Reflection groups. The study of symmetry in geometrical and abstract contexts is a central issue in such diverse areas as mathematical physics, singularity theory, algebraic geometry, quantum groups and the study of knots and braids. Group theory provides the mathematical framework for the analysis of symmetry. Reflection groups, simple examples of which are the symmetry groups of the five platonic solids, play a key role in all of the areas mentioned above. Thus an improved understanding of ref ....Reflection groups. The study of symmetry in geometrical and abstract contexts is a central issue in such diverse areas as mathematical physics, singularity theory, algebraic geometry, quantum groups and the study of knots and braids. Group theory provides the mathematical framework for the analysis of symmetry. Reflection groups, simple examples of which are the symmetry groups of the five platonic solids, play a key role in all of the areas mentioned above. Thus an improved understanding of reflection groups will significantly enhance the development of several important theories.
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The language complexity of problems in algebra and logic. This project focuses on a major problem at the intersection of algebra, logic and computer science, concerning equations over free groups and free monoids. Expected outcomes include a language-theoretic characterisation of solutions of equations in a wide class of groups and monoids, a language-theoretic understanding of the existential and first-order theories of free groups, and a classification of groups with indexed multiplication tab ....The language complexity of problems in algebra and logic. This project focuses on a major problem at the intersection of algebra, logic and computer science, concerning equations over free groups and free monoids. Expected outcomes include a language-theoretic characterisation of solutions of equations in a wide class of groups and monoids, a language-theoretic understanding of the existential and first-order theories of free groups, and a classification of groups with indexed multiplication tables and EDT0L word problem. The project is designed to expand the frontiers of knowledge in theoretical computer science and pure mathematics, but in the longer term to deepen our understanding of computers, their computational power and intrinsic limitations.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.
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.
Geometric structures in representation theory. Mathematics underpins every aspect of people's interactions with nature (e.g. physics) and with each other (e.g. finance). Its uses range from formulating physical laws in order to understand and predict nature, to analysis of financial concepts and transactions. This project will formulate and develop three new fundamental mathematical concepts: cellular algebras, eigenspace geometries, and diagram algebras. Benefits include enhancement of Australi ....Geometric structures in representation theory. Mathematics underpins every aspect of people's interactions with nature (e.g. physics) and with each other (e.g. finance). Its uses range from formulating physical laws in order to understand and predict nature, to analysis of financial concepts and transactions. This project will formulate and develop three new fundamental mathematical concepts: cellular algebras, eigenspace geometries, and diagram algebras. Benefits include enhancement of Australia's position at the very frontier of world class mathematical research, and a myriad of potential applications to physics, coding theory, information technology, electronic security and experimental design.Read moreRead less
The geometry of exotic nilpotent cones. This research will describe the geometry of some important objects which sit at the boundary of algebra, geometry, and combinatorics. It has intrinsic value as a significant addition to the heritage of mathematical thought, and will strengthen Australian traditions in these areas of mathematics.
Pyramids and decomposition numbers for the symmetric and general linear groups. This project takes a novel approach to the decomposition number problem for the symmetric and general linear groups by setting up a new framework for computing them using the combinatorics of pyramids. The decomposition numbers of an algebra are an important statistic which gives detailed structural information about its representations. These numbers can be used to compute the dimensions of the irreducible represen ....Pyramids and decomposition numbers for the symmetric and general linear groups. This project takes a novel approach to the decomposition number problem for the symmetric and general linear groups by setting up a new framework for computing them using the combinatorics of pyramids. The decomposition numbers of an algebra are an important statistic which gives detailed structural information about its representations. These numbers can be used to compute the dimensions of the irreducible representations of the algebra and they play an important role in the applications of representation theory to other fields such as knot theory and statistical mechanics.Read moreRead less
Algebras with Frobenius morphisms and quantum groups. In this digitalized world, our life relies on mathematics more than ever. Counting and numbers are just one example of this. Another is the public key codes for online payments and transactions. Mathematics is of enormous importance in this technology dominated age. This proposal is to carry out high level mathematical research in Australia. Basic research on quantum groups underpins applied research and certain areas such as quantum mechanic ....Algebras with Frobenius morphisms and quantum groups. In this digitalized world, our life relies on mathematics more than ever. Counting and numbers are just one example of this. Another is the public key codes for online payments and transactions. Mathematics is of enormous importance in this technology dominated age. This proposal is to carry out high level mathematical research in Australia. Basic research on quantum groups underpins applied research and certain areas such as quantum mechanics and string theory. Some structure of quantum groups is too complicated to be seen by even a professional mathematician. A possible interpretation by using representations over a finite field would make it more usable and accessible by computer.Read moreRead less
New Geometric and Entropy Techniques for Differential Equations. The three main practical outcomes of this mathematical research will be better predictability of salt movement responsible for land degradation, better predictability of surface evolution of microelectronic components in nanoscale technology and an open source computer package that harnesses new and powerful geometrical techniques to solve differential equations. The project will train the next generation of researchers in the math ....New Geometric and Entropy Techniques for Differential Equations. The three main practical outcomes of this mathematical research will be better predictability of salt movement responsible for land degradation, better predictability of surface evolution of microelectronic components in nanoscale technology and an open source computer package that harnesses new and powerful geometrical techniques to solve differential equations. The project will train the next generation of researchers in the mathematical modelling of critical physical processes and it will bring international experts to Australia to work on these vital problems.Read moreRead less
Modular representations of cyclotomic algebras. This project addresses cutting edge questions in the representation theory of cyclotomic Hecke algebras. Our main focus will be computing decomposition matrices for these algebras. We approach this question from several different directions, each of which will give new insights and lead to significant advances in the theory. The decomposition number problem is important because its' solution gives deep structural information about these algebras wh ....Modular representations of cyclotomic algebras. This project addresses cutting edge questions in the representation theory of cyclotomic Hecke algebras. Our main focus will be computing decomposition matrices for these algebras. We approach this question from several different directions, each of which will give new insights and lead to significant advances in the theory. The decomposition number problem is important because its' solution gives deep structural information about these algebras which can then be applied in other areas. This project will have high impact because cyclotomic Hecke algebras have applications in many different areas and they are currently a hot topic of research in mathematics.Read moreRead less