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Representations of arithmetic groups and their associated zeta functions. This project aims to investigate deep connections between number theory and group theory by studying linear actions of arithmetic groups. Arithmetic groups are used in geometry, dynamics, number theory and other areas of pure mathematics. This project will study their representations from two perspectives. First, it will establish properties of the associated zeta functions to resolve open problems about the asymptotic beh ....Representations of arithmetic groups and their associated zeta functions. This project aims to investigate deep connections between number theory and group theory by studying linear actions of arithmetic groups. Arithmetic groups are used in geometry, dynamics, number theory and other areas of pure mathematics. This project will study their representations from two perspectives. First, it will establish properties of the associated zeta functions to resolve open problems about the asymptotic behaviour of the dimensions of the irreducible representations. Second, it will explore the evolution of representations across families of groups under new induction and restriction functors, in analogy with creation and annihilation operators in physics. The project will enhance Australia's capacity in representation theory and group theory, the mathematics that underline symmetry in nature.Read moreRead less
Algorithmic and computational advances in geometric group theory. This project aims to combine new algorithmic ideas, high performance computing and experimental mathematics to answer many outstanding questions in the field of geometric group theory. This project will put Australia at the forefront of new computer-assisted research, and give new insights into complex mathematical problems.
New reactivity from unusual main group compounds. This project will develop new, fundamentally important, yet unusual main group compounds and investigate their reactivity. The project will lead to important fundamental advance in main group chemistry and will form the basis for cheaper and cleaner future synthetic methodologies and technologies.
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
Categorical representation theory and applications. Symmetry is everywhere, and nature is designed symmetrically: Snails make their shells, spiders design their webs, and bees build hexagonal honeycombs, all based on the concept of symmetry. Symmetry is a general principle which plays an important role in various areas of knowledge and perception, ranging from arts to natural sciences and mathematics.
The 21th century way of the study of symmetries is categorical representation theory. The proj ....Categorical representation theory and applications. Symmetry is everywhere, and nature is designed symmetrically: Snails make their shells, spiders design their webs, and bees build hexagonal honeycombs, all based on the concept of symmetry. Symmetry is a general principle which plays an important role in various areas of knowledge and perception, ranging from arts to natural sciences and mathematics.
The 21th century way of the study of symmetries is categorical representation theory. The project aims are to strengthen this young field by advancing the theory and by finding applications from where its significance arises. The outcome will be new results on categorical representations and this will have benefits within mathematics, cryptography and also in physics/chemistry in the long run.Read moreRead less
Finite geometry from an algebraic point of view. Bannai and Munemasa stated that Delsarte’s way of looking at many combinatorial problems in the framework of association schemes and combining design theory and coding theory in a single framework was a remarkable new approach and has been extremely successful. This project will apply the power of algebraic combinatorics to analyse finite geometric structures.
Representation theory in exactly solvable systems. This project aims to develop the representation theory of Lie and generalised Lie algebras related to exactly solvable models. The project will exploit several innovative ideas on the structure of quadratic algebras, Casimir invariants, differential operator realisations, roots systems, characters and indecomposable representations. This will give fundamental mathematical insight and allow the construction of new, exactly solvable models. This w ....Representation theory in exactly solvable systems. This project aims to develop the representation theory of Lie and generalised Lie algebras related to exactly solvable models. The project will exploit several innovative ideas on the structure of quadratic algebras, Casimir invariants, differential operator realisations, roots systems, characters and indecomposable representations. This will give fundamental mathematical insight and allow the construction of new, exactly solvable models. This will have an impact on theoretical physics as exactly solvable models play a central role in our understanding of a plethora of physical phenomena.Read moreRead less
Springer fibres, nilpotent cones and representation theory. This project will address new ideas and famous unsolved problems in the field of algebra known as representation theory, by studying the geometry of spaces called Springer fibres and nilpotent cones. This will keep Australian mathematics in the forefront of developments in this internationally active field, which is central to modern mathematics.
Synthesis, Synergy and Sustainability: Development of active-metal reagents. The design and realisation of new and important molecules requires innovative and efficient methods. This project will create a new store of active-metal molecular tools for the selective, catalytic and atom efficient construction of a diverse library of phosphorus heterocyclic scaffolds and chemical feedstocks relevant to biological, medicinal, and materials chemistry, and the fine chemical industry. Parallel studies e ....Synthesis, Synergy and Sustainability: Development of active-metal reagents. The design and realisation of new and important molecules requires innovative and efficient methods. This project will create a new store of active-metal molecular tools for the selective, catalytic and atom efficient construction of a diverse library of phosphorus heterocyclic scaffolds and chemical feedstocks relevant to biological, medicinal, and materials chemistry, and the fine chemical industry. Parallel studies employing environmentally friendly and benign deep eutectic solvents will allow for replacement of traditional hazardous volatile organic solvents, putting the newly created active-metal reagents at the forefront of the necessary shift towards a more sustainable and 'green' polar organometallic chemistry. Read moreRead less
A top-down approach to synthesising high-value fluorocarbons. Fluorocarbons' ability to impart high stability, solubility, and unique reactivity to host molecules renders them invaluable in agrochemicals, pharmaceuticals, polymers and surfactants. Their robustness also renders them environmentally persistent. There are no industrially utilised methods for the re-purposing or recycling of fluorocarbons. This project aims to generate new methods for the selective activation of carbon-fluorine bond ....A top-down approach to synthesising high-value fluorocarbons. Fluorocarbons' ability to impart high stability, solubility, and unique reactivity to host molecules renders them invaluable in agrochemicals, pharmaceuticals, polymers and surfactants. Their robustness also renders them environmentally persistent. There are no industrially utilised methods for the re-purposing or recycling of fluorocarbons. This project aims to generate new methods for the selective activation of carbon-fluorine bonds in polyfluorocarbons, allowing their incorporation or repurposing into high value chemicals and/or easy derivitisation to access a plethora of new fluorocarbon products. Expected outcomes will allow new processing methods to value add to fluorocarbons while preventing their environmental release. Read moreRead less