Noncommutative geometry in representation theory and quantum physics. One of the most important problems in natural science is to understand the structure of spacetime at the Planck scale. Mathematical investigations in recent years have predicted that at this scale, spacetime becomes noncommutative. Taking this noncommutativity into account, the project brings together geometry, algebra and quantum mechanics to develop new mathematical theories required for addressing the problem. It promises ....Noncommutative geometry in representation theory and quantum physics. One of the most important problems in natural science is to understand the structure of spacetime at the Planck scale. Mathematical investigations in recent years have predicted that at this scale, spacetime becomes noncommutative. Taking this noncommutativity into account, the project brings together geometry, algebra and quantum mechanics to develop new mathematical theories required for addressing the problem. It promises to make fundamental contributions to both mathematics and theoretical physics. Read moreRead less
Geometry and representations of classical and quantum Lie supergroups. The physical notion of supersymmetry is a unifying principle which ensures that bosonic and fermionic particles in quantum physics obey the same fundamental laws. It has permeated the forefront of mathematical research since 1980s, leading to the creation of some of the deepest theories in diverse areas. The mathematical foundation of supersymmetry lies in the theory of Lie supergroups. This project addresses major outstandin ....Geometry and representations of classical and quantum Lie supergroups. The physical notion of supersymmetry is a unifying principle which ensures that bosonic and fermionic particles in quantum physics obey the same fundamental laws. It has permeated the forefront of mathematical research since 1980s, leading to the creation of some of the deepest theories in diverse areas. The mathematical foundation of supersymmetry lies in the theory of Lie supergroups. This project addresses major outstanding problems in the geometry and representations of Lie supergroups and their quantum analogues. Results will be important to the quest for a consistent quantum theory of all the four interactions in nature.Read moreRead less
Infinite Dimensional Unitarizable Representations of Lie Superalgebras. The project addresses major outstanding mathematical problems, which are of fundamental importance to the development of a unified theory of all four interactions in quantum physics. Mathematics is essential for the understanding of our own rationality. Advances in the field promised by this project are of intrinsic value. Physics is the foundation of modern technology. Success of the project will help to create a scientif ....Infinite Dimensional Unitarizable Representations of Lie Superalgebras. The project addresses major outstanding mathematical problems, which are of fundamental importance to the development of a unified theory of all four interactions in quantum physics. Mathematics is essential for the understanding of our own rationality. Advances in the field promised by this project are of intrinsic value. Physics is the foundation of modern technology. Success of the project will help to create a scientific environment in Australia that fosters technological creativity and innovation. Results of the project will greatly enhance the scientific reputation of Australia internationally, attracting foreign researchers and Ph.D students to Australian shores. Read moreRead less
Quantum algebras: their symmetries, invariants and representations. The project addresses major outstanding mathematical problems, which are of fundamental importance to theoretical physics. The algebraic structures originated from statistical mechanics will be investigated by methods of modern mathematics. Successful completion of the project will provide physicists with important new tools for investigating the symmetry of phenomena such as quantum gravity and spinor reflections. Success of th ....Quantum algebras: their symmetries, invariants and representations. The project addresses major outstanding mathematical problems, which are of fundamental importance to theoretical physics. The algebraic structures originated from statistical mechanics will be investigated by methods of modern mathematics. Successful completion of the project will provide physicists with important new tools for investigating the symmetry of phenomena such as quantum gravity and spinor reflections. Success of the project will help to create a scientific environment in Australia that fosters technological creativity and innovation. Results of the project will greatly enhance the scientific reputation of Australia internationally, attracting foreign researchers and PhD students to Australia.Read moreRead less
Special Research Initiatives - Grant ID: SR0354716
Funder
Australian Research Council
Funding Amount
$10,000.00
Summary
Energetically Open Systems Research Network Study. Conceptual frameworks arising in the physical sciences, such as non-equilibrium statistical mechanics and thermodynamics, synergetics, chaos and dynamical systems theory, are seminal in the emerging science of complexity. This study will lay the groundwork for a network to link Australian and overseas research on these fundamental concepts, and their application within the context of entropy-producing systems vital to the long-term sustainabilit ....Energetically Open Systems Research Network Study. Conceptual frameworks arising in the physical sciences, such as non-equilibrium statistical mechanics and thermodynamics, synergetics, chaos and dynamical systems theory, are seminal in the emerging science of complexity. This study will lay the groundwork for a network to link Australian and overseas research on these fundamental concepts, and their application within the context of entropy-producing systems vital to the long-term sustainability of the earth - oceans, atmosphere, biosphere, CO2-free energy production, space and solar environment. The network would facilitate the development of young investigators and be linked into wider complex systems networks such as the CSIRO Centre for Complex Systems Science.Read moreRead less
Super Duality and Deformations in the Representation Theory of Lie Superalgebras. Supersymmetry has remained in a central stage of fundamental research in both physics and mathematics for the last forty years. It is currently being tested by experiments of massive scales conducted on the Large Hadron Collider at CERN in Geneva. The present project aims to create new mathematical concepts and techniques for addressing fundamental issues of supersymmetry. Expected outcomes include new types of Bos ....Super Duality and Deformations in the Representation Theory of Lie Superalgebras. Supersymmetry has remained in a central stage of fundamental research in both physics and mathematics for the last forty years. It is currently being tested by experiments of massive scales conducted on the Large Hadron Collider at CERN in Geneva. The present project aims to create new mathematical concepts and techniques for addressing fundamental issues of supersymmetry. Expected outcomes include new types of Bose-Fermi correspondence, a deformation theory of Lie superalgebra representations, algebraic and geometric treatments of Jantzen filtration of parabolic Verma modules of Lie superalgebras, and quantum field theoretical models for the topological invariants of knots and 3-manifolds arising from quantum supergroups. Read moreRead less
Geometric themes in the theory of Lie supergroups and their quantisations. This project aims to develop mathematics on the geometry of super spaces and the algebra of super transformations, which are the cornerstones of the mathematical foundation of supersymmetry. The Large Hadron Collider at the European Organization for Nuclear Research is investigating supersymmetry as a possible symmetry of fundamental physics. Its empirical verification would confirm the existence of new constituents of ma ....Geometric themes in the theory of Lie supergroups and their quantisations. This project aims to develop mathematics on the geometry of super spaces and the algebra of super transformations, which are the cornerstones of the mathematical foundation of supersymmetry. The Large Hadron Collider at the European Organization for Nuclear Research is investigating supersymmetry as a possible symmetry of fundamental physics. Its empirical verification would confirm the existence of new constituents of matter, and reveal deep structures of space-time beyond the framework of Einstein's general relativity. Results of the project are expected to be directly applicable to high energy physics.Read moreRead less
Quantised algebras, supersymmetry and invariant theory. The discriminant of a quadratic equation is an invariant which most high school students learn about; it does not change under linear substitution of the variables. This project will develop new theorems about quantum invariants, which occur in quantum and super symmetry. Links will be forged with physics and quantum computing.
Modular Index Theory. This project capitilises on Australian advances in mathematics, particularly noncommutative geometry. It will maintain and extend Australia's prominence in this subject, providing excellent opportunities for young researchers via the research networks this project will establish. Being at the interface of ideas in mathematics and physics, there is potential for future technological spin offs for Australia.
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