Billiards within confocal quadrics and beyond. This project aims to analyse mathematical billiards within domains bounded by confocal conics. Mathematical billiards have applications in any situation that involves collisions and reflections, and any phenomenon that includes reflections and collisions can be modelled using mathematical billiards. This project aims to revolutionise the analysis of billiards within domains bounded by several confocal conics by exploring the relations of such billia ....Billiards within confocal quadrics and beyond. This project aims to analyse mathematical billiards within domains bounded by confocal conics. Mathematical billiards have applications in any situation that involves collisions and reflections, and any phenomenon that includes reflections and collisions can be modelled using mathematical billiards. This project aims to revolutionise the analysis of billiards within domains bounded by several confocal conics by exploring the relations of such billiards with polygonal billiards, and making research advances with the higher-dimensional generalisations within confocal quadrics and their relations with billiards within polyhedra. The project will link several significant areas of scientific work including polygonal billiards, classical integrable systems, Teichmuller spaces, and relativity theory. The project outcomes will have impact across areas of mathematics such as geometry, algebraic geometry, and dynamical systems.Read moreRead less
Quantum algebras with supersymmetries. The project aims to make fundamental advances in the theory of quantum algebras. It will develop explicit
structure and representation theory of major classes of quantum algebras which are of great importance to
quantum field theory and integrable models with supersymmetries. The intended outcomes include a solution of
the outstanding classification problem for representations of quantum algebras with supersymmetries, which has
remained open for the last tw ....Quantum algebras with supersymmetries. The project aims to make fundamental advances in the theory of quantum algebras. It will develop explicit
structure and representation theory of major classes of quantum algebras which are of great importance to
quantum field theory and integrable models with supersymmetries. The intended outcomes include a solution of
the outstanding classification problem for representations of quantum algebras with supersymmetries, which has
remained open for the last two decades. It will involve newly-developed methods within the theory of quantum
groups, and both the methods and classification will bring new mathematical instruments for the advance of
supesymmetric conformal field theory and soliton spin chain models.Read moreRead less
Multi-dimensionally consistent integrable systems in geometry and algebra. This project aims to address in an innovative manner a long-standing open problem in nonlinear mathematics, namely the determination of the algebraic and geometric origin of integrable systems. It is expected to make a fundamental contribution towards integrable systems theory. The latter provides unique access to the analytic treatment of nonlinear phenomena not only in physics but also a remarkably diverse range of area ....Multi-dimensionally consistent integrable systems in geometry and algebra. This project aims to address in an innovative manner a long-standing open problem in nonlinear mathematics, namely the determination of the algebraic and geometric origin of integrable systems. It is expected to make a fundamental contribution towards integrable systems theory. The latter provides unique access to the analytic treatment of nonlinear phenomena not only in physics but also a remarkably diverse range of areas in mathematics. Expected outcomes include extended, unified and novel key mathematical concepts in a discrete setting and their applications in algebraic and geometric contexts. Due to the choice of participants, it is anticipated that Australia will benefit from strengthened research collaborations with Germany.Read moreRead less
Quadratic fusion categories: A frontier in subfactor theory. This project aims to investigate the quantum symmetries of the quadratic fusion categories. Fusion categories are mathematical structures that generalise the symmetries of finite groups. These structures arise as invariants of subfactors in operator algebras and in mathematical models of conformal field theory. The quadratic fusion categories encompass most known subfactors that do not come from finite or quantum groups and form a vast ....Quadratic fusion categories: A frontier in subfactor theory. This project aims to investigate the quantum symmetries of the quadratic fusion categories. Fusion categories are mathematical structures that generalise the symmetries of finite groups. These structures arise as invariants of subfactors in operator algebras and in mathematical models of conformal field theory. The quadratic fusion categories encompass most known subfactors that do not come from finite or quantum groups and form a vast frontier about which little is known. By uncovering the symmetries of the quadratic fusion categories, the project will advance subfactor theory and provide new models for conformal field theory. Progress in these fields will have applications to the emerging technology of quantum computing.Read moreRead less
New mathematics to improve understanding of anomalously diffusing reactions. Standard mathematical models for particles that diffuse and react are based on assumptions that improving technologies have revealed do not always hold. This project aims to create a mathematical framework that generalises existing approaches, taking into account observations of complicated transport behaviour at many scales, and including the impact of this anomalous transport on reactions. The development of the fram ....New mathematics to improve understanding of anomalously diffusing reactions. Standard mathematical models for particles that diffuse and react are based on assumptions that improving technologies have revealed do not always hold. This project aims to create a mathematical framework that generalises existing approaches, taking into account observations of complicated transport behaviour at many scales, and including the impact of this anomalous transport on reactions. The development of the framework will involve innovative approaches utilising mathematical techniques, including dynamical systems, fractional calculus, and stochastic processes. This project aims to deliver new mathematical models that can be adopted in applications across different discipline areas, and especially in biological systems. Read moreRead less
