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
The Sakai scheme-Askey table correspondence, analogues of isomonodromy and determinantal point processes. The Australian mathematical sciences enjoys two research groups with active interests on Painleve equations in applied mathematics which are able to address difficult problems. Such a problem is to give a formulation of Sakai's 2001 classification of the Painleve equations in a form most suitable for applications. For this links will be made with a seemingly distinct area of mathematics - t ....The Sakai scheme-Askey table correspondence, analogues of isomonodromy and determinantal point processes. The Australian mathematical sciences enjoys two research groups with active interests on Painleve equations in applied mathematics which are able to address difficult problems. Such a problem is to give a formulation of Sakai's 2001 classification of the Painleve equations in a form most suitable for applications. For this links will be made with a seemingly distinct area of mathematics - the Askey table from the theory of hypergeometric orthogonal polynomials. A number of tractable PhD projects are suggested by this proposal.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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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
Proper Group Actions in Complex Geometry. The results of the project will enhance Australia's performance in several key mathematical areas as well as mathematical applications to physics critical for expanding Australia's knowledge base and research capability. The project has strong international aspects, it will foster the international competitiveness of Australian research and establish long-term collaborations between Australian researchers and high profile world experts in the area of the ....Proper Group Actions in Complex Geometry. The results of the project will enhance Australia's performance in several key mathematical areas as well as mathematical applications to physics critical for expanding Australia's knowledge base and research capability. The project has strong international aspects, it will foster the international competitiveness of Australian research and establish long-term collaborations between Australian researchers and high profile world experts in the area of the proposal. It will create an opportunity for a Ph.D. graduate to be involved in top-class research as a Research Associate, and will attract Ph.D. and honours students thus enabling research training in a high-quality mathematical environment.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
ARC Centre in Bioinformatics. The Australian Centre for Genome-Phenome Bioinformatics will examine how the genome comes to life in the mammalian cell during differentiation and development. We will model, visualise and experimentally validate the complex cellular systems and regulatory networks that control the transformation of genomic information into biological structure and function. We will develop novel approaches and tools to improve health, optimise agricultural production and exploit ne ....ARC Centre in Bioinformatics. The Australian Centre for Genome-Phenome Bioinformatics will examine how the genome comes to life in the mammalian cell during differentiation and development. We will model, visualise and experimentally validate the complex cellular systems and regulatory networks that control the transformation of genomic information into biological structure and function. We will develop novel approaches and tools to improve health, optimise agricultural production and exploit new cell technologies. The Centre will build critical mass and national focus in bioinformatics to generate the human capital and intellectual property that Australia needs to compete in advanced bioscience and biotechnology.Read moreRead less
Statistical methods and tools for integrative microarray analysis. Tools used for biological and medical research have been evolving and there has been an increase in high-throughput technologies such as genome sequencing and DNA microarray. The growing number of entries and the increasing availability of public microarray repositories and other sequence databases have generated the new challenge of developing tools to efficiently integrate data by different research groups. This research provi ....Statistical methods and tools for integrative microarray analysis. Tools used for biological and medical research have been evolving and there has been an increase in high-throughput technologies such as genome sequencing and DNA microarray. The growing number of entries and the increasing availability of public microarray repositories and other sequence databases have generated the new challenge of developing tools to efficiently integrate data by different research groups. This research provides new statistical methods to integrate different data sets. Its application in the biomedical field will allow researchers to effectively interpret the myriad of data generated within the community.Read moreRead less
Classification of Microarray Gene-Expression Data. The broad aim is to provide statistical methodology for the classification of microarray gene-expression data. Microarrays are part of a new biotechnology that allows the monitoring of expression levels for thousands of genes simultaneously. The explosion in microarrays has produced massive quantities of data that require new statistical techniques for analysis in order to exploit their enormous scientific potential. One of the main uses of ....Classification of Microarray Gene-Expression Data. The broad aim is to provide statistical methodology for the classification of microarray gene-expression data. Microarrays are part of a new biotechnology that allows the monitoring of expression levels for thousands of genes simultaneously. The explosion in microarrays has produced massive quantities of data that require new statistical techniques for analysis in order to exploit their enormous scientific potential. One of the main uses of the methodology to be developed is to expedite the discovery of new subclasses of diseases. Another is to provide prediction rules for the diagnosis and treatment of diseases.Read moreRead less
Statistical Methods for Discovering Ribonucleic acids (RNAs) contributing to human diseases and phenotypes. Identifying the causative genetic factors involved in quantitative phenotypes and diseases is a major goal of biology in the 21st century and beyond. A crucial step towards this goal is identifying and classifying the functional non-protein-coding Ribonucleic acids (RNAs) encoded in the human genome. This project will make major contributions to international efforts in this area by identi ....Statistical Methods for Discovering Ribonucleic acids (RNAs) contributing to human diseases and phenotypes. Identifying the causative genetic factors involved in quantitative phenotypes and diseases is a major goal of biology in the 21st century and beyond. A crucial step towards this goal is identifying and classifying the functional non-protein-coding Ribonucleic acids (RNAs) encoded in the human genome. This project will make major contributions to international efforts in this area by identifying RNA molecules that contribute to quantitative phenotypes including susceptibility to disease. As such, it will directly benefit fundamental science via the discovery and classification of new molecules. Indirectly, it will lead to breakthroughs in biology, and consequently to major medical and pharmaceutical advances in the diagnosis and treatment of genetic disease.Read moreRead less