Dissipation and relaxation in statistical mechanics. This project studies the mathematical conditions for relaxation either to equilibrium or to steady states, which is important in predicting behaviour in diverse fields including climate modelling, materials science, nanotechnology and biology. Early career researchers will be involved in the project, gaining valuable skills in theory and simulation.
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
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
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
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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Algebraic Structures and Correlations in Quantum Many-Body Systems. Algebraic structures such as quantized superalgebras are among the most important discoveries in mathematics and have applications in a wide range of physics. Internationally there has been recent excitement about vertex operators and representations of these algebraic structures and their applications to integrable systems and quantum field theory. I have made significant contributions to this rapidly expanding field, and will ....Algebraic Structures and Correlations in Quantum Many-Body Systems. Algebraic structures such as quantized superalgebras are among the most important discoveries in mathematics and have applications in a wide range of physics. Internationally there has been recent excitement about vertex operators and representations of these algebraic structures and their applications to integrable systems and quantum field theory. I have made significant contributions to this rapidly expanding field, and will capitalize on this success. I will develop a comprehensive theory of these mathematical structures and their applications in the construction of correlation functions and form factors, and in so doing write a definitive monograph on the subject.Read moreRead less
Metallic nanograins: superconducting correlations, Josephson tunneling and conformal field theory. Experimental studies of aluminium grains which are a few nanometres in size have exposed unexpected physical characteristics, including pairing interactions which are responsible for bulk superconductivity. Our previous theoretical work has shown that precise information about these nanograins can be gained in the framework of the exact solution of the BCS model. This project will continue our work ....Metallic nanograins: superconducting correlations, Josephson tunneling and conformal field theory. Experimental studies of aluminium grains which are a few nanometres in size have exposed unexpected physical characteristics, including pairing interactions which are responsible for bulk superconductivity. Our previous theoretical work has shown that precise information about these nanograins can be gained in the framework of the exact solution of the BCS model. This project will continue our work in this area with an emphasis on investigating the nature of Josephson tunneling between coupled nanograins. The results of this project will have important applications in emerging technologies such as the implementation of Josephson junctions of nanoscale size.Read moreRead less
Quantum integrable models in nano and mesoscopic physics. The current advances in nanoscale and mesoscopic physics are
generating a wealth of activity with many exciting applications. This
project aims to study several theoretical aspects in three key areas;
the theory of ultrasmall metallic grains of dimensions of a few nanometres,
which through experimental work have shown characteristics which
are similar to macroscopic superconductors, the Nobel Prize
winning phenomenon of Bose-Einstei ....Quantum integrable models in nano and mesoscopic physics. The current advances in nanoscale and mesoscopic physics are
generating a wealth of activity with many exciting applications. This
project aims to study several theoretical aspects in three key areas;
the theory of ultrasmall metallic grains of dimensions of a few nanometres,
which through experimental work have shown characteristics which
are similar to macroscopic superconductors, the Nobel Prize
winning phenomenon of Bose-Einstein condensation in dilute alkali gases and the effects of magnetic impurities in strongly interacting electron systems. The approach of the project is to use the mathematical theory of exactly solvable systems to study these important areas in contemporary physics.
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Indecomposable representation theory. The project aims to develop a systematic approach to the study and application of indecomposable representations in pure mathematics and mathematical physics. Indecomposability is a central concept in representation theory and is thus fundamental to a wide range of applications in science. Examples of important contexts considered are diagram algebras and finite and infinite-dimensional Lie algebras including the Virasoro algebra underlying conformal field t ....Indecomposable representation theory. The project aims to develop a systematic approach to the study and application of indecomposable representations in pure mathematics and mathematical physics. Indecomposability is a central concept in representation theory and is thus fundamental to a wide range of applications in science. Examples of important contexts considered are diagram algebras and finite and infinite-dimensional Lie algebras including the Virasoro algebra underlying conformal field theory. Linear algebra is a ubiquitous mathematical tool playing a pivotal role in representation theory, and the project aims to resolve outstanding fundamental issues concerning families of so-called non-diagonalisable matrices.Read moreRead less