Australian Laureate Fellowships - Grant ID: FL170100020
Funder
Australian Research Council
Funding Amount
$1,638,060.00
Summary
Advances in index theory and applications. The project aims to develop novel techniques to investigate Geometric analysis on infinite dimensional bundles, as well as Geometric analysis of pathological spaces with Cantor set as fibre, that arise in models for the fractional quantum Hall effect and topological matter, areas recognised with the 1998 and 2016 Nobel Prizes. Building on the applicant’s expertise in the area, the project will involve postgraduate and postdoctoral training in order to e ....Advances in index theory and applications. The project aims to develop novel techniques to investigate Geometric analysis on infinite dimensional bundles, as well as Geometric analysis of pathological spaces with Cantor set as fibre, that arise in models for the fractional quantum Hall effect and topological matter, areas recognised with the 1998 and 2016 Nobel Prizes. Building on the applicant’s expertise in the area, the project will involve postgraduate and postdoctoral training in order to enhance Australia’s position at the forefront of international research in Geometric Analysis. Ultimately, the project will enhance Australia's leading position in the area of Index Theory by developing novel techniques to solve challenging conjectures, and mentoring HDR students and ECRs.Read moreRead less
New techniques and invariants in low-dimensional topology. The aim of this project is to introduce and apply new methods and invariants in the field of low-dimensional topology by developing parametrised and equivariant enhancements of Seiberg-Witten theory and Floer homology. These new refined invariants, made possible by recent advances in gauge theory, will be more powerful than existing ones, enabling the detection of new exotic phenomena. Expected outcomes include effective means for distin ....New techniques and invariants in low-dimensional topology. The aim of this project is to introduce and apply new methods and invariants in the field of low-dimensional topology by developing parametrised and equivariant enhancements of Seiberg-Witten theory and Floer homology. These new refined invariants, made possible by recent advances in gauge theory, will be more powerful than existing ones, enabling the detection of new exotic phenomena. Expected outcomes include effective means for distinguishing families of spaces, measuring their complexity and new obstructions for their existence. The new invariants and techniques will lead to the resolution of some open problems in low-dimensional topology and enhance Australia's reputation as a world leader in this field.Read moreRead less
A Novel Geometric Approach to Shocks in Reaction-Nonlinear Diffusion Models. Reaction-nonlinear diffusion models play a vital role in the study of cell migration and population dynamics. However, the presence of aggregation, or backward diffusion, leads to the formation of shock waves - distinct, sharp interfaces between different populations of densities of cells - and the breakdown of the model. This project will develop new geometric methods to explain the formation and temporal evolution of ....A Novel Geometric Approach to Shocks in Reaction-Nonlinear Diffusion Models. Reaction-nonlinear diffusion models play a vital role in the study of cell migration and population dynamics. However, the presence of aggregation, or backward diffusion, leads to the formation of shock waves - distinct, sharp interfaces between different populations of densities of cells - and the breakdown of the model. This project will develop new geometric methods to explain the formation and temporal evolution of these shock waves, while simultaneously unifying existing regularisation techniques under a single, geometric banner. It will devise innovative tools in singular perturbation theory and stability analysis that will identify key parameters in the creation of shock waves, as well as their dynamic behaviour.Read moreRead less
Twisted K-theory, higher geometry and operator algebras. This project aims to develop new theory and techniques linking twisted K-theory, higher-geometry and operator algebras. These are all fundamental areas of mathematics with applications both within mathematics itself and to mathematical physics, particularly in string theory. Anticipated outcomes are fundamental advances in knowledge in mathematics and mathematical physics, enhancement of Australia's international mathematical reputation an ....Twisted K-theory, higher geometry and operator algebras. This project aims to develop new theory and techniques linking twisted K-theory, higher-geometry and operator algebras. These are all fundamental areas of mathematics with applications both within mathematics itself and to mathematical physics, particularly in string theory. Anticipated outcomes are fundamental advances in knowledge in mathematics and mathematical physics, enhancement of Australia's international mathematical reputation and collaborative linkages, and the training of the next generation of Australian mathematicians.Read moreRead less
Multiscale modelling of systems with complex microscale detail. In modern science and engineering many complex systems are described by distinctly different microscale physical models within different regions of space. This project is to develop systematic mathematical and computational methods for the compact and accurate macroscale modelling and computation of such systems for application in industrial research and development. Our sparse simulations, justified with mathematical analysis, use ....Multiscale modelling of systems with complex microscale detail. In modern science and engineering many complex systems are described by distinctly different microscale physical models within different regions of space. This project is to develop systematic mathematical and computational methods for the compact and accurate macroscale modelling and computation of such systems for application in industrial research and development. Our sparse simulations, justified with mathematical analysis, use small bursts of particle/agent simulations, PDEs, or difference equations, to efficiently evaluate macroscale system-level behaviour. The objective is to accurately interface between disparate microscale models and establish provable predictions on how the microscale parameter spaces resolve at the macroscale.Read moreRead less
