Curvature flows and spectral estimates. Curvature flows are a class of geometrically motivated equations, modelled on the heat equation. Recently, researchers have developed new methods for studying the regularity of solutions to these equations, and applied them to a different problem, that of estimating quantities depending on the smaller eigenvalues of a Schroedinger operator. This project builds on the early success of this research and will produce a new understanding of the behaviour of ei ....Curvature flows and spectral estimates. Curvature flows are a class of geometrically motivated equations, modelled on the heat equation. Recently, researchers have developed new methods for studying the regularity of solutions to these equations, and applied them to a different problem, that of estimating quantities depending on the smaller eigenvalues of a Schroedinger operator. This project builds on the early success of this research and will produce a new understanding of the behaviour of eigenvalues, establish sharp estimates for spectral quantities, particularly on manifolds with curvature bounds, and find optimal conditions under which non-compact solutions to curvature flows are stable.Read moreRead less
Variational theory for fully nonlinear elliptic equations. This project aims to develop new methods and techniques to solve challenging mathematical problems in fully nonlinear partial differential equations arising in important applications. The project will develop methods and techniques to study these equations’ regularity and variational properties. This project is expected to establish comprehensive theories and enhance and promote Australian participation and leadership in this area of mat ....Variational theory for fully nonlinear elliptic equations. This project aims to develop new methods and techniques to solve challenging mathematical problems in fully nonlinear partial differential equations arising in important applications. The project will develop methods and techniques to study these equations’ regularity and variational properties. This project is expected to establish comprehensive theories and enhance and promote Australian participation and leadership in this area of mathematics.Read moreRead less
Harmonic analysis in rough contexts. Harmonic analysis is a set of mathematical techniques aimed at decomposing complex signals into simple pieces in a way that is reminiscent of the decomposition of sounds into harmonics. It is highly efficient in analysing signals in homogeneous media such as wave propagation through the air that underpins wireless communication technology. However, wave propagation through inhomogeneous media, such as the human body in medical imaging or the Earth in geophysi ....Harmonic analysis in rough contexts. Harmonic analysis is a set of mathematical techniques aimed at decomposing complex signals into simple pieces in a way that is reminiscent of the decomposition of sounds into harmonics. It is highly efficient in analysing signals in homogeneous media such as wave propagation through the air that underpins wireless communication technology. However, wave propagation through inhomogeneous media, such as the human body in medical imaging or the Earth in geophysical imaging, is much harder to model. Phenomena with random components, as considered in finance for instance, are also problematic. This project is an important part of an intense international research effort to develop harmonic analysis in such rough contexts.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE120101167
Funder
Australian Research Council
Funding Amount
$375,000.00
Summary
Canonical metrics on Kahler manifolds and Monge-Ampere equations. This project will introduce new ideas and techniques to study the existence of canonical metrics on Kahler manifolds, which is a fundamental problem in geometry. Advances in this research will have influence on other areas of science such as mechanics, string theory and mathematical physics.
Symmetry and geometric structures. This is a fundamental research project in mathematics, especially concerned with the interaction between symmetry, differential equations, and geometry. Based on many classical and recently discovered instances, the aim of the project is to use symmetries to build and understand curved geometric structures from their flat counterparts.
Higher order curvature flow of curves and hypersurfaces. This project aims to analyse higher order geometric partial differential equations that have important mathematical applications in differential geometry of submanifolds as well as practical applications in physics and mathematical biology. The project aims to prove new general principles that reveal properties of these higher order elliptic and parabolic partial differential equations, producing a unified framework with applications to va ....Higher order curvature flow of curves and hypersurfaces. This project aims to analyse higher order geometric partial differential equations that have important mathematical applications in differential geometry of submanifolds as well as practical applications in physics and mathematical biology. The project aims to prove new general principles that reveal properties of these higher order elliptic and parabolic partial differential equations, producing a unified framework with applications to various specific problems. This project aims to increase Australia's research capacity in geometric evolution problems, provide training for some of Australia's next generation of mathematicians and build Australia's international reputation for significant research in geometric analysis.Read moreRead less
Fully nonlinear elliptic equations and applications. This project aims to develop new methods to solve challenging problems in fully nonlinear elliptic equations, and to confirm and enhance Australia as a world leader in this very active area. In addition to high impact publications, this highly innovative research also provides continued building of expertise and training in the area.
Discovery Early Career Researcher Award - Grant ID: DE200101834
Funder
Australian Research Council
Funding Amount
$418,410.00
Summary
The structure of singularities in geometric flows. The proposed research aims to develop our understanding of the structure of singularities in mean curvature and related flows, with certain applications in mind.
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
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