Discovery Early Career Researcher Award - Grant ID: DE140100223
Funder
Australian Research Council
Funding Amount
$385,735.00
Summary
Diophantine approximation, transcendence, and related structures. Sequences produced by low-complexity structures are objects of importance to mathematics, linguistics and theoretical computer science. In the 1960s, Chomsky and Schützenberger formalised and popularised a hierarchy of such objects. In the 1920s, Mahler provided a corresponding analytic framework, which has proven extremely useful for analysing the algebraic character of low-complexity real numbers. This project will further devel ....Diophantine approximation, transcendence, and related structures. Sequences produced by low-complexity structures are objects of importance to mathematics, linguistics and theoretical computer science. In the 1960s, Chomsky and Schützenberger formalised and popularised a hierarchy of such objects. In the 1920s, Mahler provided a corresponding analytic framework, which has proven extremely useful for analysing the algebraic character of low-complexity real numbers. This project will further develop Mahler's method in order to investigate the connection between the algebraic and arithmetic properties of real numbers and the various Chomskian complexity measures of those numbers. The results of this proposal will advance our knowledge of the nature of "randomness" in low-complexity arithmetic sequences.Read moreRead less
Normal forms and Chern-Moser connection in the study of Cauchy-Riemann Manifolds. This research project is aimed at a systematic study of Cauchy-Riemann manifolds, their holomorphic mappings and automorphisms, by means of a unifying approach based on
Chern-Moser type normal forms. The importance of Cauchy-Riemann manifolds stems from the fact that they bridge complex structure and holomorphy with the Riemannian nature of real manifolds. Construction of an analogue of the Chern-Moser normal form ....Normal forms and Chern-Moser connection in the study of Cauchy-Riemann Manifolds. This research project is aimed at a systematic study of Cauchy-Riemann manifolds, their holomorphic mappings and automorphisms, by means of a unifying approach based on
Chern-Moser type normal forms. The importance of Cauchy-Riemann manifolds stems from the fact that they bridge complex structure and holomorphy with the Riemannian nature of real manifolds. Construction of an analogue of the Chern-Moser normal form for multicodimensional Levi-nondegenerate CR-manifolds and extension of CR-mappings between them are major goals in complex analysis. Identification of Chern-Moser chains and equivariant linearisation of isotropy automorphisms are major goals in geometry.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE160100173
Funder
Australian Research Council
Funding Amount
$315,000.00
Summary
Partial Differential Equations in Several Complex Variables. This project aims to make advances in partial differential equations (PDEs) in several complex variables. PDEs in several complex variables are important in modern analysis and geometry, especially harmonic analysis, operator theory, geometric analysis and PDE with rough coefficients. The project aims to study the relationship between geometric curvature conditions and regularity properties of the solutions of complex partial different ....Partial Differential Equations in Several Complex Variables. This project aims to make advances in partial differential equations (PDEs) in several complex variables. PDEs in several complex variables are important in modern analysis and geometry, especially harmonic analysis, operator theory, geometric analysis and PDE with rough coefficients. The project aims to study the relationship between geometric curvature conditions and regularity properties of the solutions of complex partial differential equations: specifically the D-bar-Neumann problem, linear operators associated to pseudoconvex domains, and the complex Monge-Ampere equation. These areas find applications in the physical sciences and mathematical finance.Read moreRead less
The canonical stratification of jet spaces. Singularities occur everywhere in nature, from the formation and collapse of stars to the morphology of living embryos. They appear whenever the geometry of surfaces or spaces undergoes a process of twisting, folding, or collapsing on itself. Singularity Theory is the study of such phenomena, an important branch of modern mathematics which has close connections with many other branches of mathematics and applied sciences. Singularity Theory lies at the ....The canonical stratification of jet spaces. Singularities occur everywhere in nature, from the formation and collapse of stars to the morphology of living embryos. They appear whenever the geometry of surfaces or spaces undergoes a process of twisting, folding, or collapsing on itself. Singularity Theory is the study of such phenomena, an important branch of modern mathematics which has close connections with many other branches of mathematics and applied sciences. Singularity Theory lies at the crossroads of the paths connecting the most important areas of applications of mathematics with its most abstract parts. Analytic Singularity Theory is a central part of Singularity Theory. This project would lead to substantially new advancements in Analytic Singularity Theory.Read moreRead less
Low-order dynamical models for non-linear fluid behaviour in quasi two-dimensional plasmas. Two complex systems in which a magnetic field imposes two-dimensional fluid motions are turbulent fusion plasmas and magnetospheric plasmas. A distinctive property of 2D flows is the inverse energy cascade, whereby energy streaming into large-scale vortices, coherent structures, or sheared flows gives a remarkable propensity for self-organizing behaviour. This can be exploited to govern or guide our respo ....Low-order dynamical models for non-linear fluid behaviour in quasi two-dimensional plasmas. Two complex systems in which a magnetic field imposes two-dimensional fluid motions are turbulent fusion plasmas and magnetospheric plasmas. A distinctive property of 2D flows is the inverse energy cascade, whereby energy streaming into large-scale vortices, coherent structures, or sheared flows gives a remarkable propensity for self-organizing behaviour. This can be exploited to govern or guide our response to such systems. We propose to investigate the dynamics of momentum and energy exchange in these plasmas, using reduced dynamical models and bifurcation and stability mathematics. Expected outcomes are improved prediction of magnetospheric substorms and confinement of fusion plasmas.