Geometric analysis of nonlinear systems. Modern science derives its power from mathematics. The project aims to capture, identify and describe pivotal, transcendental solutions of nonlinear systems that are universal in science, in the sense that they always arise as mathematical models under certain physical limits. The project expects to produce new mathematical methods to describe such functions by using a newly discovered geometric framework. Expected outcomes include the description of elus ....Geometric analysis of nonlinear systems. Modern science derives its power from mathematics. The project aims to capture, identify and describe pivotal, transcendental solutions of nonlinear systems that are universal in science, in the sense that they always arise as mathematical models under certain physical limits. The project expects to produce new mathematical methods to describe such functions by using a newly discovered geometric framework. Expected outcomes include the description of elusive solutions of discrete and higher-dimensional nonlinear systems. This should provide significant benefits, such as new mathematical knowledge, innovative techniques, enhanced scientific capacity in Australia.Read moreRead less
Biophysics of the brain’s waste disposal system: Understanding why we sleep. This project aims to develop a new biophysical model of the brain, founded on the recently discovered glymphatic system responsible for waste disposal during sleep. It sets out to formulate, analyse, and validate rigorous new multiscale quantitative modelling – to advance the study of sleep and brain clearance dynamics, at timescales from hours to decades. Among expected outcomes are powerful models ready for applicatio ....Biophysics of the brain’s waste disposal system: Understanding why we sleep. This project aims to develop a new biophysical model of the brain, founded on the recently discovered glymphatic system responsible for waste disposal during sleep. It sets out to formulate, analyse, and validate rigorous new multiscale quantitative modelling – to advance the study of sleep and brain clearance dynamics, at timescales from hours to decades. Among expected outcomes are powerful models ready for application at both population and individual level, and testable predictions concerning the sleep patterns that lead to aggregation of waste in the brain and eventual cognitive decline. Project outcomes should also benefit society and the economy though translation into interventions for sleep disturbance – in future applied research.Read moreRead less
Reaching new frontiers of quantum fields and gravity through deformations. This project aims to reach new frontiers in quantum field and gravity theories. These underpin systems ranging from semi-conductors to particle collisions and the quantum behavior of black holes. An obstacle is that these theories are notoriously hard to solve. This project proposes to tackle this longstanding problem by using new deformations, symmetries and dualities that have attracted widespread attention. Expected ou ....Reaching new frontiers of quantum fields and gravity through deformations. This project aims to reach new frontiers in quantum field and gravity theories. These underpin systems ranging from semi-conductors to particle collisions and the quantum behavior of black holes. An obstacle is that these theories are notoriously hard to solve. This project proposes to tackle this longstanding problem by using new deformations, symmetries and dualities that have attracted widespread attention. Expected outcomes will include innovative techniques that will greatly enhance and interconnect our knowledge of field theories and quantum gravity, together with new discoveries in quantum-corrected geometries. A new network of domestic and international experts will largely benefit the fields of theoretical and mathematical physics.Read moreRead less
Symmetry via braiding, diagrammatics and cellularity. Symmetry is a basic organising tool for humans to understand their environment. Invariants are the mathematical embodiment of symmetry, and their study is as ancient as thought itself. This project aims to use the tools of braided tensor categories and cellular structure, to analyse the invariants occurring in several fundamental areas of mathematics, particularly relating to physics. The endomorphism algebras in certain tensor categories, pa ....Symmetry via braiding, diagrammatics and cellularity. Symmetry is a basic organising tool for humans to understand their environment. Invariants are the mathematical embodiment of symmetry, and their study is as ancient as thought itself. This project aims to use the tools of braided tensor categories and cellular structure, to analyse the invariants occurring in several fundamental areas of mathematics, particularly relating to physics. The endomorphism algebras in certain tensor categories, particularly those for quantised superalgebras, will be realised as diagram algebras, and analysed using cellular theory. The intended output include criteria for semisimplicity, a new theory of diagram algebras, and decomposition theory which are expected to permit the determination of multiplicities of composition factors.Read moreRead less
Quantum vertex algebras. The project aims to address major mathematical problems on the structure and representations of the families of quantum groups and vertex algebras associated with Lie algebras. Originating from solvable lattice models in statistical mechanics, the theory of quantum groups has important connections with, and applications to, a wide range of subjects in mathematics and physics. The project will extend and develop explicit theory of both the classical and quantum versions o ....Quantum vertex algebras. The project aims to address major mathematical problems on the structure and representations of the families of quantum groups and vertex algebras associated with Lie algebras. Originating from solvable lattice models in statistical mechanics, the theory of quantum groups has important connections with, and applications to, a wide range of subjects in mathematics and physics. The project will extend and develop explicit theory of both the classical and quantum versions of the vertex algebras which are of importance to conformal field theory and soliton spin-chain models.Read moreRead less