Mathematics of the quantum-classical mechanics interface. Nanotechnology focusses increasing attention on the interface between quantum and classical mechanics. Semiclassical approximations have long been studied, as a means to describe classical systems with 'small' actions as this interface is approached from the classical side. I have recently shown that classical mechanics can be formulated in complex Hilbert space, as a pseudo-quantum theory. This establishes a framework for the developme ....Mathematics of the quantum-classical mechanics interface. Nanotechnology focusses increasing attention on the interface between quantum and classical mechanics. Semiclassical approximations have long been studied, as a means to describe classical systems with 'small' actions as this interface is approached from the classical side. I have recently shown that classical mechanics can be formulated in complex Hilbert space, as a pseudo-quantum theory. This establishes a framework for the development of 'semiquantum' approximations, to enable the description of quantum systems with 'large' actions as the quantum-classical interface is approached from the quantum side. The project aims to explore some ramifications of this theoretical breakthrough.Read moreRead less
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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Discovery Early Career Researcher Award - Grant ID: DE130101067
Funder
Australian Research Council
Funding Amount
$302,540.00
Summary
New constructions of superintegrable systems and the connection with Painlevé transcendents. The research of this project will lead to deep discoveries in the field of superintegrability and expand our knowledge of their related algebraic structures, supersymmetric quantum mechanics and Painlevé transcendents. The project will generate new techniques that will be utilised in future applications of mathematical and theoretical physics.
Unified theory of Richardson-Gaudin integrability. Richardson-Gaudin systems form a class of mathematical models of interacting particles that serve as a foundation to understand important phenomena in modern physics. Being integrable, these quantum systems enable deep insights. They are tractable so as to allow for exact analysis, while being elaborate enough to exhibit complex physical properties, notably phase transitions. The international team of researchers aims to merge various approaches ....Unified theory of Richardson-Gaudin integrability. Richardson-Gaudin systems form a class of mathematical models of interacting particles that serve as a foundation to understand important phenomena in modern physics. Being integrable, these quantum systems enable deep insights. They are tractable so as to allow for exact analysis, while being elaborate enough to exhibit complex physical properties, notably phase transitions. The international team of researchers aims to merge various approaches for analysing the integrability of such models. Successful outcomes are expected to produce inventive mathematical techniques, linking a diverse range of fields of current activity and growth. The resulting unified theory is expected to open the door to exciting and innovative pathways in mathematical physics research.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 Integrable Systems and Applications: From Condensed Matter to Quantum Information. Quantum integrable systems have produced exciting results and techniques vital in the efforts to achieve the ultimate goal of understanding quantum science beyond perturbation. The proposal gathers four world experts from Australia, Japan and Russia to work on highly interdisciplinary projects designed to resolve fundamental problems in the field, which will underpin the development of emerging technologie ....Quantum Integrable Systems and Applications: From Condensed Matter to Quantum Information. Quantum integrable systems have produced exciting results and techniques vital in the efforts to achieve the ultimate goal of understanding quantum science beyond perturbation. The proposal gathers four world experts from Australia, Japan and Russia to work on highly interdisciplinary projects designed to resolve fundamental problems in the field, which will underpin the development of emerging technologies. As a result, Australian science will be seen to be at the forefront internationally, and the leading status of Australia in the field will be greatly strengthened. Early career researchers and PhD students will be trained as part of the project, important in enhancing Australia's capability to develop and retain scientific talent. Read moreRead less
ARC Complex Open Systems Research Network. Complexity is the common frontier in the physical, biological and social sciences. This Network will link specialists in all three sciences through five generic conceptual and mathematical theme activities. It will promote research into how subsystems self-organise into new emergent structures when assembled into an open, non-equilibrium system. Outcomes will include new technologies and software tools and deeper understanding of fundamental questions i ....ARC Complex Open Systems Research Network. Complexity is the common frontier in the physical, biological and social sciences. This Network will link specialists in all three sciences through five generic conceptual and mathematical theme activities. It will promote research into how subsystems self-organise into new emergent structures when assembled into an open, non-equilibrium system. Outcomes will include new technologies and software tools and deeper understanding of fundamental questions in science. An essential function of the network will be introducing researchers end users to new tools and broadening the horizons of graduate students.Read moreRead less
Fluctuations in the properties of nonequilibrium fluids and the influence of thermostatting mechanisms. The behaviour of nonequilibrium fluids will be studied by combining ideas from liquid state theory, statistical mechanics and dynamical systems theory. This work will result in development and testing of mathematical expressions (Fluctuation Theorems) that are consistent with the Second Law of Thermodynamics, which determines the direction of any change in any macroscopic system, but are also ....Fluctuations in the properties of nonequilibrium fluids and the influence of thermostatting mechanisms. The behaviour of nonequilibrium fluids will be studied by combining ideas from liquid state theory, statistical mechanics and dynamical systems theory. This work will result in development and testing of mathematical expressions (Fluctuation Theorems) that are consistent with the Second Law of Thermodynamics, which determines the direction of any change in any macroscopic system, but are also applicable to microscopic systems. The expressions will determine the probability that finite sized systems will violate the Second Law for small periods of time and will therefore contribute to development of a fundamental understanding of microscopic systems and the development of nanotechnology.
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Fluid properties and chaotic dynamics in equilibrium and nonequilibrium states. Over the last decade a revolution has been taking place in nonequilibrium statistical mechanics [Physics Today, Sept, 2002]. This revolution is characterized by adapting the mathematical theory of chaos to nonequilibrium statistical mechanics. Fundamental new theorems and algorithms for computing transport coefficients have been derived. The CIs have played a key role in this revolution. We seek to broaden these dev ....Fluid properties and chaotic dynamics in equilibrium and nonequilibrium states. Over the last decade a revolution has been taking place in nonequilibrium statistical mechanics [Physics Today, Sept, 2002]. This revolution is characterized by adapting the mathematical theory of chaos to nonequilibrium statistical mechanics. Fundamental new theorems and algorithms for computing transport coefficients have been derived. The CIs have played a key role in this revolution. We seek to broaden these developments by: generalizing a theorem which relates transport coefficients to chaoticity; detailed studies of the influence of thermostatting mechanisms on nonequilibrium chaoticity and fluctuations, and by understanding the range of applicability of a nonequilibrium fluctuation theorem for non-isoenergetic systems.Read moreRead less
Integrable quantum systems: mathematical foundations for developing quantum science. Quantum science is an exciting and challenging area, one which will underpin the development of the next generation of computers and novel devices such as atom lasers. New mathematical techniques are being pursued, to formulate the frameworks that will provide deep insights into the complex nature of the physical principles governing this field, in order to fully realise the potential applications. This project ....Integrable quantum systems: mathematical foundations for developing quantum science. Quantum science is an exciting and challenging area, one which will underpin the development of the next generation of computers and novel devices such as atom lasers. New mathematical techniques are being pursued, to formulate the frameworks that will provide deep insights into the complex nature of the physical principles governing this field, in order to fully realise the potential applications. This project will enhance the scale of an established and internationally competitive program in mathematics research, producing new approaches to meet these demands. It will also provide opportunities for research training, important in ensuring that Australia is well equipped to play a leading role in future quantum science developments.
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