Symmetries in CR-geometry. This project aims at investigating symmetries of geometric objects called CR-manifolds. It is expected to open new avenues for understanding such symmetries at the infinitesimal level and lead to ground-breaking results in CR-geometry. Expected outcomes include new methodology, solving long-standing problems, and establishing international research collaborations. The benefits are in enhancing the strength of the research in analysis and geometry performed in Australia ....Symmetries in CR-geometry. This project aims at investigating symmetries of geometric objects called CR-manifolds. It is expected to open new avenues for understanding such symmetries at the infinitesimal level and lead to ground-breaking results in CR-geometry. Expected outcomes include new methodology, solving long-standing problems, and establishing international research collaborations. The benefits are in enhancing the strength of the research in analysis and geometry performed in Australia, in fostering the international competitiveness of Australian research and in high-quality research training.Read moreRead less
Nilpotent associative algebras and spherical hypersurfaces. This project concerns pure basic research in mathematics and is based on an important recently discovered relationship between certain geometric and algebraic objects. In the project, this relationship will be applied in a novel way to solve several significant long-standing problems in the research area of complex geometry.
The Reconstruction and Recognition Problems for Hypersurface Singularities. This project concerns pure basic research in mathematics. It is centred around a surprising relationship between geometric objects called quasi-homogeneous isolated hypersurface singularities, and algebraic structures described as Artinian Gorenstein algebras. This relationship has not been fully understood despite numerous attempts by internationally based experts to shed light on it. Armed with a novel approach to Arti ....The Reconstruction and Recognition Problems for Hypersurface Singularities. This project concerns pure basic research in mathematics. It is centred around a surprising relationship between geometric objects called quasi-homogeneous isolated hypersurface singularities, and algebraic structures described as Artinian Gorenstein algebras. This relationship has not been fully understood despite numerous attempts by internationally based experts to shed light on it. Armed with a novel approach to Artinian Gorenstein algebras, this project proposes to clarify the nature of this relationship and utilise it for solving related geometric and algebraic problems. In particular, it aims at obtaining a groundbreaking result in the area of classical invariant theory.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
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.
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.
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
Symmetry and geometric structures. This is a fundamental research project in mathematics, especially concerned with the interaction between symmetry, differential equations, and geometry. Based on many classical and recently discovered instances, the aim of the project is to use symmetries to build and understand curved geometric structures from their flat counterparts.
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