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An efficient approach to the computation of bacterial evolutionary distance. This project aims to apply advanced mathematical tools to improve our understanding of bacterial evolution. Bacteria account for as much total Earth biomass as all plant species put together, and have an unparalleled ability to evolve quickly and adapt to changing environments. Unfortunately, the existing mathematical models used to model bacterial evolution are generally computationally intractable. This project will r ....An efficient approach to the computation of bacterial evolutionary distance. This project aims to apply advanced mathematical tools to improve our understanding of bacterial evolution. Bacteria account for as much total Earth biomass as all plant species put together, and have an unparalleled ability to evolve quickly and adapt to changing environments. Unfortunately, the existing mathematical models used to model bacterial evolution are generally computationally intractable. This project will rectify this situation by using representation theory to transform combinatorial group theory into linear algebra, allowing for the application of advanced methods of numeric approximation. This will provide a better understanding of how bacteria evolve and improve our ability to manage their impact.Read moreRead less
Algebraic evolution and evolutionary algebra. Algebra and biology have developed in extraordinary ways over the last half century yet, to date, the use of algebraic ideas in biology has been limited. This project will address this by modelling evolutionary processes in bacteria using algebraic ideas.
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
Diagram categories and transformation semigroups. A structural understanding of diagram categories is essential in many branches of mathematics and science. Despite this, very few methods for studying such categories are available, a fact this pure mathematics project seeks to rectify. By building strong bridges between diagram categories and semigroup theory, a field of abstract algebra that models transformation and change, the structure of diagram categories may be unlocked with powerful semi ....Diagram categories and transformation semigroups. A structural understanding of diagram categories is essential in many branches of mathematics and science. Despite this, very few methods for studying such categories are available, a fact this pure mathematics project seeks to rectify. By building strong bridges between diagram categories and semigroup theory, a field of abstract algebra that models transformation and change, the structure of diagram categories may be unlocked with powerful semigroup tools developed by the applicant investigator. Diagrammatic insights will also yield new ways to study semigroups, and the many other mathematical structures they interact with. Outcomes will have a lasting impact on both theories as well as the many fields influenced by them.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
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.