Integrating rifts and swell in the mathematics of ice shelf disintegration. Antarctic ice-shelf disintegrations have the alarming potential to cause rapid sea level rise, through accelerated discharge of the Antarctic Ice Sheet and initiating runaway Ice Sheet destabilisations. The project aims to develop a mathematical model of swell-induced ice-shelf vibrations in a coupled ocean–shelf 3D framework, focusing on interactions between vibrations and the rift networks that characterise outer shelf ....Integrating rifts and swell in the mathematics of ice shelf disintegration. Antarctic ice-shelf disintegrations have the alarming potential to cause rapid sea level rise, through accelerated discharge of the Antarctic Ice Sheet and initiating runaway Ice Sheet destabilisations. The project aims to develop a mathematical model of swell-induced ice-shelf vibrations in a coupled ocean–shelf 3D framework, focusing on interactions between vibrations and the rift networks that characterise outer shelf margins before disintegration. Accurate, efficient solutions will be developed by fusing powerful approximation theories, and validated by numerical solutions. The model will be combined with state-of-the-art data to predict trends in Antarctica’s remaining ice shelves and indicate potential future disintegrations.Read moreRead less
Modeling, Mathematical Analysis, and Computation of Multiscale Systems. This project develops and implements a systematic approach, both analytic and computational, to extract compact, accurate, system level models of complex physical and engineering systems. Our wide ranging methodology is to construct computationally efficient "wrappers" around fine scale, microscopic, detailed descriptions of dynamical systems (particle or molecular simulation, or PDE or lattice equations). Comprehensively a ....Modeling, Mathematical Analysis, and Computation of Multiscale Systems. This project develops and implements a systematic approach, both analytic and computational, to extract compact, accurate, system level models of complex physical and engineering systems. Our wide ranging methodology is to construct computationally efficient "wrappers" around fine scale, microscopic, detailed descriptions of dynamical systems (particle or molecular simulation, or PDE or lattice equations). Comprehensively accounting for multiscale interactions between subgrid processes among macroscale variations ensures stability and accuracy. Based on dynamical systems theory and analysis, our approach will empower systematic analysis and understanding for optimal macroscopic simulation for forthcoming exascale computing. Read moreRead less
Mathematics the key to modern glass and polymer fibre technology. This project aims to develop fully coupled flow and energy models to determine the preform structure and fibre-drawing parameters needed to fabricate a desired microstructured optical fibre by stretching of the preform to a fibre. It will focus on polymer to develop a non-Newtonian flow model, which can handle the subset of Newtonian glass fibre drawing. It will develop fast, powerful three-dimensional predictive tools to solve th ....Mathematics the key to modern glass and polymer fibre technology. This project aims to develop fully coupled flow and energy models to determine the preform structure and fibre-drawing parameters needed to fabricate a desired microstructured optical fibre by stretching of the preform to a fibre. It will focus on polymer to develop a non-Newtonian flow model, which can handle the subset of Newtonian glass fibre drawing. It will develop fast, powerful three-dimensional predictive tools to solve the models and experimentally validate solutions. This work will direct future design of microstructured optical fibres to empower next-generation optical-fibre technologies. Expected outcomes are fibre designs for telecommunications, medicine, biotechnology, sensing and imaging.Read moreRead less
Symmetry and geometric partial differential equations. This project aims to develop tools to assist the study of partial differential equations, which are fundamental to our understanding of the physical world. Symmetries of the Laplace equation are fundamental in both finding and interpreting its solutions and can be traced to the conformal symmetries of the underlying space. Only for the most symmetric of spaces, Euclidean space and the sphere, is this correspondence well understood. Using pow ....Symmetry and geometric partial differential equations. This project aims to develop tools to assist the study of partial differential equations, which are fundamental to our understanding of the physical world. Symmetries of the Laplace equation are fundamental in both finding and interpreting its solutions and can be traced to the conformal symmetries of the underlying space. Only for the most symmetric of spaces, Euclidean space and the sphere, is this correspondence well understood. Using powerful geometric tools from conformal geometry, the project will extend this to less symmetric spaces. The knowledge generated from this project will extend to more general geometric contexts providing a concrete setting for the study of the associated natural equations in curved spaces.Read moreRead less