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
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.
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
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.
Propagation described by partial differential equations with free boundary. Cutting edge nonlinear mathematics is required to understand many important propagation phenomena in nature, such as the spreading of invasive species or nerve signals. This project aims to systematically investigate nonlinear partial differential equation models that govern the dynamics of such propagations, with emphasis on the development of new approaches that enable deeper insights on the evolution of the propagatin ....Propagation described by partial differential equations with free boundary. Cutting edge nonlinear mathematics is required to understand many important propagation phenomena in nature, such as the spreading of invasive species or nerve signals. This project aims to systematically investigate nonlinear partial differential equation models that govern the dynamics of such propagations, with emphasis on the development of new approaches that enable deeper insights on the evolution of the propagating fronts. The project aims to develop new mathematics for new applications of lasting values.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE170101410
Funder
Australian Research Council
Funding Amount
$330,324.00
Summary
Nonlinear free boundary problems: Propagation and regularity. This project aims to understand the propagation profile and regularity of two important classes of free boundary problems. Nonlinear free boundary problems arise from many applied fields, and pose great challenges to the theory of nonlinear partial differential equations, as the underlying domain of the solution to such problems has to be solved together with the solution itself. This research is expected to enhance the existing theor ....Nonlinear free boundary problems: Propagation and regularity. This project aims to understand the propagation profile and regularity of two important classes of free boundary problems. Nonlinear free boundary problems arise from many applied fields, and pose great challenges to the theory of nonlinear partial differential equations, as the underlying domain of the solution to such problems has to be solved together with the solution itself. This research is expected to enhance the existing theory of partial differential equations, and extend its applications to new situations such as flow through porous media and spreading of invasive species.Read moreRead less
Propagation and free boundary problems in nonlinear partial differential equations. Understanding the propagation of invasive species, flames and disadvantageous genes is a challenging problem in many areas of modern science. This project develops a new mathematical approach to better understand such propagation problems, where the mathematical model predicts a precise location of the propagating front for future time.