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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Algebraic and computational approaches for classical and quantum systems. This project aims to use a combination of algebraic, analytic and numerical techniques to develop computational algorithms to address a range of notoriously challenging problems in the mathematical sciences. These problems involve predicting the large-scale behaviour of strongly interacting classical and quantum spin systems originating in condensed matter physics, including models of relevance to proposals for topological ....Algebraic and computational approaches for classical and quantum systems. This project aims to use a combination of algebraic, analytic and numerical techniques to develop computational algorithms to address a range of notoriously challenging problems in the mathematical sciences. These problems involve predicting the large-scale behaviour of strongly interacting classical and quantum spin systems originating in condensed matter physics, including models of relevance to proposals for topological quantum computation and the latest progress using field theory. The project outcomes will involve advances in understanding these systems from new exact results and high precision numerical estimates.Read moreRead less
Composition tree algorithms for large matrix groups. This project aims to develop new algorithms for analysing groups. A group is a rather simple mathematical structure – an example is the set of all integers considering only the operations of addition and subtraction. Since the symmetries of an object form a group, groups are ubiquitous throughout mathematics and elsewhere in science. Because it is frequently necessary to determine a group's properties, there is great interest in finding effici ....Composition tree algorithms for large matrix groups. This project aims to develop new algorithms for analysing groups. A group is a rather simple mathematical structure – an example is the set of all integers considering only the operations of addition and subtraction. Since the symmetries of an object form a group, groups are ubiquitous throughout mathematics and elsewhere in science. Because it is frequently necessary to determine a group's properties, there is great interest in finding efficient algorithms for analysing groups. A matrix group is a common type of group whose elements are square matrices. This project plans to employ a novel approach to designing algorithms for analysing large matrix groups, a task which is currently impossible using existing algorithms.Read moreRead less
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
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.
New approaches and applications of integrable quantum field theory. This project aims to develop new mathematical approaches to the theory of integrable systems to obtain exact solutions of various non-linear models of two-dimensional quantum field theory. The project is based on an unexpected correspondence between classical and quantum systems which provides a powerful method for describing physically interesting models of integrable quantum field theory. Expected outcomes include exact soluti ....New approaches and applications of integrable quantum field theory. This project aims to develop new mathematical approaches to the theory of integrable systems to obtain exact solutions of various non-linear models of two-dimensional quantum field theory. The project is based on an unexpected correspondence between classical and quantum systems which provides a powerful method for describing physically interesting models of integrable quantum field theory. Expected outcomes include exact solutions to non-linear sigma-models which have important applications in many areas, including condensed matter physics, string and field theories and Riemannian geometry. The project expects to provide significant benefit to the advancement of knowledge in physics and mathematics.Read moreRead less
Symmetry and computation. The overall objective of the project is to explore connections between symmetry and computation, especially the theory and algorithms that facilitate the use of groups in computational science. The main outcome will be theoretically fast algorithms and implementations to drive applications in the sciences and for secure communication.
Algebraic categories and categorical algebra. Algebra is the study of operations, such as addition and multiplication, and the relationships between these operations. This project will study two exciting new branches of algebra, quantum algebra and postmodern algebra, which will lead to important advances in physics, geometry, and computing.
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