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
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.
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
New frontiers in statistical mechanics. The chiral Potts model has been introduced in 1981 as a model for commensurate-incommensurate phase transitions in a layer of atoms or molecules adsorbed to a solid surface. If the adsorbed atoms all fit to holes between the surface atoms, the added layer is frozen, commensurate with the surface. If the added atoms are unable to fit holes, the added layer is no longer commensurate with the surface and could be in a floating state. A deeper understanding of ....New frontiers in statistical mechanics. The chiral Potts model has been introduced in 1981 as a model for commensurate-incommensurate phase transitions in a layer of atoms or molecules adsorbed to a solid surface. If the adsorbed atoms all fit to holes between the surface atoms, the added layer is frozen, commensurate with the surface. If the added atoms are unable to fit holes, the added layer is no longer commensurate with the surface and could be in a floating state. A deeper understanding of this and similar phenomena in layered systems has nanotechnological implications. This may affect the design of new small electronic devices or could apply to small biological systems and the development of new medicines. The project will surely lead to new applicable mathematics.Read moreRead less
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
Structure of relations: algebra and applications. Relations and relational structures form the fundamental mathematical essence required for studying computational problems and computational systems. This project will provide new algebraic methods for solving old problems in the theory of relations, informing our understanding of computational complexity and the nature of computing.
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
Quasi-subtractive varieties: a unified framework for substructural, modal and quantum logic. An algebraic theory is proposed that provides a common umbrella for a plethora of non-classical logics. At the same time, it identifies a core that these logics share with classical algebras.