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
Ceramic matrix nanocomposites. Using a novel process developed by the applicant, this project will create and study ceramic matrix nanocomposites of two types: (i) those in which the nanoparticles are homogeneously distributed in alumina and (ii) functionally-graded nanocomposites of controlled heterogeneity, that is, nanocomposites in which the nanoparticles are distributed heterogeneously in glass. Homogeneous nanocomposites of alumina are potentially of great importance to the mining industry ....Ceramic matrix nanocomposites. Using a novel process developed by the applicant, this project will create and study ceramic matrix nanocomposites of two types: (i) those in which the nanoparticles are homogeneously distributed in alumina and (ii) functionally-graded nanocomposites of controlled heterogeneity, that is, nanocomposites in which the nanoparticles are distributed heterogeneously in glass. Homogeneous nanocomposites of alumina are potentially of great importance to the mining industry as they can increase the toughness and wear resistance of mining components. Heterogeneous nanocomposities have the potential to revolutionise the dental restoration industry by combining greatly increased toughness with the aesthetic benefit of controllable translucency.Read moreRead less
Microwave Antennas based on Metamaterials. This project concerns one of the most exciting and dynamic areas of research at present. Metamaterials have tremendous potential, with the promise of multitudinous applications in microwave, optical and optoelectronic fields. This project will contribute towards the ARC priority goal on advanced materials and frontier technologies by (a) developing new synthesized materials which have special properties not found in nature, and (b) developing new techn ....Microwave Antennas based on Metamaterials. This project concerns one of the most exciting and dynamic areas of research at present. Metamaterials have tremendous potential, with the promise of multitudinous applications in microwave, optical and optoelectronic fields. This project will contribute towards the ARC priority goal on advanced materials and frontier technologies by (a) developing new synthesized materials which have special properties not found in nature, and (b) developing new technologies to deliver practical benefits for communication systems users by exploiting these materials. Other benefits for Australia include intellectual property and patent outcomes, which may help Australia to become a leader in metamaterial-based technologies.Read moreRead less
Generalised conformal mappings. A conformal mapping preserves shape, at least at very small scale, circles are mapped to circles. The more recently introduced quasi-conformal mappings nearly preserves shape, at least at a very small scale, circles are mapped to regions which are similar to circles. This project will allow different directions to be scaled differently, and will consider mappings that send circles to ellipses of arbitrary eccentricity. The theory to be developed is mathematical an ....Generalised conformal mappings. A conformal mapping preserves shape, at least at very small scale, circles are mapped to circles. The more recently introduced quasi-conformal mappings nearly preserves shape, at least at a very small scale, circles are mapped to regions which are similar to circles. This project will allow different directions to be scaled differently, and will consider mappings that send circles to ellipses of arbitrary eccentricity. The theory to be developed is mathematical and it will provide a unified approach to important results in several areas, including Lie groups and functions of several complex variables. 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.
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
Development of Deformation-Mechanism Based Parameters for Improved Design of Hard Coatings. The use of thin hard abrasion-resistant coatings is an important method for significantly improving the operational lifetime of components in a wide range of mechanical, biomedical and sensory applications. The optimal design of these coatings is however severely restricted by a lack of detailed knowledge of their material deformation mechanisms. The proposed project will use novel nano-indentation and el ....Development of Deformation-Mechanism Based Parameters for Improved Design of Hard Coatings. The use of thin hard abrasion-resistant coatings is an important method for significantly improving the operational lifetime of components in a wide range of mechanical, biomedical and sensory applications. The optimal design of these coatings is however severely restricted by a lack of detailed knowledge of their material deformation mechanisms. The proposed project will use novel nano-indentation and electron microscope techniques to create a basis for mechanism-based deformation models. These models will then be used to develop new coating architectures with improved operational lifetimes as well as predicting coating lifetimes and developing simple tools for coating assessment.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
Fibre-Polymer Composites With Graded Microstructure. This project concerns the development of a novel fibre-polymer functionally graded composite material that mimics the structure of the spinal disk. The spinal disk comprises a soft core (nucleus) and a tough fibrous periphery (annulus). The graded composite approach of this proposal represents a significant advance over spinal fusion, and over the current generation of spinal disk protheses (metal plates with springs/polymers as shock absorber ....Fibre-Polymer Composites With Graded Microstructure. This project concerns the development of a novel fibre-polymer functionally graded composite material that mimics the structure of the spinal disk. The spinal disk comprises a soft core (nucleus) and a tough fibrous periphery (annulus). The graded composite approach of this proposal represents a significant advance over spinal fusion, and over the current generation of spinal disk protheses (metal plates with springs/polymers as shock absorbers). Chronic back pain affects a significant proportion of the population and is primarily caused by failure of the spinal disk. Therefore a strong social and economic imperative exists to develop engineering solutions to this problem.Read moreRead less