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.
Discovery Early Career Researcher Award - Grant ID: DE210100535
Funder
Australian Research Council
Funding Amount
$340,548.00
Summary
Minimal surfaces and singularities of mean curvature flow. The project aims to characterise the geometric structure of minimal surfaces in the variational theory and classify singularities of mean curvature flow. Minimal surfaces are mathematical models of soap films, and their time-varying analogue is mean curvature flow, a dynamic process by which a surface flows to decrease its area as quickly as possible. As a central topic in geometric analysis, the theory of minimal surfaces and mean curv ....Minimal surfaces and singularities of mean curvature flow. The project aims to characterise the geometric structure of minimal surfaces in the variational theory and classify singularities of mean curvature flow. Minimal surfaces are mathematical models of soap films, and their time-varying analogue is mean curvature flow, a dynamic process by which a surface flows to decrease its area as quickly as possible. As a central topic in geometric analysis, the theory of minimal surfaces and mean curvature flow has proven to be a powerful and essential tool in mathematics. The project expects to generate new and significant results in minimal surfaces and singularity analysis of mean curvature flow and enhance potential applications in related disciplines such as computer vision and probability.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE180100110
Funder
Australian Research Council
Funding Amount
$343,450.00
Summary
Analysis of fully non-linear geometric problems and differential equations. This project aims to investigate non-linear geometric evolution equations that have received considerable attention in the past decades through their use in solving outstanding problems in mathematics, such as the Poincare conjecture. By developing innovative new techniques intertwining geometry and analysis, the project endeavours to make advances in non-linear problems modelling complex phenomena. The project addresses ....Analysis of fully non-linear geometric problems and differential equations. This project aims to investigate non-linear geometric evolution equations that have received considerable attention in the past decades through their use in solving outstanding problems in mathematics, such as the Poincare conjecture. By developing innovative new techniques intertwining geometry and analysis, the project endeavours to make advances in non-linear problems modelling complex phenomena. The project addresses topics as varied as hyperbolic geometry, and a geometric approach to irregularities forming in crystal growth in materials science, focusing on developing cutting-edge mathematical tools and connections to geometry.Read moreRead less
Parabolic methods for elliptic boundary value problems. This project aims to uncover new results for second order nonlinear elliptic partial differential equations via the use of uniqueness properties of solutions for related nonlinear parabolic partial differential equations. This will build on theory for fully nonlinear equations developed over the last 30 years. The project expects to generate new knowledge in the theory that will guide future research and have direct impact to applications ....Parabolic methods for elliptic boundary value problems. This project aims to uncover new results for second order nonlinear elliptic partial differential equations via the use of uniqueness properties of solutions for related nonlinear parabolic partial differential equations. This will build on theory for fully nonlinear equations developed over the last 30 years. The project expects to generate new knowledge in the theory that will guide future research and have direct impact to applications in optimal transport, geometric problems and more applied areas including image analysis and mathematical finance. The project will enhance Australia's international reputation for research in the field and train some of the next generation of mathematical analysts.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE230100415
Funder
Australian Research Council
Funding Amount
$422,154.00
Summary
Rigidity and boundary phenomena for geometric variational problems. The proposed project aims to investigate theoretical properties of thin films and fluid interfaces, which are modelled as surfaces driven by surface tension, possibly in an enclosing container. This project is expected to generate new knowledge in the area of geometric partial differential equations, by utilising new techniques in geometric flows, and by establishing novel methods for boundary value problems. The developed techn ....Rigidity and boundary phenomena for geometric variational problems. The proposed project aims to investigate theoretical properties of thin films and fluid interfaces, which are modelled as surfaces driven by surface tension, possibly in an enclosing container. This project is expected to generate new knowledge in the area of geometric partial differential equations, by utilising new techniques in geometric flows, and by establishing novel methods for boundary value problems. The developed techniques may have far-reaching applications in other areas of mathematical analysis, and the expected results would contribute greatly to the theory of surfaces governed by mean curvature, which arise in various real-world phenomena such as soap bubbles, black hole horizons and bushfire fronts. Read moreRead less
Australian Laureate Fellowships - Grant ID: FL150100126
Funder
Australian Research Council
Funding Amount
$2,080,100.00
Summary
Geometric analysis of eigenvalues and heat flows. Geometric analysis of eigenvalues and heat flows: This fellowship project aims to build on Australia's leading position in the areas of nonlinear partial differential equations and geometric analysis to exploit new and highly innovative mathematical methods. It is expected that the methods will affect a range of related fields including stochastic modelling and finance, image processing, and the basic sciences. The project seeks to serve as a foc ....Geometric analysis of eigenvalues and heat flows. Geometric analysis of eigenvalues and heat flows: This fellowship project aims to build on Australia's leading position in the areas of nonlinear partial differential equations and geometric analysis to exploit new and highly innovative mathematical methods. It is expected that the methods will affect a range of related fields including stochastic modelling and finance, image processing, and the basic sciences. The project seeks to serve as a focal point for a developing community of Australian researchers in this field, providing a training ground for young researchers and students at the forefront of a vigorous and internationally active area of research, and bringing top international researchers to Australia to interact with the local research community.Read moreRead less
Singularity and regularity for Monge-Ampere type equations. The Monge-Ampere equation, as a premier nonlinear partial differential equation, arises in several areas including geometry, physics, and optimal transportation. Many important problems and applications are related to the regularity of solutions, which are obstructed by singularities. This project aims to classify the geometry of the singular sets, and to establish a comprehensive regularity theory for general Monge-Ampere type equation ....Singularity and regularity for Monge-Ampere type equations. The Monge-Ampere equation, as a premier nonlinear partial differential equation, arises in several areas including geometry, physics, and optimal transportation. Many important problems and applications are related to the regularity of solutions, which are obstructed by singularities. This project aims to classify the geometry of the singular sets, and to establish a comprehensive regularity theory for general Monge-Ampere type equations by using innovative approaches and developing cutting-edge technologies in partial differential equations. Expected outcomes include the resolution of outstanding open problems. This project will significantly enhance Australia’s leadership and expertise in a major area of mathematics and applications.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE180101360
Funder
Australian Research Council
Funding Amount
$328,075.00
Summary
The geometry and cohomology of moduli spaces of curves. This project aims to develop new insights on moduli spaces in algebraic geometry. Algebraic geometry is the field of mathematics that uses geometric methods to analyse algebraic equations, with wide applications ranging from cryptography to genetics. Moduli spaces in algebraic geometry provide powerful methods to geometrically analyse collections of related equations. Using innovative new techniques, the project aims to generate new knowled ....The geometry and cohomology of moduli spaces of curves. This project aims to develop new insights on moduli spaces in algebraic geometry. Algebraic geometry is the field of mathematics that uses geometric methods to analyse algebraic equations, with wide applications ranging from cryptography to genetics. Moduli spaces in algebraic geometry provide powerful methods to geometrically analyse collections of related equations. Using innovative new techniques, the project aims to generate new knowledge about fundamental moduli spaces. Expected outcomes include the establishment of an active community of algebraic geometers in Australia. These outcomes should provide significant benefits to pure mathematics and related scientific fields.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
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