and closely related question of Ricci iteration existence and convergence. Moreover, the project aims to exploit the interplay between geometry and algebra to provide new insight into the physically significant problem of classifying unitary Lie algebra representations. This project is expected to facilitate interdisciplinary interaction leading to exciting developments across a range of fields.Read moreRead less
A semiclassical approach to spectral theory. Spectral theory is the branch of mathematics dealing with natural frequencies (eigenvalues) and modes of vibration (eigenfunctions) of systems arising in geometry, quantum physics and engineering. As such, they have important applications in seismic and medical imaging, nanotechnology, and optical communications. This project aims to use recently developed mathematical tools to advance our understanding of high energy eigenvalues and eigenfunctions, a ....A semiclassical approach to spectral theory. Spectral theory is the branch of mathematics dealing with natural frequencies (eigenvalues) and modes of vibration (eigenfunctions) of systems arising in geometry, quantum physics and engineering. As such, they have important applications in seismic and medical imaging, nanotechnology, and optical communications. This project aims to use recently developed mathematical tools to advance our understanding of high energy eigenvalues and eigenfunctions, as well as new algorithms for numerically computing them.Read moreRead less