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Special Research Initiatives - Grant ID: SR0354716
Funder
Australian Research Council
Funding Amount
$10,000.00
Summary
Energetically Open Systems Research Network Study. Conceptual frameworks arising in the physical sciences, such as non-equilibrium statistical mechanics and thermodynamics, synergetics, chaos and dynamical systems theory, are seminal in the emerging science of complexity. This study will lay the groundwork for a network to link Australian and overseas research on these fundamental concepts, and their application within the context of entropy-producing systems vital to the long-term sustainabilit ....Energetically Open Systems Research Network Study. Conceptual frameworks arising in the physical sciences, such as non-equilibrium statistical mechanics and thermodynamics, synergetics, chaos and dynamical systems theory, are seminal in the emerging science of complexity. This study will lay the groundwork for a network to link Australian and overseas research on these fundamental concepts, and their application within the context of entropy-producing systems vital to the long-term sustainability of the earth - oceans, atmosphere, biosphere, CO2-free energy production, space and solar environment. The network would facilitate the development of young investigators and be linked into wider complex systems networks such as the CSIRO Centre for Complex Systems Science.Read moreRead less
Stochastic methods in mathematical geophysical fluid dynamics. We will develop analytical and numerical methods for long-term weather forecasting and climate modelling. The project deals with the mathematical aspects and fundamental mechanisms underpinning numerical
climate forecasting. We will develop new methodology for accurate modelling of the important and dominant slow global processes without explicitly resolving the precise detail of the weather of each day at all scales. Using sophisti ....Stochastic methods in mathematical geophysical fluid dynamics. We will develop analytical and numerical methods for long-term weather forecasting and climate modelling. The project deals with the mathematical aspects and fundamental mechanisms underpinning numerical
climate forecasting. We will develop new methodology for accurate modelling of the important and dominant slow global processes without explicitly resolving the precise detail of the weather of each day at all scales. Using sophisticated mathematics, this project investigates how to parameterize the fast and small processes by using stochastic processes in a controllable and adaptive way.Read moreRead less
Stochastic Methods in Mathematical Geophysical Fluid Dynamics. The project will develop analytical and numerical methods for long-term weather forecasting and climate modelling. The project deals with the mathematical aspects and fundamental mechanisms underpinning numerical climate forecasting. The project will develop new methodology for accurate modelling of the important and dominant slow global processes without explicitly resolving the precise detail of the weather of each day at all scale ....Stochastic Methods in Mathematical Geophysical Fluid Dynamics. The project will develop analytical and numerical methods for long-term weather forecasting and climate modelling. The project deals with the mathematical aspects and fundamental mechanisms underpinning numerical climate forecasting. The project will develop new methodology for accurate modelling of the important and dominant slow global processes without explicitly resolving the precise detail of the weather of each day at all scales. Using sophisticated mathematics, this project investigates how to parameterize the fast and small processes by using stochastic processes in a controllable and adaptive way.Read moreRead less
A Dynamical Systems Approach to Mapping Southern Ocean Circulation Pathways. Climate change can be expected to have complex, long-term consequences for Australia's biodiversity, for our agricultural and marine production systems, and for regional communities. The Southern Ocean is a critical driver of global climate, connecting the three major oceanic basins. Using sophisticated mathematics to analyse state-of-the-art global ocean models, this project will create a detailed picture of hitherto i ....A Dynamical Systems Approach to Mapping Southern Ocean Circulation Pathways. Climate change can be expected to have complex, long-term consequences for Australia's biodiversity, for our agricultural and marine production systems, and for regional communities. The Southern Ocean is a critical driver of global climate, connecting the three major oceanic basins. Using sophisticated mathematics to analyse state-of-the-art global ocean models, this project will create a detailed picture of hitherto invisible Southern Ocean circulation 'pathways'. The newly discovered pathways will greatly enhance our understanding of the Southern Ocean circulation, lead to improved coupled climate models, and increase our ability to predict future climate change and threats to marine populations. Read moreRead less
Symmetry in Differential Geometry. Differential geometry is a major branch of mathematics studying shape by using calculus and differential equations. This is a fundamental research project in this area, especially concerned with the interaction between geometry, differential equations, and symmetry. The mathematical notion of symmetry was already formalised early last century and nowadays lies at the very heart of mathematics and physics. Advances in this area provide essential tools in basic s ....Symmetry in Differential Geometry. Differential geometry is a major branch of mathematics studying shape by using calculus and differential equations. This is a fundamental research project in this area, especially concerned with the interaction between geometry, differential equations, and symmetry. The mathematical notion of symmetry was already formalised early last century and nowadays lies at the very heart of mathematics and physics. Advances in this area provide essential tools in basic science and unexpected technological benefits can easily arise (for example, in medical imaging). Fundamental mathematical research is absolutely necessary if Australia is to maintain a presence on the international scientific stage.
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