New mathematics for understanding complex patterns in the natural sciences. This project aims to examine the interaction of fundamental two-dimensional patterns such as spots and stripes in reaction-diffusion equations, by developing and extending mathematical techniques. These fundamental planar structures form the backbone of more complex patterns and are, for example, observed in models that describe the propagation of impulses in nerve axons and the formation of vegetation patterns. The futu ....New mathematics for understanding complex patterns in the natural sciences. This project aims to examine the interaction of fundamental two-dimensional patterns such as spots and stripes in reaction-diffusion equations, by developing and extending mathematical techniques. These fundamental planar structures form the backbone of more complex patterns and are, for example, observed in models that describe the propagation of impulses in nerve axons and the formation of vegetation patterns. The future impact of this research will have economic and environmental benefits. For example, the project will develop a deeper understanding of interacting patterns that will provide insights into the role of vegetation in ecosystems that are undergoing desertification.Read moreRead less
The mathematics of novel magnetic memory materials. Magnetic memories are the world’s principal device for storing information. The next generation will have greatly increased access speed and data-storage capacity. This project will develop the mathematical theory of these new magnetic memory materials, a crucial first step in understanding and being able to fine-tune their properties.
Curvature flows and spectral estimates. Curvature flows are a class of geometrically motivated equations, modelled on the heat equation. Recently, researchers have developed new methods for studying the regularity of solutions to these equations, and applied them to a different problem, that of estimating quantities depending on the smaller eigenvalues of a Schroedinger operator. This project builds on the early success of this research and will produce a new understanding of the behaviour of ei ....Curvature flows and spectral estimates. Curvature flows are a class of geometrically motivated equations, modelled on the heat equation. Recently, researchers have developed new methods for studying the regularity of solutions to these equations, and applied them to a different problem, that of estimating quantities depending on the smaller eigenvalues of a Schroedinger operator. This project builds on the early success of this research and will produce a new understanding of the behaviour of eigenvalues, establish sharp estimates for spectral quantities, particularly on manifolds with curvature bounds, and find optimal conditions under which non-compact solutions to curvature flows are stable.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE140100741
Funder
Australian Research Council
Funding Amount
$389,564.00
Summary
Analysis of defect driven pattern formation in mathematical models. . Defects, or heterogeneities, are common in nature and technology and therefore in mathematical models. This project will underpin the effects a defect can have on the dynamics of a model, characterise the new patterns created by a heterogeneity and see how the dynamics can be controlled by manipulating the heterogeneity. Moreover, these new insights will be applied to a model for skin cancer, resulting in a more appropriate mo ....Analysis of defect driven pattern formation in mathematical models. . Defects, or heterogeneities, are common in nature and technology and therefore in mathematical models. This project will underpin the effects a defect can have on the dynamics of a model, characterise the new patterns created by a heterogeneity and see how the dynamics can be controlled by manipulating the heterogeneity. Moreover, these new insights will be applied to a model for skin cancer, resulting in a more appropriate model and a mathematically justifiable analysis of a very important scientific problem.Read moreRead less
Variational theory for fully nonlinear elliptic equations. This project aims to develop new methods and techniques to solve challenging mathematical problems in fully nonlinear partial differential equations arising in important applications. The project will develop methods and techniques to study these equations’ regularity and variational properties. This project is expected to establish comprehensive theories and enhance and promote Australian participation and leadership in this area of mat ....Variational theory for fully nonlinear elliptic equations. This project aims to develop new methods and techniques to solve challenging mathematical problems in fully nonlinear partial differential equations arising in important applications. The project will develop methods and techniques to study these equations’ regularity and variational properties. This project is expected to establish comprehensive theories and enhance and promote Australian participation and leadership in this area of mathematics.Read moreRead less
Geometric methods in mathematical physiology. This project will develop new geometric methods for the analysis of multiple-scales models of physiological rhythms and patterns, and will design diagnostic tools to identify key parameters that cause and control these signals. Thus, this project will deliver powerful mathematics for detecting and understanding fundamental issues of physiological systems.
Non-linear partial differential equations: Bubbles, layers and stability. This project aims to investigate non-linear elliptic partial differential equations in well-established models in applied sciences. The treatment of them challenges the existing mathematical theory. This project will enrich and expand the mathematical theory in semi-linear elliptic equations to understand the equations under investigation.
Comprehensive Study of Kahler-Ricci Flows. The intended outcome of the project is to reveal the deep relation between geometry and topology of underlying spaces. Ricci flow has attracted major attention in pure mathematics over the past 30 years, including ground-breaking contributions by Perelman on Ricci flow regarding the famous Poincare and Thurston's Geometrisation Conjectures. The project focuses on the complex version of Ricci flow: Kahler-Ricci flow. The project plans to explore the Kahl ....Comprehensive Study of Kahler-Ricci Flows. The intended outcome of the project is to reveal the deep relation between geometry and topology of underlying spaces. Ricci flow has attracted major attention in pure mathematics over the past 30 years, including ground-breaking contributions by Perelman on Ricci flow regarding the famous Poincare and Thurston's Geometrisation Conjectures. The project focuses on the complex version of Ricci flow: Kahler-Ricci flow. The project plans to explore the Kahler-Ricci flow in the closed and complete non-compact settings and the corresponding versions of Geometric Minimal Model Program; and the Kahler-Ricci flow in the Fano manifold setting and stability conditions.Read moreRead less
Harmonic analysis in rough contexts. Harmonic analysis is a set of mathematical techniques aimed at decomposing complex signals into simple pieces in a way that is reminiscent of the decomposition of sounds into harmonics. It is highly efficient in analysing signals in homogeneous media such as wave propagation through the air that underpins wireless communication technology. However, wave propagation through inhomogeneous media, such as the human body in medical imaging or the Earth in geophysi ....Harmonic analysis in rough contexts. Harmonic analysis is a set of mathematical techniques aimed at decomposing complex signals into simple pieces in a way that is reminiscent of the decomposition of sounds into harmonics. It is highly efficient in analysing signals in homogeneous media such as wave propagation through the air that underpins wireless communication technology. However, wave propagation through inhomogeneous media, such as the human body in medical imaging or the Earth in geophysical imaging, is much harder to model. Phenomena with random components, as considered in finance for instance, are also problematic. This project is an important part of an intense international research effort to develop harmonic analysis in such rough contexts.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE120101167
Funder
Australian Research Council
Funding Amount
$375,000.00
Summary
Canonical metrics on Kahler manifolds and Monge-Ampere equations. This project will introduce new ideas and techniques to study the existence of canonical metrics on Kahler manifolds, which is a fundamental problem in geometry. Advances in this research will have influence on other areas of science such as mechanics, string theory and mathematical physics.