Geometric evolution problems in nonlinear partial differential equations. This project aims to address important problems key to the understanding of geometric evolution equations and certain other nonlinear partial differential equations. The problems to be tackled lie in a very active area of mathematics: harmonic maps, liquid crystals and Yang-Mills theory. Special aims are to exploit new methods to settle open problems in harmonic maps and Yang-Mills equations, and to improve understanding o ....Geometric evolution problems in nonlinear partial differential equations. This project aims to address important problems key to the understanding of geometric evolution equations and certain other nonlinear partial differential equations. The problems to be tackled lie in a very active area of mathematics: harmonic maps, liquid crystals and Yang-Mills theory. Special aims are to exploit new methods to settle open problems in harmonic maps and Yang-Mills equations, and to improve understanding of practical questions such as the mathematical modelling of liquid crystals via the celebrated Ericksen-Leslie and Landau-de Gennes theories. The expected outcomes are fundamental results in mathematics, with applications in other sciences.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE220100919
Funder
Australian Research Council
Funding Amount
$345,000.00
Summary
Distinguished Geometric Structures with Symmetry in Four Dimensions. The Ricci flow is a geometric evolution equation having significant applications in geometry, topology, as well as in physics, biology and image processing. This project aims to provide a complete description and classification of highly symmetric, self-similar solutions to the Ricci Flow in four dimensions. Such a classification is essential to understanding the behaviour of the flow, but has so far evaded discovery. This proj ....Distinguished Geometric Structures with Symmetry in Four Dimensions. The Ricci flow is a geometric evolution equation having significant applications in geometry, topology, as well as in physics, biology and image processing. This project aims to provide a complete description and classification of highly symmetric, self-similar solutions to the Ricci Flow in four dimensions. Such a classification is essential to understanding the behaviour of the flow, but has so far evaded discovery. This project intends to combine techniques from pure mathematics with computational techniques to complete this classification. Such an outcome would greatly improve the understanding of the geometry of four-dimensional manifolds, potentially leading to applications in several areas of science as well as image processing.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
A Novel Geometric Approach to Shocks in Reaction-Nonlinear Diffusion Models. Reaction-nonlinear diffusion models play a vital role in the study of cell migration and population dynamics. However, the presence of aggregation, or backward diffusion, leads to the formation of shock waves - distinct, sharp interfaces between different populations of densities of cells - and the breakdown of the model. This project will develop new geometric methods to explain the formation and temporal evolution of ....A Novel Geometric Approach to Shocks in Reaction-Nonlinear Diffusion Models. Reaction-nonlinear diffusion models play a vital role in the study of cell migration and population dynamics. However, the presence of aggregation, or backward diffusion, leads to the formation of shock waves - distinct, sharp interfaces between different populations of densities of cells - and the breakdown of the model. This project will develop new geometric methods to explain the formation and temporal evolution of these shock waves, while simultaneously unifying existing regularisation techniques under a single, geometric banner. It will devise innovative tools in singular perturbation theory and stability analysis that will identify key parameters in the creation of shock waves, as well as their dynamic behaviour.Read moreRead less
A geometric theory for travelling waves in advection-reaction-diffusion models. Cell migration patterns often develop distinct sharp interfaces between identifiably different cell populations within a tissue. This research will develop new geometric methods for the mathematical analysis of cell migration models, and will design diagnostic tools to identify key parameters that cause and control these patterns and interfaces.
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