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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Geodetic groups: foundational problems in algebra and computer science. The project aims to resolve important and longstanding open problems in Geometric Group Theory and Theoretical Computer Science. Since the 1980s researchers have conjectured that the geometric property of being geodetic is equivalent to several purely algebraic, algorithmic, and language-theoretic characterisations.
The project team's expertise in geodesic properties of groups, the interaction between formal languages and g ....Geodetic groups: foundational problems in algebra and computer science. The project aims to resolve important and longstanding open problems in Geometric Group Theory and Theoretical Computer Science. Since the 1980s researchers have conjectured that the geometric property of being geodetic is equivalent to several purely algebraic, algorithmic, and language-theoretic characterisations.
The project team's expertise in geodesic properties of groups, the interaction between formal languages and groups, and the theory of rewriting systems, together with recent breakthroughs by the team ensures that significant results can be expected.
Benefits include training research students and postdoctoral researchers in cutting-edge techniques, and advancing fundamental knowledge in mathematics and computer science.Read moreRead less
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.
Ring constructions and algorithms for enhancing performance of BCH codes. BCH codes form a major class of codes used in modern communication systems. The aim of this project is to enhance the efficiency of this class of codes by combining them in constructions enabling correction of deletion and insertion errors, and develop efficient implementations of encoding and decoding algorithms incorporating soft decision methods for enhanced error correction. Significance of the project is explained by ....Ring constructions and algorithms for enhancing performance of BCH codes. BCH codes form a major class of codes used in modern communication systems. The aim of this project is to enhance the efficiency of this class of codes by combining them in constructions enabling correction of deletion and insertion errors, and develop efficient implementations of encoding and decoding algorithms incorporating soft decision methods for enhanced error correction. Significance of the project is explained by the role of fast, secure and reliable communications in modern information and communication technology. Expected outcomes include new efficient algorithms and commercial modules available for symbolic computation systems with applications in telecommunications industry.
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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.
Expander graphs, isoperimetric numbers, and forwarding indices. Expanders are sparse but well connected networks. With numerous applications to modern technology, they have attracted many world leaders in mathematics and computer science. This project aims at substantial advancement on some important problems on expanders and related areas. It will put Australia at the forefront of this topical field.
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