Groups: statistics, structure, and algorithms. Science today relies on digital technologies using quantised and digital information. Because of the discrete nature of digital information, much of the mathematics underpinning these advances comes from the core disciplines of algebra and combinatorics within which this proposal falls. All aspects of the proposal focus on strengthening theoretical understanding of algebraic and combinatorial structures, and increasing computational power for workin ....Groups: statistics, structure, and algorithms. Science today relies on digital technologies using quantised and digital information. Because of the discrete nature of digital information, much of the mathematics underpinning these advances comes from the core disciplines of algebra and combinatorics within which this proposal falls. All aspects of the proposal focus on strengthening theoretical understanding of algebraic and combinatorial structures, and increasing computational power for working with them. The fundamental research outcomes, in terms of theorems, algorithms, and the training of young research mathematicians, will thus both enhance the high international standing of Australian mathematics, and strengthen Australia's capabilities in these important areas.Read moreRead less
Factorisation of Finite Groups and Graphs. The combinatorial structure of a graph is strongly influenced by its
symmetry, and the symmetry is described precisely by its group of
automorphisms. Interplay between actions of the automorphism group on
vertices, edges, and other configurations, reveals important graph
structure, especially the existence of graph factorisations. In turn, a group factorisation arises whenever a group has two
independent transitive actions, and these arise in parti ....Factorisation of Finite Groups and Graphs. The combinatorial structure of a graph is strongly influenced by its
symmetry, and the symmetry is described precisely by its group of
automorphisms. Interplay between actions of the automorphism group on
vertices, edges, and other configurations, reveals important graph
structure, especially the existence of graph factorisations. In turn, a group factorisation arises whenever a group has two
independent transitive actions, and these arise in particular while
determining graph automorphism groups, and graph factorisations. We will classify families of group factorisations, especially for simple groups, and apply this to establish a theory of symmetrical graph
factorisations, and to study Cayley graphs and 2-closures of permutation groups.
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Efficient computation in finite groups with applications in algebra and graph theory. The cutting-edge research of the project will further strengthen Australia's prominent role in computational group theory and algebraic graph theory. Besides the theoretical advances, the project includes the implementation and wide distribution of matrix group algorithms, benefiting immediately the algebraic research community and undergraduate mathematical education.
Finite permutation groups and flag-transitive incidence structures. Mathematics is the enabling discipline for all the sciences and so a strong mathematical research community in Australia provides the foundations for future discoveries in science and technology. By developing new theory for permutation groups, producing a new paradigm for the study of Buekenhout geometries and classifying certain families of flag-transitive incidence structures, we will enhance Australia's leading position in P ....Finite permutation groups and flag-transitive incidence structures. Mathematics is the enabling discipline for all the sciences and so a strong mathematical research community in Australia provides the foundations for future discoveries in science and technology. By developing new theory for permutation groups, producing a new paradigm for the study of Buekenhout geometries and classifying certain families of flag-transitive incidence structures, we will enhance Australia's leading position in Permutation Group Theory, Algebraic Graph Theory and Finite Geometry. This will attract international and Australian postgraduate students and visitors, and strengthen the research activities of Australia by enhancing the collaboration between UWA and leading international universities.Read moreRead less
Symmetry: Groups, Graphs, Number Fields and Loops. Exploiting symmetry can greatly simplify complex mathematical problems. This project aims to apply the powerful Classification of Finite Simple Groups to advance our understanding of the internal structure of number fields, highly symmetric graphs, and algebraic structures associated with Latin squares. The project expects to generate new constructions and classifications utilising group theory. Expected outcomes include resolutions of major ope ....Symmetry: Groups, Graphs, Number Fields and Loops. Exploiting symmetry can greatly simplify complex mathematical problems. This project aims to apply the powerful Classification of Finite Simple Groups to advance our understanding of the internal structure of number fields, highly symmetric graphs, and algebraic structures associated with Latin squares. The project expects to generate new constructions and classifications utilising group theory. Expected outcomes include resolutions of major open problems in each area as well as innovative methods for studying algebraic and combinatorial structures based on group actions. Expected benefits include enhanced international collaboration, and highly trained mathematicians to strengthen Australia’s research standing in fundamental science.Read moreRead less
Symmetries of finite digraphs. Highly symmetrical graphs are well-studied and, in many respects, the theory for dealing with them is well-established. By comparison, our understanding of symmetrical digraphs is much poorer. There are some rather basic questions about these about which we know shamefully little. The aim of this project is to remedy this shortage of knowledge by extending many important results and theories about symmetrical graphs to digraphs.
