Graded semisimple deformations. Recent advances in representation theory have revealed beautiful new structures in the classical representation theory of the symmetric groups and Hecke algebras. These discoveries have provided us with new algebras, the cyclotomic KLR algebras, that encode deep properties of fundamental objects in algebraic combinatorics and geometric representation theory. The cyclotomic quiver Hecke algebras are central to several open problems in mathematics but they are still ....Graded semisimple deformations. Recent advances in representation theory have revealed beautiful new structures in the classical representation theory of the symmetric groups and Hecke algebras. These discoveries have provided us with new algebras, the cyclotomic KLR algebras, that encode deep properties of fundamental objects in algebraic combinatorics and geometric representation theory. The cyclotomic quiver Hecke algebras are central to several open problems in mathematics but they are still poorly understood, with even basic properties like their dimensions being unknown. This project will establish a new framework for studying these algebras that will remove the current obstacles in this field and alllow us to prove substantial new results that advance the theory.Read moreRead less
Computing with Lie groups and algebras: nilpotent orbits and applications. This project aims to advance knowledge of an important class of Lie algebras, for which recent work has shown that a deeper mathematical theory, and better computational tools are required. Lie theory is a mathematical area with rich applications in the physical sciences. Expected outcomes from this project include the first systematic treatment of these algebras, new powerful algorithms to compute with them, and explicit ....Computing with Lie groups and algebras: nilpotent orbits and applications. This project aims to advance knowledge of an important class of Lie algebras, for which recent work has shown that a deeper mathematical theory, and better computational tools are required. Lie theory is a mathematical area with rich applications in the physical sciences. Expected outcomes from this project include the first systematic treatment of these algebras, new powerful algorithms to compute with them, and explicit nilpotent orbit classifications that will solve open problems in black hole theory. This should significantly enhance fundamental mathematical research and the Lie functionality of leading computer algebra systems, and is expected to strengthen international linkages.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE210100180
Funder
Australian Research Council
Funding Amount
$400,475.00
Summary
Effective classification of closed vertex-transitive groups acting on trees. Symmetry is a fundamental organising principle in mathematics and human endeavour. This project aims to advance our knowledge of zero-dimensional symmetry, a frontier in symmetry research. In the longer term, advancements in fundamental knowledge in this area have the potential to inform the usage and development of digital structures in more practical contexts, such as data networks and information processing. The proj ....Effective classification of closed vertex-transitive groups acting on trees. Symmetry is a fundamental organising principle in mathematics and human endeavour. This project aims to advance our knowledge of zero-dimensional symmetry, a frontier in symmetry research. In the longer term, advancements in fundamental knowledge in this area have the potential to inform the usage and development of digital structures in more practical contexts, such as data networks and information processing. The project is expected to develop new tools of both theoretical and computational nature that will accelerate ongoing research across the field and enable new approaches. This will cement Australia's position at the forefront of research in symmetry and its use in the digital age.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE150100308
Funder
Australian Research Council
Funding Amount
$283,536.00
Summary
Branching and self-similarity in group actions. This project aims to develop the theory of groups of symmetries that have self-similarity (part of the object has the same structure as the whole) and branching (transformations may be performed on parts of the object independently of one another while preserving the overall structure). The focus will be on a class of topological groups in which these properties frequently occur, building on methods recently developed and their actions on trees and ....Branching and self-similarity in group actions. This project aims to develop the theory of groups of symmetries that have self-similarity (part of the object has the same structure as the whole) and branching (transformations may be performed on parts of the object independently of one another while preserving the overall structure). The focus will be on a class of topological groups in which these properties frequently occur, building on methods recently developed and their actions on trees and on the Cantor set. The project aims to significantly advance the theory of locally compact groups, as well as giving insights into the phenomena of self-similarity and branching as they occur in group theory and dynamical systems.Read moreRead less
Algorithmic approaches to braids and their generalisations. This project combines theoretical methods from pure mathematics with computational experiments in order to gain new knowledge. The objects of interest, so-called braid groups and generalisations, are important for many fields of mathematics, but also have applications for data security. Both the theoretical outcomes of this project and the algorithms developed will strengthen Australia as a centre of cutting-edge research in computatio ....Algorithmic approaches to braids and their generalisations. This project combines theoretical methods from pure mathematics with computational experiments in order to gain new knowledge. The objects of interest, so-called braid groups and generalisations, are important for many fields of mathematics, but also have applications for data security. Both the theoretical outcomes of this project and the algorithms developed will strengthen Australia as a centre of cutting-edge research in computational algebra. Moreover, the results can lead to new technologies for protecting confidential data, which are more efficient and hence cheaper to implement than existing alternatives. Secure identification of legitimate users in the context of online banking is one possible field of application.Read moreRead less
