Discovery Early Career Researcher Award - Grant ID: DE190100666
Funder
Australian Research Council
Funding Amount
$381,000.00
Summary
Extremal combinatorics meets finite geometry. This project aims to investigate important open problems lying at the intersection of two areas of mathematics, extremal combinatorics and finite geometry. The project will focus on the area of discrete mathematics, which has been at the centre of some of recent developments in mathematics and computer science. This project proposes new methods, derived from algebra, geometry and computer science, to tackle important extremal problems in finite geome ....Extremal combinatorics meets finite geometry. This project aims to investigate important open problems lying at the intersection of two areas of mathematics, extremal combinatorics and finite geometry. The project will focus on the area of discrete mathematics, which has been at the centre of some of recent developments in mathematics and computer science. This project proposes new methods, derived from algebra, geometry and computer science, to tackle important extremal problems in finite geometry. The project will provide answers to a number of open problems in extremal combinatorics and finite geometry. Moreover, new methods will be developed which will have an interdisciplinary impact.Read moreRead less
Symmetrical graphs, generalized polygons and expanders. This project proposes to study a class of highly symmetrical graphs -- locally s-arc-transitive graphs. Studying the class of graphs has been one of the central topics in algebraic graph theory for over 50 years. This class of graphs has been effectively used in computer science, communication network, group theory, geometry, and other areas. This project will develop new methods to solve several fundamental problems regarding locally s-arc ....Symmetrical graphs, generalized polygons and expanders. This project proposes to study a class of highly symmetrical graphs -- locally s-arc-transitive graphs. Studying the class of graphs has been one of the central topics in algebraic graph theory for over 50 years. This class of graphs has been effectively used in computer science, communication network, group theory, geometry, and other areas. This project will develop new methods to solve several fundamental problems regarding locally s-arc-transitive graphs, and apply the outcomes to solve important problems in communication networks, graph theory, group theory, and geometry.Read moreRead less
Exact structure in graphs and matroids. One of the main goals of mathematics is to understand and describe the structure of the mathematical world. This project will contribute to this goal, and deepen our understanding of the fundamental mathematical structures called graphs and matroids, by providing exact structural descriptions of a number of important minor-closed classes.
Real chromatic roots of graphs and matroids. This project will develop the theory of real chromatic roots of graphs, especially as it applies to minor-closed classes of graphs, with the aim of extending this theory to minor-closed classes of matroids. One of the fundamental results from the study of real chromatic roots in graph theory is that any minor-closed class of graphs has an absolute upper bound on its chromatic roots. However, while many results on minor-closed classes on graphs have cl ....Real chromatic roots of graphs and matroids. This project will develop the theory of real chromatic roots of graphs, especially as it applies to minor-closed classes of graphs, with the aim of extending this theory to minor-closed classes of matroids. One of the fundamental results from the study of real chromatic roots in graph theory is that any minor-closed class of graphs has an absolute upper bound on its chromatic roots. However, while many results on minor-closed classes on graphs have close analogues or mild variants for minor-closed classes of matroids, this upper bound on real chromatic roots appears, somewhat mysteriously, to apply only to graphs. By studying the upper root-free intervals of minor-closed classes of matroids, this project aims to shed light on this phenomenon.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
The synchronisation hierarchy of permutation groups. This project aims to make significant advances in understanding finite primitive permutation groups, which are the basic building blocks of the mathematical study of symmetry. A recently-developed perspective, inspired by the notion of a synchronising automaton, has revealed that these groups fall into a natural hierarchy. While the outline of this synchronisation hierarchy is known, many questions remain about exactly which primitive groups l ....The synchronisation hierarchy of permutation groups. This project aims to make significant advances in understanding finite primitive permutation groups, which are the basic building blocks of the mathematical study of symmetry. A recently-developed perspective, inspired by the notion of a synchronising automaton, has revealed that these groups fall into a natural hierarchy. While the outline of this synchronisation hierarchy is known, many questions remain about exactly which primitive groups lie in which layers. Answering these questions using techniques from group theory, graph theory and finite geometry will substantially deepen our understanding. The benefits of this include new knowledge and enhanced insight into this fundamental class of groups and new tools for their analysis.Read moreRead less