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
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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Totally disconnected groups in algebra and geometry. Mathematics research creates and develops new concepts for understanding the world. Group theory is a branch of mathematics based on our innate sense of symmetry. It was invented 200 hundred years ago and has grown into a language for analysing and classifying things ranging from wallpaper patterns to crystals, the fundamental particles of physics and Rubik's cube. The chief investigators have made significant breakthroughs in the study of sym ....Totally disconnected groups in algebra and geometry. Mathematics research creates and develops new concepts for understanding the world. Group theory is a branch of mathematics based on our innate sense of symmetry. It was invented 200 hundred years ago and has grown into a language for analysing and classifying things ranging from wallpaper patterns to crystals, the fundamental particles of physics and Rubik's cube. The chief investigators have made significant breakthroughs in the study of symmetry groups of networks, giving Australia an international lead in this research area. The project will develop the insights gained to make Australia a centre of expertise on these symmetry groups, which have applications to many areas including information and communication technology.Read moreRead less
Applications of Group Theory to Finite Geometry. Group theory and geometry have influenced one another for over a century. The most important structures in geometry are the symmetric ones and the most important groups act on geometries. Recent developments in finite geometry, although informed by symmetry, have used a minimum of group theory. The project aims to redress this, by applying results from a broad range of finite group theory to the presently hot topics in finite geometry. Our aim is ....Applications of Group Theory to Finite Geometry. Group theory and geometry have influenced one another for over a century. The most important structures in geometry are the symmetric ones and the most important groups act on geometries. Recent developments in finite geometry, although informed by symmetry, have used a minimum of group theory. The project aims to redress this, by applying results from a broad range of finite group theory to the presently hot topics in finite geometry. Our aim is to achieve a paradigm shift, by finding substantively different structures than those presently known. Should it succeed, much activity in geometry would follow, seeking geometric interpretation of these group theoretic results. Our focus is necessitated by the lack of a result characterising the underlying groups of symmetric generalised quadrangles.Read moreRead less
Permutation groups and their interplay with symmetry in finite geometry and graph theory. A strong mathematical community in Australia provides the foundations for future discoveries in technology, science and business. The use of group theory to characterise symmetric generalised quadrangles, partial quadrangles, and strongly regular graphs, and the construction of new examples of such objects, will enhance Australia's leading position in Group Theory, Algebraic Graph Theory and Finite Geometry ....Permutation groups and their interplay with symmetry in finite geometry and graph theory. A strong mathematical community in Australia provides the foundations for future discoveries in technology, science and business. The use of group theory to characterise symmetric generalised quadrangles, partial quadrangles, and strongly regular graphs, and the construction of new examples of such objects, will enhance Australia's leading position in Group Theory, Algebraic Graph Theory and Finite Geometry. This project will also strengthen the collaboration between Australian, Belgian and Italian Universities and support young researchers, developing expertise in a world-leading research group, to drive Australia's future in mathematics.Read moreRead less