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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Classification and Invariants in Complex Differential Geometry. Differential geometry is the study of shape using calculus and differential equations. This is a fundamental research project in this area. Complex differential geometry refers to geometry based on the complex numbers, generally a rich and intriguing setting. Geometries will be distinguished by the construction of suitable invariants, both algebraic and analytic. Classification problems will be solved by these means. Of particular i ....Classification and Invariants in Complex Differential Geometry. Differential geometry is the study of shape using calculus and differential equations. This is a fundamental research project in this area. Complex differential geometry refers to geometry based on the complex numbers, generally a rich and intriguing setting. Geometries will be distinguished by the construction of suitable invariants, both algebraic and analytic. Classification problems will be solved by these means. Of particular interest are geometries with a high degree of symmetry, a critical feature that pervades both mathematics and physics. Twistor theory provides the unifying theme for this project.Read moreRead less
Homotopical structures in algebraic, analytic, and equivariant geometry. This is a project for fundamental research in pure mathematics. It is focused on an emerging subfield of complex geometry concerned with spaces and maps that exhibit exceptional flexibility properties, which often go hand-in-hand with a high degree of symmetry. The project aims to develop the foundations of this new area, solve several open problems, and pursue interconnections with and applications to algebraic geometry, c ....Homotopical structures in algebraic, analytic, and equivariant geometry. This is a project for fundamental research in pure mathematics. It is focused on an emerging subfield of complex geometry concerned with spaces and maps that exhibit exceptional flexibility properties, which often go hand-in-hand with a high degree of symmetry. The project aims to develop the foundations of this new area, solve several open problems, and pursue interconnections with and applications to algebraic geometry, complex analysis, geometric invariant theory, and topology.Read moreRead less
Symmetries in real and complex geometry. This project concerns an important area of abstract modern geometry. The results and techniques of the project will lead to significant progress in this area. It will benefit the national scientific reputation, strengthen the research profile of the home institutions, and provide training to young researchers.
Flexibility and symmetry in complex geometry. Differential equations play a fundamental role in science and technology. The aim of the project is to study important differential equations that arise in geometry, their symmetries, and obstructions to solving them.
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
Behavioural syndromes and social networks in sleepy lizards. Fauna in Australian ecosystems are threatened by habitat fragmentation, changing environments and the spread of exotic pathogens. To manage these threats we need to understand the behavioural flexibility of wildlife populations. This project focuses on how individual behavioural differences can influence social networks and consequently pathogen transmission. It will help to protect our fauna from invasive diseases and contribute to su ....Behavioural syndromes and social networks in sleepy lizards. Fauna in Australian ecosystems are threatened by habitat fragmentation, changing environments and the spread of exotic pathogens. To manage these threats we need to understand the behavioural flexibility of wildlife populations. This project focuses on how individual behavioural differences can influence social networks and consequently pathogen transmission. It will help to protect our fauna from invasive diseases and contribute to sustaining biodiversity. With better knowledge of how diseases spread we can develop more effective controls of those diseases, thereby protecting wildlife species, animal populations and Australian ecosystems. Read moreRead less
Lizard social networks and the spread of parasites. Australian ecosystems are continually threatened by new epidemics of diseases and parasites, some local, others from overseas. Examples include the facial tumours of Tasmanian devils and the fungus that threatens many native frog species. To manage these epidemics effectively, we must understand how they spread through animal populations. This project will help to protect our fauna from invasive diseases. It contributes to sustaining the biodiv ....Lizard social networks and the spread of parasites. Australian ecosystems are continually threatened by new epidemics of diseases and parasites, some local, others from overseas. Examples include the facial tumours of Tasmanian devils and the fungus that threatens many native frog species. To manage these epidemics effectively, we must understand how they spread through animal populations. This project will help to protect our fauna from invasive diseases. It contributes to sustaining the biodiversity of the country. With better knowledge of how diseases of wildlife spread, we can develop more effective control of those diseases thereby protecting wildlife species, animal populations and, ultimately, Australian ecosystems.Read moreRead less
Phylogeography, evolution and taxonomy of humanity's greatest pest, Rattus rattus: Epidemiological, archaeological and conservation implications. This project will characterise a major threat to Australian biosecurity and health, and identify the range of likely disease risks associated with introductions of different 'strains' of black rat. It will provide critical data for management efforts around the world, especially for strategic partners in neighbouring Southeast Asian nations, as well as ....Phylogeography, evolution and taxonomy of humanity's greatest pest, Rattus rattus: Epidemiological, archaeological and conservation implications. This project will characterise a major threat to Australian biosecurity and health, and identify the range of likely disease risks associated with introductions of different 'strains' of black rat. It will provide critical data for management efforts around the world, especially for strategic partners in neighbouring Southeast Asian nations, as well as for conservation efforts within Australia. The data will also provide novel means to track the timing and routes of human prehistoric movements throughout the area. It will establish strategic research collaborations between researchers in zoological, medical, epidemiological, genetics, and conservation fields in a unique multi-disciplinary study.Read moreRead less
New methods for integrating population structure and stochasticity into models of disease dynamics. Epidemics, such as the 2007 equine 'flu outbreak and 2009 swine 'flu pandemic, highlight the need to make informed decisive responses. This project will develop new methods that incorporate two important aspects of disease dynamics---host structure and chance---into mathematical models, and determine their impact in terms of controlling infections.