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
Harmonic analysis of Laplacians in curved spaces. Harmonic Analysis is a branch of mathematics which is interrelated to other fields of mathematics like complex analysis, number theory and partial differential equations (pdes) with many applications in engineering and technology. This project aims to solve a number of difficult fundamental problems at the frontier of harmonic analysis in understanding Laplacians in curved spaces. Such Laplacians control the propagation of heat and waves on manif ....Harmonic analysis of Laplacians in curved spaces. Harmonic Analysis is a branch of mathematics which is interrelated to other fields of mathematics like complex analysis, number theory and partial differential equations (pdes) with many applications in engineering and technology. This project aims to solve a number of difficult fundamental problems at the frontier of harmonic analysis in understanding Laplacians in curved spaces. Such Laplacians control the propagation of heat and waves on manifolds and Lie groups, arising in mathematical physics and quantum mechanics. Expected outcomes are the solutions of dispersive equations and the framework of singular integrals in curved spaces; new ideas and techniques in harmonic analysis developed; and training of Australian future mathematicians.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
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 Role Of Melanoma Tumour Antigen P97 (Melanotransferrin) In Melanoma Tumourigenesis.
Funder
National Health and Medical Research Council
Funding Amount
$563,242.00
Summary
The Role of Melanoma Tumour Antigen p97 (Melanotransferrin) in Melanoma Tumourigenesis Melanotransferrin (MTf) is a homologue of the iron transport protein, transferrin, and was one of the first well characterised melanoma tumour antigens. Our published studies have shown that MTf plays an important role in melanoma tumourigenesis in vivo. In this proposal, we will assess if it is associated with melanoma progression in patient samples and examine its role in melanoma growth and metastasis.
The Immunoregulatory Role of the Endogenous Cannabinoid Anandamide. Anandamides are naturally occurring fatty acids that act at the cannabinoid receptor expressed in the brain and periphery. A new proposal by our research group challenges traditional models of the disease process by suggesting that the anandamide system is an important imunoregulatory system that can be targeted by invading pathogens. We propose that disruption to the anandamide system, by bacteria or viruses acting at the rece ....The Immunoregulatory Role of the Endogenous Cannabinoid Anandamide. Anandamides are naturally occurring fatty acids that act at the cannabinoid receptor expressed in the brain and periphery. A new proposal by our research group challenges traditional models of the disease process by suggesting that the anandamide system is an important imunoregulatory system that can be targeted by invading pathogens. We propose that disruption to the anandamide system, by bacteria or viruses acting at the receptor to block immunological responses, contributes to chronic illness states. At this point we have good evidence that anandamides are immunoregulators, however, we have very little knowledge of their precise physiological role. The aim of this research is to begin to characterise the immunoregulatory role of the anandamide system. This project will provide a comprehensive understanding of this endogenous control system, the immunological properties of which have not previously been described. The outcome of this research may lead to the identification of new avenues for the development of pharmaceutical interventions that can target this system.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE130101591
Funder
Australian Research Council
Funding Amount
$375,000.00
Summary
Novel postsynaptic functions of the microtubule-associated protein tau. The protein tau is present in abnormal deposits in brains of individuals with dementia. The main aim of this project is to unravel and understand in detail new roles of tau in neurons and thus shed new light into normal brain function. Understanding these new functions of tau will aid in identifying new ways to treat these debilitating diseases.