Discovery Early Career Researcher Award - Grant ID: DE240100386
Funder
Australian Research Council
Funding Amount
$435,875.00
Summary
Anti-racist neuroethics for epistemic justice in mental health research. Racial/ethnic minorities are underrepresented in brain and mental health (BMH) research, risking inadequate healthcare for the 9.5 million minorities in Australia. With the $73 billion annual cost of BMH disorders to the country, all Australians should equally benefit from BMH research. This project aims to develop recommendations to make BMH research more diverse and inclusive. It will audit representation of minorities in ....Anti-racist neuroethics for epistemic justice in mental health research. Racial/ethnic minorities are underrepresented in brain and mental health (BMH) research, risking inadequate healthcare for the 9.5 million minorities in Australia. With the $73 billion annual cost of BMH disorders to the country, all Australians should equally benefit from BMH research. This project aims to develop recommendations to make BMH research more diverse and inclusive. It will audit representation of minorities in Australian BMH publications and will conduct surveys, interviews, and workshops with scientists to determine institutional barriers to the inclusion of and engagement with minorities in research. This project will draw from concepts of epistemic justice and anti-racism to develop ethical frameworks for BMH racial equity.Read moreRead less
Harmonic analysis: function spaces and partial differential equations. This project aims to solve a number of important problems at the frontier of harmonic analysis on metric measure spaces. Harmonic analysis has been instrumental to several fields of mathematics including complex analysis and partial differential equations which have had many applications in engineering and technology. This project will solve a number of important problems as well as develop new approaches and techniques for r ....Harmonic analysis: function spaces and partial differential equations. This project aims to solve a number of important problems at the frontier of harmonic analysis on metric measure spaces. Harmonic analysis has been instrumental to several fields of mathematics including complex analysis and partial differential equations which have had many applications in engineering and technology. This project will solve a number of important problems as well as develop new approaches and techniques for research in harmonic analysis and related topics. The project will maintain and enhance the strength of Australian mathematical research in harmonic analysis and contribute to the training of the next generation of mathematical researchers in Australia.Read moreRead less
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
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
Nonlinear harmonic analysis and dispersive partial differential equations. This proposal is devoted to linear and nonlinear harmonic analysis. It aims to unify the most significant attributes of harmonic analysis such as restriction estimates, dispersive properties of differential operators, spectral multipliers, uniform Sobolev estimates and sharp Weyl formula. Such unification will strongly improve tools for mathematical modelling in all areas of technology and science. Notable applications in ....Nonlinear harmonic analysis and dispersive partial differential equations. This proposal is devoted to linear and nonlinear harmonic analysis. It aims to unify the most significant attributes of harmonic analysis such as restriction estimates, dispersive properties of differential operators, spectral multipliers, uniform Sobolev estimates and sharp Weyl formula. Such unification will strongly improve tools for mathematical modelling in all areas of technology and science. Notable applications include medical imaging, fluid dynamics and subatomic modelling using quantum interpretation.
It will solve several important open problems in spectral analysis of partial differential operators and develop new cutting-edge techniques in harmonic analysis with application to nonlinear partial differential equations.Read moreRead less
Australian Laureate Fellowships - Grant ID: FL220100072
Funder
Australian Research Council
Funding Amount
$2,490,704.00
Summary
Mathematical Breakthroughs in Wave Propagation. This Fellowship proposal in theoretical mathematics aims to solve three major open problems in wave propagation. These are the long-time behaviour of nonlinear waves, including the behaviour and interaction of solitary waves; the propagation of waves in rough media; and the small-scale behaviour of interacting waves under the assumption of chaotic ray dynamics. The research aims to analyse wave equations that model problems in optical media and wav ....Mathematical Breakthroughs in Wave Propagation. This Fellowship proposal in theoretical mathematics aims to solve three major open problems in wave propagation. These are the long-time behaviour of nonlinear waves, including the behaviour and interaction of solitary waves; the propagation of waves in rough media; and the small-scale behaviour of interacting waves under the assumption of chaotic ray dynamics. The research aims to analyse wave equations that model problems in optical media and waveguides, medical and seismic imaging, and nano-electronic devices. Outcomes and benefits are expected in new mathematical theory, Australian research capability, better algorithms for numerically computing waves, and technological advances in communications, medical imaging, and seismic imaging.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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Discovery Early Career Researcher Award - Grant ID: DE200101802
Funder
Australian Research Council
Funding Amount
$354,016.00
Summary
Combinatorial and Representation Theoretic Methods in Number Theory. This Project aims to explore connections of Number Theory and Representation Theory by utilising tools of Algebraic Combinatorics. Symmetries and constructions of crucial number theoretic objects such as Whittaker functions are underpinned by models for Lie algebras and root systems. The Project expects to advance the algebraic framework of the constructions. Expected outcomes include a unified combinatorial model of these obje ....Combinatorial and Representation Theoretic Methods in Number Theory. This Project aims to explore connections of Number Theory and Representation Theory by utilising tools of Algebraic Combinatorics. Symmetries and constructions of crucial number theoretic objects such as Whittaker functions are underpinned by models for Lie algebras and root systems. The Project expects to advance the algebraic framework of the constructions. Expected outcomes include a unified combinatorial model of these objects, and an extension of the costructions to the infinite dimensional setting. This will benefit the applications in Number Theory and strengthen nascent connections with Mathematical Physics. Read moreRead less
Australian Laureate Fellowships - Grant ID: FL170100052
Funder
Australian Research Council
Funding Amount
$2,107,500.00
Summary
Breakthrough methods for noncommutative calculus. This project aims to solve hard, outstanding problems which have impeded our ability to progress in the area of quantum or noncommutative calculus. Calculus has provided an invaluable tool to science, enabling scientific and technological revolutions throughout the past two centuries. The project will initiate a program of collaboration among top mathematical researchers from around the world and bring together two separate mathematical areas int ....Breakthrough methods for noncommutative calculus. This project aims to solve hard, outstanding problems which have impeded our ability to progress in the area of quantum or noncommutative calculus. Calculus has provided an invaluable tool to science, enabling scientific and technological revolutions throughout the past two centuries. The project will initiate a program of collaboration among top mathematical researchers from around the world and bring together two separate mathematical areas into a powerful new set of tools. The outcomes from the project will impact research at the forefront of mathematical physics and other sciences and enhance Australia’s reputation and standing.Read moreRead less
Machine learning, group theory and combinatorics. This project aims to investigate group theory and combinatorics using machine learning techniques. This project expects to generate new knowledge concerning symmetric groups and symmetric functions, using an innovative approach from reinforcement learning. Expected outcomes of this project include a clarification of the types of difficult problems in pure mathematics that can be gainfully attacked via machine learning, and an understanding of the ....Machine learning, group theory and combinatorics. This project aims to investigate group theory and combinatorics using machine learning techniques. This project expects to generate new knowledge concerning symmetric groups and symmetric functions, using an innovative approach from reinforcement learning. Expected outcomes of this project include a clarification of the types of difficult problems in pure mathematics that can be gainfully attacked via machine learning, and an understanding of the role of group theory in machine learning. This should provide significant benefits, such as progress on long standing open problems, the development of an emerging technology with significant implications for mathematics, and the training of Australian scientists in a vital area of research.Read moreRead less