Partial differential equation: Schrodinger operator and long-time dynamics. This project aims to develop new analysis methods associated to the Schrodinger operator, and to solve several challenging problems regarding dispersive partial differential equations (PDE). Long-time dynamics of PDE solutions are a key goal in both pure and applied mathematics, and have been extensively studied by leading mathematicians and mathematical physicists. However, it is unknown how to investigate large soluti .... Partial differential equation: Schrodinger operator and long-time dynamics. This project aims to develop new analysis methods associated to the Schrodinger operator, and to solve several challenging problems regarding dispersive partial differential equations (PDE). Long-time dynamics of PDE solutions are a key goal in both pure and applied mathematics, and have been extensively studied by leading mathematicians and mathematical physicists. However, it is unknown how to investigate large solutions when the order of the PDE's nonlinearity is low. This project expects to develop new methods to attack such problems. The results of the project will be of great importance in mathematics and physics, as many fundamental physical models in areas such as optics, fluid mechanics and quantum mechanics fit the paradigm.Read moreRead less
Problems in harmonic analysis: decoupling and Bourgain-Brezis inequalities. This project in mathematics aims to study two recent, promising developments in harmonic analysis, namely Fourier decoupling and Bourgain-Brezis inequalities. The former captures how waves interfere upon superposition; the latter arose initially in the study of the Ginzburg-Landau theory of superconductors. This exciting project seeks to deliver deep insights into how different frequencies interact, and aims to develop p ....Problems in harmonic analysis: decoupling and Bourgain-Brezis inequalities. This project in mathematics aims to study two recent, promising developments in harmonic analysis, namely Fourier decoupling and Bourgain-Brezis inequalities. The former captures how waves interfere upon superposition; the latter arose initially in the study of the Ginzburg-Landau theory of superconductors. This exciting project seeks to deliver deep insights into how different frequencies interact, and aims to develop powerful new tools to advance the study of partial differential equations and analytic number theory. This Future Fellowship should benefit Australia by improving our scientific capability. It will bring world-class researchers to Australia for collaboration, and put Australia at the forefront of first rate research.
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Reducing the social, economic and health burden associated with obesity-related chronic diseases among socio-economically disadvantaged populations. This project will develop new methods and approaches for reducing obesity-related chronic diseases (OCDs) among socially disadvantaged populations in Australia, using prevention models. These prevention models will improve the evidence base in this field as well as inform public health policy and practice in Australia (and other industrialised count ....Reducing the social, economic and health burden associated with obesity-related chronic diseases among socio-economically disadvantaged populations. This project will develop new methods and approaches for reducing obesity-related chronic diseases (OCDs) among socially disadvantaged populations in Australia, using prevention models. These prevention models will improve the evidence base in this field as well as inform public health policy and practice in Australia (and other industrialised countries).Read moreRead less
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
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
Inverse Problems For Partial Differential Equations - A Geometric Analysis Perspective. This project will study mathematical models of various medical imaging techniques. These problems are formulated as inverse problems in partial differential equations (PDE) where one wishes to obtain information about a differential equation from data about its solutions. This problem is not well understood in the geometric setting where the PDE is taking place on a manifold and the goal of this research is t ....Inverse Problems For Partial Differential Equations - A Geometric Analysis Perspective. This project will study mathematical models of various medical imaging techniques. These problems are formulated as inverse problems in partial differential equations (PDE) where one wishes to obtain information about a differential equation from data about its solutions. This problem is not well understood in the geometric setting where the PDE is taking place on a manifold and the goal of this research is to advance the field in this direction. This project will introduce novel and innovative ideas from geometry and topology to overcome some of these difficulties. This project will enrich mathematics by providing links between different fields. Furthermore, it will enable the application of imaging techniques in a broader geometric setting to provide more efficient and accurate non-invasive detection techniques.Read moreRead less
Holonomy groups and special structures in pseudo-Riemannian geometry. The project studies mathematical models used in physical theories, such as general relativity and string theory, to create a global picture of the universe. The outcomes will enhance the role that Australia plays in these developments and contribute to the mathematical knowledge which lies at the foundations of modern technologies.
Understanding and addressing racism in Australia. Despite the well recognised need to understand and address racism, it remains a globally significant issue. Encompassing a range of internationally novel research, this project aims to enhance conceptual understandings of racism and anti-racism and investigate empirical data on the health and social effects of racism. It will design, implement, evaluate and analyse findings from anti-racism interventions, examine the interplay between racism and ....Understanding and addressing racism in Australia. Despite the well recognised need to understand and address racism, it remains a globally significant issue. Encompassing a range of internationally novel research, this project aims to enhance conceptual understandings of racism and anti-racism and investigate empirical data on the health and social effects of racism. It will design, implement, evaluate and analyse findings from anti-racism interventions, examine the interplay between racism and the impact of organisational diversity and shape policy and practice relating to racism, anti-racism and diversity across various sectors. This multi-faceted approach will considerably enhance our understanding of, and capacity to address, racism both in Australia and internationally.Read moreRead less