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.
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
Permutation groups: factorisations, structure and applications. Group theory is the mathematical study of symmetry. This project aims to improve our understanding of the structure of groups by studying their factorisations and the structure of certain subgroups and elements. The obtained knowledge will be applied to study embeddings of graphs on surfaces and regular subgroups of important families of groups. The main impact would be in areas of pure mathematics such as group theory and graph the ....Permutation groups: factorisations, structure and applications. Group theory is the mathematical study of symmetry. This project aims to improve our understanding of the structure of groups by studying their factorisations and the structure of certain subgroups and elements. The obtained knowledge will be applied to study embeddings of graphs on surfaces and regular subgroups of important families of groups. The main impact would be in areas of pure mathematics such as group theory and graph theory by strengthening our knowledge of the finite simple groups.Read moreRead less
Cayley graphs and their associated geometric and combinatorial objects. As for everything else, so for a mathematical theory: beauty can be perceived but not explained' -- Arthur Cayley. This project will exploit the profound interaction between algebra, networks and geometry in order to construct and analyse inherently symmetric geometric structures.
Finite geometry from an algebraic point of view. Bannai and Munemasa stated that Delsarte’s way of looking at many combinatorial problems in the framework of association schemes and combining design theory and coding theory in a single framework was a remarkable new approach and has been extremely successful. This project will apply the power of algebraic combinatorics to analyse finite geometric structures.
Permutation groups and their interrelationship with the symmetry of graphs, codes and geometric configurations. Symmetry is important in simplifying complex structures and in efficiently encoding information. This project will develop new fundamental theory for the mathematics behind symmetry. It will apply this to construct and classify families of networks lying on surfaces and structures that arise from set theory and projective geometry.
Discovery Early Career Researcher Award - Grant ID: DE130101521
Funder
Australian Research Council
Funding Amount
$302,845.00
Summary
Permutation groups with finite subdegrees and the structure of totally disconnected locally compact groups. Understanding the symmetries of a structure is one way of determining its nature. Our lack of knowledge of these symmetries is holding back research in combinatorics and topological group theory. This project aims to understand the symmetries of an important collection of infinite structures that arise naturally in other areas of mathematics.