Discovery Early Career Researcher Award - Grant ID: DE130101191
Funder
Australian Research Council
Funding Amount
$375,000.00
Summary
Formation of the osteocyte network in bone matrix. The formation of new bone, which occurs throughout life for bone renewal and acutely after fractures, entraps a network of cells that can detect micro-damage and direct repair mechanisms. Mathematical and computational methods will be used to understand how this network can lead to a self-detecting and self-repairing biomaterial.
Discovery Early Career Researcher Award - Grant ID: DE180100957
Funder
Australian Research Council
Funding Amount
$339,328.00
Summary
Partial differential equations, free boundaries and applications. This project aims to investigate fundamental problems in the analysis of partial differential equations and free boundary theory, to develop advanced mathematical theories with the possibility of important applications. The expected outcome is the establishment of a regularity and classification theory for nonlocal equations and for free boundary problems in linear and nonlinear settings. The benefit of the project lies in a concr ....Partial differential equations, free boundaries and applications. This project aims to investigate fundamental problems in the analysis of partial differential equations and free boundary theory, to develop advanced mathematical theories with the possibility of important applications. The expected outcome is the establishment of a regularity and classification theory for nonlocal equations and for free boundary problems in linear and nonlinear settings. The benefit of the project lies in a concrete advancement of the mathematical research with advantages for a deeper understanding of complex phenomena in physics and biology. Some of the problems also provide results useful for industrial applications.Read moreRead less
Symmetrical graphs, generalized polygons and expanders. This project proposes to study a class of highly symmetrical graphs -- locally s-arc-transitive graphs. Studying the class of graphs has been one of the central topics in algebraic graph theory for over 50 years. This class of graphs has been effectively used in computer science, communication network, group theory, geometry, and other areas. This project will develop new methods to solve several fundamental problems regarding locally s-arc ....Symmetrical graphs, generalized polygons and expanders. This project proposes to study a class of highly symmetrical graphs -- locally s-arc-transitive graphs. Studying the class of graphs has been one of the central topics in algebraic graph theory for over 50 years. This class of graphs has been effectively used in computer science, communication network, group theory, geometry, and other areas. This project will develop new methods to solve several fundamental problems regarding locally s-arc-transitive graphs, and apply the outcomes to solve important problems in communication networks, graph theory, group theory, and geometry.Read moreRead less
Numerical Algorithms for Constructing Feedback Control Laws. Many decision making problems in engineering, finance and management are governed by optimal feedback control systems. These systems are normally too complex to be solved by conventional numerical methods. In this project, we propose to develop novel numerical algorithms for constructing feedback control laws. We will also investigate the procatical significance of these algorithms for solving real-world problems. The outcome of the pr ....Numerical Algorithms for Constructing Feedback Control Laws. Many decision making problems in engineering, finance and management are governed by optimal feedback control systems. These systems are normally too complex to be solved by conventional numerical methods. In this project, we propose to develop novel numerical algorithms for constructing feedback control laws. We will also investigate the procatical significance of these algorithms for solving real-world problems. The outcome of the project will provide efficient and accurate tools for constructing feedback laws in high dimensions.Read moreRead less
Exact structure in graphs and matroids. One of the main goals of mathematics is to understand and describe the structure of the mathematical world. This project will contribute to this goal, and deepen our understanding of the fundamental mathematical structures called graphs and matroids, by providing exact structural descriptions of a number of important minor-closed classes.
Real chromatic roots of graphs and matroids. This project will develop the theory of real chromatic roots of graphs, especially as it applies to minor-closed classes of graphs, with the aim of extending this theory to minor-closed classes of matroids. One of the fundamental results from the study of real chromatic roots in graph theory is that any minor-closed class of graphs has an absolute upper bound on its chromatic roots. However, while many results on minor-closed classes on graphs have cl ....Real chromatic roots of graphs and matroids. This project will develop the theory of real chromatic roots of graphs, especially as it applies to minor-closed classes of graphs, with the aim of extending this theory to minor-closed classes of matroids. One of the fundamental results from the study of real chromatic roots in graph theory is that any minor-closed class of graphs has an absolute upper bound on its chromatic roots. However, while many results on minor-closed classes on graphs have close analogues or mild variants for minor-closed classes of matroids, this upper bound on real chromatic roots appears, somewhat mysteriously, to apply only to graphs. By studying the upper root-free intervals of minor-closed classes of matroids, this project aims to shed light on this phenomenon.Read moreRead less
Complex dynamical systems: inferring form and function of interacting biological systems. Often in biology a large number of simple parts interacting according to simple rules can result in behaviour that is rich and varied. This project aims to develop the mathematics of complex systems theory to describe how such collections of simple interacting parts can form large complicated structures, and to deduce what dynamical behaviour can result.
Symmetries of finite digraphs. Highly symmetrical graphs are well-studied and, in many respects, the theory for dealing with them is well-established. By comparison, our understanding of symmetrical digraphs is much poorer. There are some rather basic questions about these about which we know shamefully little. The aim of this project is to remedy this shortage of knowledge by extending many important results and theories about symmetrical graphs to digraphs.
Finite linearly representable geometries and symmetry. Finite geometry has profound mathematical connections to the theory of symmetry. Advances in finite geometry and in symmetry have historically led to advances in diverse areas such as algebra, computing, and theoretical physics. The project aims to characterise basic geometric objects called "projective planes'' and "generalised polygons'' using their symmetry properties. To achieve these aims, conceptual links between certain elements in cl ....Finite linearly representable geometries and symmetry. Finite geometry has profound mathematical connections to the theory of symmetry. Advances in finite geometry and in symmetry have historically led to advances in diverse areas such as algebra, computing, and theoretical physics. The project aims to characterise basic geometric objects called "projective planes'' and "generalised polygons'' using their symmetry properties. To achieve these aims, conceptual links between certain elements in classical symmetry groups and geometric planes and polygons must be developed. The density of these certain elements has important applications to probabilistic geometric algorithms.Read moreRead less
HYBRID METHODS FOR SOLVING LARGE-SCALE OPTIMISATION PROBLEMS. Mathematical modelling and optimisation plays a crucial role in the advancement of modern business, science and technology. A significant benefit of this project is the development of a range of powerful computational tools for improving the productivity of Australian industry, including: agriculture; communications; defence; manufacturing; mining and petroleum; transport and logistics. These tools will be built upon advances in the f ....HYBRID METHODS FOR SOLVING LARGE-SCALE OPTIMISATION PROBLEMS. Mathematical modelling and optimisation plays a crucial role in the advancement of modern business, science and technology. A significant benefit of this project is the development of a range of powerful computational tools for improving the productivity of Australian industry, including: agriculture; communications; defence; manufacturing; mining and petroleum; transport and logistics. These tools will be built upon advances in the fundamental theory developed by the research team. The resulting high quality publications and associated algorithms will greatly enhance Australia's international scientific reputation and provide Australian industry with new cutting-edge optimisation technology.Read moreRead less