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
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
Graph symmetry and simple groups. This project aims to use knowledge of finite simple groups to tackle problems in graph symmetry. The symmetry of an object is encoded by a group, and this allows tools from algebra to be used to study graphs. The main impact will be in areas of pure mathematics such as graph theory and group theory by obtaining new classifications and constructions. Expected outcomes include providing new constructions and classifications of highly symmetric graphs, and an impro ....Graph symmetry and simple groups. This project aims to use knowledge of finite simple groups to tackle problems in graph symmetry. The symmetry of an object is encoded by a group, and this allows tools from algebra to be used to study graphs. The main impact will be in areas of pure mathematics such as graph theory and group theory by obtaining new classifications and constructions. Expected outcomes include providing new constructions and classifications of highly symmetric graphs, and an improved knowledge and understanding of local symmetries for graphs of higher valencies so that they become as well understood as the valency three case.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
Complexity of group algorithms and statistical fingerprints of groups. This project aims to shape the next generation of efficient randomised algorithms in the field of group theory, the mathematics of symmetry. Fundamental mathematics underpins modern technological tasks such as web searches, sorting and data compression. This project aims to determine characteristic statistical fingerprints of key building-block groups. These group statistics lead to much faster procedures to essentially facto ....Complexity of group algorithms and statistical fingerprints of groups. This project aims to shape the next generation of efficient randomised algorithms in the field of group theory, the mathematics of symmetry. Fundamental mathematics underpins modern technological tasks such as web searches, sorting and data compression. This project aims to determine characteristic statistical fingerprints of key building-block groups. These group statistics lead to much faster procedures to essentially factor huge groups into smaller building-block groups in a manner akin to factoring an integer into its prime factors. The anticipated goal is to include the outcomes in publicly available symbolic algebra computer packages. As the theory of symmetry has broad applications in the mathematical and physical sciences, there is the potential for far reaching benefits.Read moreRead less
Industrial Transformation Training Centres - Grant ID: IC190100017
Funder
Australian Research Council
Funding Amount
$3,703,664.00
Summary
ARC Training Centre for Integrated Operations for Complex Resources. This Training Centre aims to increase value in mining through clever applications of ‘lean processing’ and train the next generation of scientists and engineers in advanced sensors and data analytics in complex resources; knowledge priorities for the mining industry. Sensor information will be linked to the resource’s in-place knowledge to enable data analytics of all embedded knowledge. Processing can then be tuned to resource ....ARC Training Centre for Integrated Operations for Complex Resources. This Training Centre aims to increase value in mining through clever applications of ‘lean processing’ and train the next generation of scientists and engineers in advanced sensors and data analytics in complex resources; knowledge priorities for the mining industry. Sensor information will be linked to the resource’s in-place knowledge to enable data analytics of all embedded knowledge. Processing can then be tuned to resource attributes, maximising value ‘on the fly’. Benefits will include increasing certainty on product quality and maximising throughput and recovery. Outcomes will include new tools to rapidly model geological and geometallurgical uncertainty with sensor inputs, to track the resource to product and enhance interpretation.Read moreRead less