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.
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
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
Counting points on algebraic surfaces. This project aims to develop algorithms for calculating the number of solutions to polynomial equations and to compute zeta functions of certain types of algebraic varieties. Existing algorithms cannot solve these problems. The new algorithms will enable researchers in number theory to test and refine conjectures on generalisations of many famous problems, such as the Sato-Tate conjecture, the Lang-Trotter conjecture and the Birch-Swinnerton-Dyer conjecture ....Counting points on algebraic surfaces. This project aims to develop algorithms for calculating the number of solutions to polynomial equations and to compute zeta functions of certain types of algebraic varieties. Existing algorithms cannot solve these problems. The new algorithms will enable researchers in number theory to test and refine conjectures on generalisations of many famous problems, such as the Sato-Tate conjecture, the Lang-Trotter conjecture and the Birch-Swinnerton-Dyer conjecture. The project will also have a flow-on effect in other areas of mathematics and computer science where zeta functions play a central role, including cryptography, coding theory and mathematical physics.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE220100859
Funder
Australian Research Council
Funding Amount
$354,000.00
Summary
New techniques for exponential sums over low degree polynomials. This project aims to obtain new quantitative estimates for Weyl sums over low degree polynomials. Such estimates are fundamental to several areas of number theory. By interfacing techniques from diverse areas of mathematics, including algebraic geometry, analytic number theory, the geometry of numbers and harmonic analysis, this project will provide the first progress on estimating Weyl sums over low degree polynomials in over a ce ....New techniques for exponential sums over low degree polynomials. This project aims to obtain new quantitative estimates for Weyl sums over low degree polynomials. Such estimates are fundamental to several areas of number theory. By interfacing techniques from diverse areas of mathematics, including algebraic geometry, analytic number theory, the geometry of numbers and harmonic analysis, this project will provide the first progress on estimating Weyl sums over low degree polynomials in over a century. The expected outcomes include a deeper understanding of Weyl sums and enhanced international collaborations. Such progress will place Australia at the forefront of this important branch of number theory.Read moreRead less