Hypergraph models for complex discrete systems. This project aims to better understand the structure and properties of very large hypergraphs of various kinds. Hypergraphs are very general mathematical objects which can be used to model complex discrete systems. They arise naturally in many areas such as ecology, chemistry and computer science. Despite this, our theoretical understanding of very large, or random, hypergraphs lags far behind the intensely-studied special case of graphs. This proj ....Hypergraph models for complex discrete systems. This project aims to better understand the structure and properties of very large hypergraphs of various kinds. Hypergraphs are very general mathematical objects which can be used to model complex discrete systems. They arise naturally in many areas such as ecology, chemistry and computer science. Despite this, our theoretical understanding of very large, or random, hypergraphs lags far behind the intensely-studied special case of graphs. This project will answer many fundamental questions about large, random hypergraphs. The expected outcomes of the project also include new tools for working with hypergraphs, such as efficient algorithms for sampling hypergraphs. These outcomes will benefit researchers who use hypergraphs in their work and will enhance Australia's reputation for research in this area.Read moreRead less
A new model for random discrete structures: distributions, counting and sampling. Random discrete structures are used in countless applications across science for modelling complex systems. This project will study a new, very general model of random discrete structures which encapsulates both random networks and random matrices. This project will develop general tools for working with this model, thereby unlocking the model for use by practitioners in areas such as physics, biology, statistics a ....A new model for random discrete structures: distributions, counting and sampling. Random discrete structures are used in countless applications across science for modelling complex systems. This project will study a new, very general model of random discrete structures which encapsulates both random networks and random matrices. This project will develop general tools for working with this model, thereby unlocking the model for use by practitioners in areas such as physics, biology, statistics and cryptography. The questions that will be tackled are fundamental problems in probability, and include as special cases the analysis of subgraph distribution in models of random networks, and the joint distribution of entries of contingency tables, which are important in statistics.Read moreRead less
A new approach to compressed sensing. Compressed sensing is an exciting new paradigm promising vastly improved signal sampling and reconstruction in a wide variety of applications including digital cameras, mobile phones and MRI machines. This project will explore a newly discovered approach to compressed sensing which uses mathematical arrays known as hash families.
Novel dissimilarity techniques for characterising noisy spatial networks. This project will invent new and widely applicable ways of summarising fundamental characteristics of noisy spatial networks that change slightly in space or time. The techniques developed will be applied to solve important problems in two diverse applications - predicting disease spread in wildlife and protecting human biometric information.
Assuring dependability of complex adaptive multi-agent systems using time bands. As the complexity of computer-based systems rapidly increases, we need new methods for assuring their correct behaviour. This project will provide a means of relating behaviour at different timescales, enabling us to understand how the long-term behaviour of a system results from the short-term interactions between its components.
Relaxed correctness criteria for modern multi-core architectures. This project seeks to lay groundwork for fully exploiting the potential of multicore computers. Multicore computers have become ubiquitous over the last decade, now being standard in everything from laptops to mobile phones. Their benefits are clear – better performance leading to more sophisticated applications. Key to ensuring those benefits are complex, and often subtle, algorithms that exploit the parallelism that multicore co ....Relaxed correctness criteria for modern multi-core architectures. This project seeks to lay groundwork for fully exploiting the potential of multicore computers. Multicore computers have become ubiquitous over the last decade, now being standard in everything from laptops to mobile phones. Their benefits are clear – better performance leading to more sophisticated applications. Key to ensuring those benefits are complex, and often subtle, algorithms that exploit the parallelism that multicore computers offer. This project aims to lay foundations for extending those benefits to applications where high reliability is a concern. It plans to do so by developing theoretical results about the correctness of algorithms on standard multicore computers, and practical tools and techniques to help programmers of multicore computers to better understand the behaviour of their code.Read moreRead less
Analysis of the structure of latin squares. The benefits to Australia of fundamental research in core disciplines such as mathematics are well documented. Discrete mathematics and combinatorics are boom disciplines of the computer age and this project seeks new knowledge concerning basic building blocks of combinatorial mathematics. The outcomes will be of interest to theoretical discrete mathematicians around the world, enhancing Australia's already high research profile in this important area ....Analysis of the structure of latin squares. The benefits to Australia of fundamental research in core disciplines such as mathematics are well documented. Discrete mathematics and combinatorics are boom disciplines of the computer age and this project seeks new knowledge concerning basic building blocks of combinatorial mathematics. The outcomes will be of interest to theoretical discrete mathematicians around the world, enhancing Australia's already high research profile in this important area of pure mathematical research. Importantly, the problems under investigation offer substantial opportunity for excellent postgraduate training, critical for the future of Australian research. Read moreRead less
Cycle decompositions of graphs. The benefits to Australia of fundamental research in core disciplines such as mathematics are well documented. This project aims to solve long-standing and significant open problems in the field of mathematics known as graph theory. Solving such problems will undoubtedly bring Australian research in this field to the fore, and help to enhance Australia's international research profile generally. The project offers substantial postgraduate training in the form of t ....Cycle decompositions of graphs. The benefits to Australia of fundamental research in core disciplines such as mathematics are well documented. This project aims to solve long-standing and significant open problems in the field of mathematics known as graph theory. Solving such problems will undoubtedly bring Australian research in this field to the fore, and help to enhance Australia's international research profile generally. The project offers substantial postgraduate training in the form of three excellent PhD projects in discrete mathematics. The computer age has ensured that this is a booming discipline and an increasing component of undergraduate syllabi around the world. It is thus a crucial area in which to be providing quality research training.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
Matchings in Combinatorial Structures. The theory of matching in graphs concerns the problem of pairing up objects, subject to constraints on which objects may be paired. It is a well-developed theory that is not only of tremendous mathematical importance, but is also widely applied to efficiently deal with allocation and scheduling problems. Much less is known, however, about the equally important but harder problem of dividing objects into collections of three or more. This project aims to add ....Matchings in Combinatorial Structures. The theory of matching in graphs concerns the problem of pairing up objects, subject to constraints on which objects may be paired. It is a well-developed theory that is not only of tremendous mathematical importance, but is also widely applied to efficiently deal with allocation and scheduling problems. Much less is known, however, about the equally important but harder problem of dividing objects into collections of three or more. This project aims to address this deficiency by developing the theory of matching in important combinatorial objects. The problems it expects to solve are of great significance in their own right, and when considered together may help to lay a foundation for a more general theory of matching.Read moreRead less