Discrete Projective Differential Geometry: Comprehensive Theory and Integrable Structure. Differential geometry has been developed over centuries by the most distinguished of mathematicians and its applicability in the mathematical and physical sciences is beyond doubt. However, both natural and man-made structures are inherently discrete. Discrete differential geometry constitutes a relatively new and active research area located between pure and applied mathematics which is more fundamental th ....Discrete Projective Differential Geometry: Comprehensive Theory and Integrable Structure. Differential geometry has been developed over centuries by the most distinguished of mathematicians and its applicability in the mathematical and physical sciences is beyond doubt. However, both natural and man-made structures are inherently discrete. Discrete differential geometry constitutes a relatively new and active research area located between pure and applied mathematics which is more fundamental than differential geometry in that it aims to establish an autonomous discrete analogue from which differential geometry may be derived via an appropriate continuum limit. Even though discrete differential geometry has reached a high degree of sophistication, this project seeks to deliver the first comprehensive theory in this area. Read moreRead less
Algebraic algorithms for investigating the space of bacterial genomes. Understanding evolutionary processes and the way organisms are related is a fundamental objective of the biological sciences. This project brings the power of group theory and computation to bear on these problems, developing new ways of understanding them and new tools to address them.
Explicit methods in number theory: Computation, theory and application. This project aims to use explicit estimates to unify three problems in number theory: primitive roots, Diophantine quintuples, and linear independence of zeroes of the Riemann zeta-function. It will use computational and analytic number theory to reduce the quintuples problem to a soluble level. Pursuing relations between the zeta zeroes will overhaul many current results. This project will apply its findings about primitive ....Explicit methods in number theory: Computation, theory and application. This project aims to use explicit estimates to unify three problems in number theory: primitive roots, Diophantine quintuples, and linear independence of zeroes of the Riemann zeta-function. It will use computational and analytic number theory to reduce the quintuples problem to a soluble level. Pursuing relations between the zeta zeroes will overhaul many current results. This project will apply its findings about primitive roots to signal processing, cryptography and cybersecurity.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.
Variational theory for fully nonlinear elliptic equations. This project aims to develop new methods and techniques to solve challenging mathematical problems in fully nonlinear partial differential equations arising in important applications. The project will develop methods and techniques to study these equations’ regularity and variational properties. This project is expected to establish comprehensive theories and enhance and promote Australian participation and leadership in this area of mat ....Variational theory for fully nonlinear elliptic equations. This project aims to develop new methods and techniques to solve challenging mathematical problems in fully nonlinear partial differential equations arising in important applications. The project will develop methods and techniques to study these equations’ regularity and variational properties. This project is expected to establish comprehensive theories and enhance and promote Australian participation and leadership in this area of mathematics.Read moreRead less
Propagation described by partial differential equations with free boundary. Cutting edge nonlinear mathematics is required to understand many important propagation phenomena in nature, such as the spreading of invasive species or nerve signals. This project aims to systematically investigate nonlinear partial differential equation models that govern the dynamics of such propagations, with emphasis on the development of new approaches that enable deeper insights on the evolution of the propagatin ....Propagation described by partial differential equations with free boundary. Cutting edge nonlinear mathematics is required to understand many important propagation phenomena in nature, such as the spreading of invasive species or nerve signals. This project aims to systematically investigate nonlinear partial differential equation models that govern the dynamics of such propagations, with emphasis on the development of new approaches that enable deeper insights on the evolution of the propagating fronts. The project aims to develop new mathematics for new applications of lasting values.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE170101410
Funder
Australian Research Council
Funding Amount
$330,324.00
Summary
Nonlinear free boundary problems: Propagation and regularity. This project aims to understand the propagation profile and regularity of two important classes of free boundary problems. Nonlinear free boundary problems arise from many applied fields, and pose great challenges to the theory of nonlinear partial differential equations, as the underlying domain of the solution to such problems has to be solved together with the solution itself. This research is expected to enhance the existing theor ....Nonlinear free boundary problems: Propagation and regularity. This project aims to understand the propagation profile and regularity of two important classes of free boundary problems. Nonlinear free boundary problems arise from many applied fields, and pose great challenges to the theory of nonlinear partial differential equations, as the underlying domain of the solution to such problems has to be solved together with the solution itself. This research is expected to enhance the existing theory of partial differential equations, and extend its applications to new situations such as flow through porous media and spreading of invasive species.Read moreRead less
Propagation and free boundary problems in nonlinear partial differential equations. Understanding the propagation of invasive species, flames and disadvantageous genes is a challenging problem in many areas of modern science. This project develops a new mathematical approach to better understand such propagation problems, where the mathematical model predicts a precise location of the propagating front for future time.