Classical and affine W-algebras. The project aims to address major mathematical problems on representations of the families of quantum groups and vertex algebras associated with Lie algebras. It aims to create new connections between representation theory and mathematical physics. The theory of quantum groups originated from solvable lattice models in statistical mechanics and has turned out to have important connections with and applications to a wide range of subjects in mathematics and physic ....Classical and affine W-algebras. The project aims to address major mathematical problems on representations of the families of quantum groups and vertex algebras associated with Lie algebras. It aims to create new connections between representation theory and mathematical physics. The theory of quantum groups originated from solvable lattice models in statistical mechanics and has turned out to have important connections with and applications to a wide range of subjects in mathematics and physics. The project aims to rely on these connections to extend and develop explicit theory of both the classical and quantum versions of the vertex algebras which are of great importance to conformal field theory and soliton spin chain models.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
Vertex algebras and representations of quantum groups. The project will tackle mathematical problems involving algebraic structures that have fascinated scientists for several decades, and which are of fundamental importance to theoretical physics. The research will attract talented PhD students and visiting researchers, and will enhance Australia's scientific reputation.
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
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.
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
Symmetries of subfactors. A subfactor is a mathematical object that encodes "quantum" symmetries which may be thought of as generalisations of group symmetries. This project will study subfactors and classify families of subfactor symmetries which include the exotic subfactors of small index. It will also develop computational tools for analysing and cataloguing these symmetries. This project contributes to the development of operator algebra theory, and the new mathematical fields of quantum al ....Symmetries of subfactors. A subfactor is a mathematical object that encodes "quantum" symmetries which may be thought of as generalisations of group symmetries. This project will study subfactors and classify families of subfactor symmetries which include the exotic subfactors of small index. It will also develop computational tools for analysing and cataloguing these symmetries. This project contributes to the development of operator algebra theory, and the new mathematical fields of quantum algebra and quantum topology; it also has applications to physical models.Read moreRead less