Group algorithms: Complexity, Theory and Practice. The symmetry of a mathematical or physical system is often best described by an abstract structure called a group, and groups are commonly represented as groups of permutations or matrices. In this project we shall design and analyse a general algorithmic framework for computing with finite groups. In the context of permutation groups and matrix groups we will produce prototype implementations. The proposed framework has the potential to revolut ....Group algorithms: Complexity, Theory and Practice. The symmetry of a mathematical or physical system is often best described by an abstract structure called a group, and groups are commonly represented as groups of permutations or matrices. In this project we shall design and analyse a general algorithmic framework for computing with finite groups. In the context of permutation groups and matrix groups we will produce prototype implementations. The proposed framework has the potential to revolutionise algorithmic group theory as it draws together theoretical and computational models of groups.Read moreRead less
Computing with large groups: probability distributions and fast randomised algorithms. Fast algorithms produced by the project will impact on the practical management of symmetry in large scale searches, which have important industrial applications. Hence the project addresses the Priority Goals Breakthrough Science and Smart Information Use. The project will enhance Australia's leading position in Computational Algebra. Implementations of our algorithms will be incorporated in the Computer Alge ....Computing with large groups: probability distributions and fast randomised algorithms. Fast algorithms produced by the project will impact on the practical management of symmetry in large scale searches, which have important industrial applications. Hence the project addresses the Priority Goals Breakthrough Science and Smart Information Use. The project will enhance Australia's leading position in Computational Algebra. Implementations of our algorithms will be incorporated in the Computer Algebra system Magma, based at the University of Sydney, distributed world-wide, and used intensively in research and teaching. The project will attract international and Australian graduate students and postdoctoral researchers, and strengthen research activities in Australia by enhancing already strong international collaborations. Read moreRead less
Scale-Multiplicative Semigroups and Geometry. Symmetry is treated mathematically through the algebraic concept of a group. Conversely, geometric representations play a crucial role in group theory. Many classes of groups, such as the connected groups that arise in physics, have useful geometric representations, but such a representation is lacking in the case of general disconnected groups. Certain disconnected groups, closely related in algebraic terms to the connected ones, do have a geometric ....Scale-Multiplicative Semigroups and Geometry. Symmetry is treated mathematically through the algebraic concept of a group. Conversely, geometric representations play a crucial role in group theory. Many classes of groups, such as the connected groups that arise in physics, have useful geometric representations, but such a representation is lacking in the case of general disconnected groups. Certain disconnected groups, closely related in algebraic terms to the connected ones, do have a geometric representation called a 'building'. This project aims to address the lack of a representation for general disconnected groups by extending the notion of a building to create combinatorial structures on which these groups act as symmetries.Read moreRead less
Totally disconnected groups and their algebras. Groups are algebraic objects which convey symmetry much as
numbers convey size. For example, the symmetries of a
crystal form a crystallographic group and the classification of
crystallographic groups describes all possible crystal
structures. Totally disconnected groups arise as
symmetries of network structures having nodes and a `neighbour'
relation, as models of crystals do, but which are not rigid like
crystals. Powerful techniques for a ....Totally disconnected groups and their algebras. Groups are algebraic objects which convey symmetry much as
numbers convey size. For example, the symmetries of a
crystal form a crystallographic group and the classification of
crystallographic groups describes all possible crystal
structures. Totally disconnected groups arise as
symmetries of network structures having nodes and a `neighbour'
relation, as models of crystals do, but which are not rigid like
crystals. Powerful techniques for analysing totally
disconnected groups have recently been discovered and this
project aims to develop those techniques. The resulting
significant advances in the understanding of symmetry will
extend the range of applications of
group theory.
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Lie-type methods for totally disconnected groups. Groups are algebraic objects which convey symmetry, much as numbers convey size. For example, the rotations of a sphere form a group. This rotation group is one of a class known as the Lie groups that is well understood and has important applications. Totally disconnected groups arise as symmetries of network structures having nodes and a `neighbour' relation between nodes. The Australian investigator has discovered powerful methods for analysing ....Lie-type methods for totally disconnected groups. Groups are algebraic objects which convey symmetry, much as numbers convey size. For example, the rotations of a sphere form a group. This rotation group is one of a class known as the Lie groups that is well understood and has important applications. Totally disconnected groups arise as symmetries of network structures having nodes and a `neighbour' relation between nodes. The Australian investigator has discovered powerful methods for analysing totally disconnected groups which have parallels with Lie group techniques. This project will develop these parallels and establish links with international researchers on Lie groups.Read moreRead less