Towards logarithmic representation theory of W-algebras. Aims: To construct and analyse indecomposable representations of significance in conformal field theory.
Significance: Conformal field theory plays a key role in many developments in mathematics and physics. Logarithmic conformal field theories govern important systems such as two-dimensional critical percolation. This proposal aims to develop the representation theory necessary for understanding salient features of critical systems des ....Towards logarithmic representation theory of W-algebras. Aims: To construct and analyse indecomposable representations of significance in conformal field theory.
Significance: Conformal field theory plays a key role in many developments in mathematics and physics. Logarithmic conformal field theories govern important systems such as two-dimensional critical percolation. This proposal aims to develop the representation theory necessary for understanding salient features of critical systems described by logarithmic conformal field theory.
Expected Outcomes: Novel representations of fundamental importance in logarithmic conformal field theory.
Benefit: Resolution of open problems in logarithmic conformal field theory, thus continuing the strong tradition in the field in Australia.
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Transformative simulation techniques for complex polymer networks. The study of long chain polymers like DNA using computer simulations has uncovered exciting insights over many years. Generally these have been limited to simple topologies, interactions, and environments. This project aims to develop the next generation of simulation techniques to tackle a new frontier of polymer models, including those with complex topologies like stars, knots, and links, which have hitherto been inaccessible. ....Transformative simulation techniques for complex polymer networks. The study of long chain polymers like DNA using computer simulations has uncovered exciting insights over many years. Generally these have been limited to simple topologies, interactions, and environments. This project aims to develop the next generation of simulation techniques to tackle a new frontier of polymer models, including those with complex topologies like stars, knots, and links, which have hitherto been inaccessible. Expected outcomes include new simulation methods which harness modern computational clusters, leading to greater understanding of polymers with complex topologies and in complicated environments. Important elements of biological processes may be discovered, such as how polymer structure affects DNA transcription.Read moreRead less
Quantum control designed from broken integrability. This Project aims to open new avenues in quantum device engineering design. This will be achieved through the use of advanced mathematical methodologies developed around the notion of quantum integrability, and the breaking of that integrability. The expert team of Investigators will capitalise on their recent achievements in this field, which includes a first example of a quantum switch designed through broken integrability. The expected outco ....Quantum control designed from broken integrability. This Project aims to open new avenues in quantum device engineering design. This will be achieved through the use of advanced mathematical methodologies developed around the notion of quantum integrability, and the breaking of that integrability. The expert team of Investigators will capitalise on their recent achievements in this field, which includes a first example of a quantum switch designed through broken integrability. The expected outcomes will encompass novel applications of abstract mathematical physics towards the concrete control of quantum mechanical architectures. These outcomes will promote new opportunities for the construction of atomtronic devices, which are rising as a foundation for next-generation quantum technologies.
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Free parafermions: a challenge for non-Hermitian physics. This project aims to calculate and understand the physical properties of free parafermions. Parafermions have attracted interest in topological schemes for quantum computation because they are computationally more powerful than Majorana fermions. The core of this project is a fundamental model of free parafermions, which has been shown to exhibit unexplained puzzling properties. The project outcomes include an in-depth understanding of th ....Free parafermions: a challenge for non-Hermitian physics. This project aims to calculate and understand the physical properties of free parafermions. Parafermions have attracted interest in topological schemes for quantum computation because they are computationally more powerful than Majorana fermions. The core of this project is a fundamental model of free parafermions, which has been shown to exhibit unexplained puzzling properties. The project outcomes include an in-depth understanding of this model by taking the non-Hermitian features into account, establishing a connection with open quantum systems. Non-Hermitian systems are also of increasing relevance in physics, especially in quantum optics. The project also aims to contribute to training researchers in the mathematical sciences.
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Shuffle algebras and vertex models. Shuffle algebras are important new mathematical structures that offer a new approaches and techniques to solve outstanding open problems in a variety of branches of mathematics, including mathematical physics, algebraic geometry and combinatorics. This project proposes to find solutions to key open problems using connections between shuffle algebras and integrable lattice models. The expected outcomes include (i) a new framework of shuffle algebra techniques t ....Shuffle algebras and vertex models. Shuffle algebras are important new mathematical structures that offer a new approaches and techniques to solve outstanding open problems in a variety of branches of mathematics, including mathematical physics, algebraic geometry and combinatorics. This project proposes to find solutions to key open problems using connections between shuffle algebras and integrable lattice models. The expected outcomes include (i) a new framework of shuffle algebra techniques to solve challenging research problems in mathematical physics and statistical mechanics, (ii) practical and computationally feasible constructions of shuffle algebras using vertex models, (iii) solutions to unresolved spectral problems of open quantum systems.Read moreRead less
Matrix product multi-variable polynomials from quantum algebras. This project aims to expand the theory of polynomials and develop generalised polynomial families using connections to affine and toroidal algebras. Many combinatorial and computational problems in pure and applied mathematics as well as mathematical physics can be solved using polynomials in many variables, such as Macdonald polynomials. This project is anticipated to address the current difficulty of implementing symmetric and no ....Matrix product multi-variable polynomials from quantum algebras. This project aims to expand the theory of polynomials and develop generalised polynomial families using connections to affine and toroidal algebras. Many combinatorial and computational problems in pure and applied mathematics as well as mathematical physics can be solved using polynomials in many variables, such as Macdonald polynomials. This project is anticipated to address the current difficulty of implementing symmetric and non-symmetric polynomials in symbolic algebra packages by developing completely new algorithms. New understanding from the project is expected to facilitate challenging computational problems of measurable quantities in quantum systems.Read moreRead less
Increasing the efficiency and interpretability of stepped wedge trials. Stepped wedge cluster randomised trials are increasingly being used to test interventions, across many disciplines. This project aims to develop highly efficient trial designs and new methods for the estimation of causally interpretable effects when adherence to interventions is not perfect. This project expects to generate new design types that reduce resources required to test interventions, and methods to understand how t ....Increasing the efficiency and interpretability of stepped wedge trials. Stepped wedge cluster randomised trials are increasingly being used to test interventions, across many disciplines. This project aims to develop highly efficient trial designs and new methods for the estimation of causally interpretable effects when adherence to interventions is not perfect. This project expects to generate new design types that reduce resources required to test interventions, and methods to understand how these interventions work. Expected outcomes include tools to help researchers develop cheaper and more appealing trials, tools to estimate causal effects, the methodology underpinning these tools, and new collaborations. This should provide significant benefits by allowing more interventions to be tested and understood.Read moreRead less
From superintegrability to quasi-exact solvability: theory and application. This project aims to develop mathematical techniques to resolve longstanding problems in the area of integrability and exact solvability. Quantum integrable systems and exact solvable models are of central importance for understanding the correct behaviours of complex quantum problems without approximation. This project aims to construct sophisticated mathematical tools to settle key questions across a variety of models ....From superintegrability to quasi-exact solvability: theory and application. This project aims to develop mathematical techniques to resolve longstanding problems in the area of integrability and exact solvability. Quantum integrable systems and exact solvable models are of central importance for understanding the correct behaviours of complex quantum problems without approximation. This project aims to construct sophisticated mathematical tools to settle key questions across a variety of models such as superintegrable systems, quantum spin chains, and spin-boson models. Anticipated applications of the proposed research include the accurate prediction of physical phenomena, from energy spectra to quantum correlations. Such advances should have significant ramifications, and provide benefits, well beyond the mathematical discipline itself.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