Symmetry and computation. The overall objective of the project is to explore connections between symmetry and computation, especially the theory and algorithms that facilitate the use of groups in computational science. The main outcome will be theoretically fast algorithms and implementations to drive applications in the sciences and for secure communication.
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
Discovery Early Career Researcher Award - Grant ID: DE140100088
Funder
Australian Research Council
Funding Amount
$378,628.00
Summary
Computing with matrix groups and Lie algebras: new concepts and applications. Computational algebra combines symbolic computation and pure research in algebra, and is concerned with the design of algorithms for solving mathematical problems endowed with algebraic structure. Matrix groups and Lie algebras are prominent algebraic objects describing the natural concept of symmetry. Despite being very common and important in science, there is a paucity of algorithms to study their structure. This pr ....Computing with matrix groups and Lie algebras: new concepts and applications. Computational algebra combines symbolic computation and pure research in algebra, and is concerned with the design of algorithms for solving mathematical problems endowed with algebraic structure. Matrix groups and Lie algebras are prominent algebraic objects describing the natural concept of symmetry. Despite being very common and important in science, there is a paucity of algorithms to study their structure. This project will develop deep new mathematical theories for computing with these objects, leading to ground-breaking advances in computational algebra, and providing powerful tools facilitating new research, including in other sciences. The new functionality will be used to solve two classification problems in group and Lie theory.Read moreRead less
Composition tree algorithms for large matrix groups. This project aims to develop new algorithms for analysing groups. A group is a rather simple mathematical structure – an example is the set of all integers considering only the operations of addition and subtraction. Since the symmetries of an object form a group, groups are ubiquitous throughout mathematics and elsewhere in science. Because it is frequently necessary to determine a group's properties, there is great interest in finding effici ....Composition tree algorithms for large matrix groups. This project aims to develop new algorithms for analysing groups. A group is a rather simple mathematical structure – an example is the set of all integers considering only the operations of addition and subtraction. Since the symmetries of an object form a group, groups are ubiquitous throughout mathematics and elsewhere in science. Because it is frequently necessary to determine a group's properties, there is great interest in finding efficient algorithms for analysing groups. A matrix group is a common type of group whose elements are square matrices. This project plans to employ a novel approach to designing algorithms for analysing large matrix groups, a task which is currently impossible using existing algorithms.Read moreRead less