Permanents, permutations and polynomials. 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 theoreticians around the world, enhancing Australia's already high research profile in this crucial area. Importantly, the project ....Permanents, permutations and polynomials. 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 theoreticians around the world, enhancing Australia's already high research profile in this crucial area. Importantly, the project also offers substantial postgraduate training in mathematics, an area in which Australia has an identified skill shortage.Read moreRead less
The Structure and Geometry of Graphs. Graphs are ubiquitous mathematical structures that model relational information such as information flows, transportation networks, and biochemical pathways. It is often desirable to have a geometric representation of a graph. For example, a programmer will better understand a computer program if the flow of information within the program is represented by a visually appealing drawing. The focus of the project will be the interplay between graph structure th ....The Structure and Geometry of Graphs. Graphs are ubiquitous mathematical structures that model relational information such as information flows, transportation networks, and biochemical pathways. It is often desirable to have a geometric representation of a graph. For example, a programmer will better understand a computer program if the flow of information within the program is represented by a visually appealing drawing. The focus of the project will be the interplay between graph structure theory and geometric properties of graphs. Moreover, the project will have significant applications to other area of mathematics and computer science, including computational complexity, analysis of data structures, and three-dimensional information visualisation.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
Generalised topological spaces. Pure mathematics creates abstractions of real-world entities; one such is the idea of a 'topological space', which abstracts from geometric forms like cubes and toruses. But topological spaces fail to capture geometric structures arising in areas like quantum physics; and this project seeks to rectify this, by developing a new more general notion.
Random Structures and Asymptotics. Discrete random structures have many uses in algorithms in computer science (for instance, random networks modelling computer link-ups), biology (for instance, random sequences modelling DNA) and engineering. New techniques for studying these structures will lead to powerful new results on their properties. The emphasis will be on the behaviour of the random structures when their size becomes large. With the advent of
more powerful computing techniques, it is ....Random Structures and Asymptotics. Discrete random structures have many uses in algorithms in computer science (for instance, random networks modelling computer link-ups), biology (for instance, random sequences modelling DNA) and engineering. New techniques for studying these structures will lead to powerful new results on their properties. The emphasis will be on the behaviour of the random structures when their size becomes large. With the advent of
more powerful computing techniques, it is often the large-scale behaviour which has relevance to the more diffucult computations being undertaken. The results are also of potential application to other areas of mathematics.Read moreRead less
Interconnection Network Routing and Graph Symmetry. Efficient routing schemes are of fundamental importance to both
traditional and optical interconnection networks. To achieve high
performance it is recommended that the graph modelling the network be vertex-transitive, meaning that it looks the same viewed from any vertex. In this project we will conduct a systematic study of the routing problem for such networks. We will focus on the effect of vertex-transitivity and some other symmetry pro ....Interconnection Network Routing and Graph Symmetry. Efficient routing schemes are of fundamental importance to both
traditional and optical interconnection networks. To achieve high
performance it is recommended that the graph modelling the network be vertex-transitive, meaning that it looks the same viewed from any vertex. In this project we will conduct a systematic study of the routing problem for such networks. We will focus on the effect of vertex-transitivity and some other symmetry properties on the efficiency of routing schemes measured by the vertex- and edge-congestions, and the minimum number of wavelengths needed in optical networks.Read moreRead less
Chromatic polynomials, random graphs, and error-correcting codes: a unified approach to graph colouring problems. Through a unified approach involving cutting-edge results on chromatic polynomials, random graphs, matroids, and error-correcting codes, this project will establish the foundations for a rigorous mathematical framework for attempting to provide a short, transparent and illuminating solution to the Four Colour Problem. The project will support developments in computer science and sta ....Chromatic polynomials, random graphs, and error-correcting codes: a unified approach to graph colouring problems. Through a unified approach involving cutting-edge results on chromatic polynomials, random graphs, matroids, and error-correcting codes, this project will establish the foundations for a rigorous mathematical framework for attempting to provide a short, transparent and illuminating solution to the Four Colour Problem. The project will support developments in computer science and statistical mechanics and is likely to have flow-on effects in real-world disciplines such as network communication. This project will also strengthen Australia's international presence in discrete mathematics and will further strengthen ties between Australian and international mathematicians.Read moreRead less
Green functions, correlation functions and differential equations. Classical and quantum exact solutions are established cornerstones in Australian applied mathematical research. In this project, we will:- 1). Address long standing open problems, whose resolution will add to mathematical knowledge and enhance Australia's reputation as a leading contributor to these topics; 2). List concrete and tractable sub-projects that will engage young scientists, whose training we are particularly keen on, ....Green functions, correlation functions and differential equations. Classical and quantum exact solutions are established cornerstones in Australian applied mathematical research. In this project, we will:- 1). Address long standing open problems, whose resolution will add to mathematical knowledge and enhance Australia's reputation as a leading contributor to these topics; 2). List concrete and tractable sub-projects that will engage young scientists, whose training we are particularly keen on, in vigorous and internationally competitive research; 3). Facilitate collaborations between various Australian research groups, all of whom are very well positioned to contribute to it; 4). Bring leading scientists to visit Australia to the benefit of the entire Australian mathematical community.Read moreRead less
The theory of natural dualities: old questions, new techniques. Natural dualities provide a powerful toolkit in algebra and logic, which form part of the formal language that underpins modern advances in information technology. This project will help to define the limits of where the toolkit can be applied, and where it can be applied in a user-friendly way. The project will enable Australia's La Trobe University to maintain its place as a world leader in the theory of natural dualities, and to ....The theory of natural dualities: old questions, new techniques. Natural dualities provide a powerful toolkit in algebra and logic, which form part of the formal language that underpins modern advances in information technology. This project will help to define the limits of where the toolkit can be applied, and where it can be applied in a user-friendly way. The project will enable Australia's La Trobe University to maintain its place as a world leader in the theory of natural dualities, and to attract leading mathematicians and talented students from Australia and overseas.Read moreRead less