Discovery Early Career Researcher Award - Grant ID: DE230100579
Funder
Australian Research Council
Funding Amount
$445,754.00
Summary
The existence and abundance of small bases of permutation groups. This project aims to study bases for permutation groups, which are the mathematical formalisation of symmetry. Bases are crucial to encoding and computing with groups in diverse areas of science. Small bases are desirable for efficiency, but can be hard to find. This project expects to combine techniques from areas of algebra and probability to determine the existence and abundance of bases. Expected outcomes of this project inclu ....The existence and abundance of small bases of permutation groups. This project aims to study bases for permutation groups, which are the mathematical formalisation of symmetry. Bases are crucial to encoding and computing with groups in diverse areas of science. Small bases are desirable for efficiency, but can be hard to find. This project expects to combine techniques from areas of algebra and probability to determine the existence and abundance of bases. Expected outcomes of this project include new methods to address enduring open problems in the study of bases, as well as novel applications of existing techniques. This should provide significant benefits, such as creating and strengthening international collaborations, and building on Australia’s reputation as a powerhouse of finite group theory.Read moreRead less
Finite linearly representable geometries and symmetry. Finite geometry has profound mathematical connections to the theory of symmetry. Advances in finite geometry and in symmetry have historically led to advances in diverse areas such as algebra, computing, and theoretical physics. The project aims to characterise basic geometric objects called "projective planes'' and "generalised polygons'' using their symmetry properties. To achieve these aims, conceptual links between certain elements in cl ....Finite linearly representable geometries and symmetry. Finite geometry has profound mathematical connections to the theory of symmetry. Advances in finite geometry and in symmetry have historically led to advances in diverse areas such as algebra, computing, and theoretical physics. The project aims to characterise basic geometric objects called "projective planes'' and "generalised polygons'' using their symmetry properties. To achieve these aims, conceptual links between certain elements in classical symmetry groups and geometric planes and polygons must be developed. The density of these certain elements has important applications to probabilistic geometric algorithms.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE160100081
Funder
Australian Research Council
Funding Amount
$306,000.00
Summary
Structure theory for permutation groups and local graph theory conjectures. The focus of this project is on graphs, which are mathematical descriptions of networks, and it seeks to answer fundamental questions about how many symmetries such objects possess. This question is important since the symmetries of an object reveal its deepest structure. One of the main aims of this project it to convert local information into global properties of graphs. To make progress on the investigation of graphs, ....Structure theory for permutation groups and local graph theory conjectures. The focus of this project is on graphs, which are mathematical descriptions of networks, and it seeks to answer fundamental questions about how many symmetries such objects possess. This question is important since the symmetries of an object reveal its deepest structure. One of the main aims of this project it to convert local information into global properties of graphs. To make progress on the investigation of graphs, this project aims to classify the symmetry groups which arise from the local viewpoint. This classification is expected to provide new insight into symmetrical structures and have further impact on other areas of group theory.Read moreRead less
Exceptionally symmetric combinatorial designs. Advances in digital technologies are underpinned by powerful mathematics; use of symmetry greatly simplifies complex problems. This project aims to exploit the mathematical theory of groups to advance our understanding of combinatorial designs with exceptional symmetry. New designs have become prominent through links with networks and error correcting codes. The project expects to generate constructions and classifications in these areas by utilisin ....Exceptionally symmetric combinatorial designs. Advances in digital technologies are underpinned by powerful mathematics; use of symmetry greatly simplifies complex problems. This project aims to exploit the mathematical theory of groups to advance our understanding of combinatorial designs with exceptional symmetry. New designs have become prominent through links with networks and error correcting codes. The project expects to generate constructions and classifications in these areas by utilising powerful group theory. As well as innovative methods for studying designs with symmetry based on group actions, expected outcomes include enhanced international collaboration, and highly trained combinatorial mathematicians to strengthen Australia’s research standing in fundamental science. Read moreRead less