Discovery Early Career Researcher Award - Grant ID: DE230100579
Funder
Australian Research Council
Funding Amount
$445,754.00
Summary
The existence and abundance of small bases of permutation groups. This project aims to study bases for permutation groups, which are the mathematical formalisation of symmetry. Bases are crucial to encoding and computing with groups in diverse areas of science. Small bases are desirable for efficiency, but can be hard to find. This project expects to combine techniques from areas of algebra and probability to determine the existence and abundance of bases. Expected outcomes of this project inclu ....The existence and abundance of small bases of permutation groups. This project aims to study bases for permutation groups, which are the mathematical formalisation of symmetry. Bases are crucial to encoding and computing with groups in diverse areas of science. Small bases are desirable for efficiency, but can be hard to find. This project expects to combine techniques from areas of algebra and probability to determine the existence and abundance of bases. Expected outcomes of this project include new methods to address enduring open problems in the study of bases, as well as novel applications of existing techniques. This should provide significant benefits, such as creating and strengthening international collaborations, and building on Australia’s reputation as a powerhouse of finite group theory.Read moreRead less
Partial differential equation: Schrodinger operator and long-time dynamics. This project aims to develop new analysis methods associated to the Schrodinger operator, and to solve several challenging problems regarding dispersive partial differential equations (PDE). Long-time dynamics of PDE solutions are a key goal in both pure and applied mathematics, and have been extensively studied by leading mathematicians and mathematical physicists. However, it is unknown how to investigate large soluti .... Partial differential equation: Schrodinger operator and long-time dynamics. This project aims to develop new analysis methods associated to the Schrodinger operator, and to solve several challenging problems regarding dispersive partial differential equations (PDE). Long-time dynamics of PDE solutions are a key goal in both pure and applied mathematics, and have been extensively studied by leading mathematicians and mathematical physicists. However, it is unknown how to investigate large solutions when the order of the PDE's nonlinearity is low. This project expects to develop new methods to attack such problems. The results of the project will be of great importance in mathematics and physics, as many fundamental physical models in areas such as optics, fluid mechanics and quantum mechanics fit the paradigm.Read moreRead less
Fractional decomposition of graphs and the Nash-Williams conjecture. Nash-Williams' conjecture is a famous unsolved problem about decomposing graphs (abstract networks). Breakthrough results achieved in recent years have shown that the conjecture, along with other major graph decomposition problems, could be solved if only more were known about fractional decomposition. This project aims to clear this bottleneck to progress by dramatically expanding the state of knowledge on fractional decomposi ....Fractional decomposition of graphs and the Nash-Williams conjecture. Nash-Williams' conjecture is a famous unsolved problem about decomposing graphs (abstract networks). Breakthrough results achieved in recent years have shown that the conjecture, along with other major graph decomposition problems, could be solved if only more were known about fractional decomposition. This project aims to clear this bottleneck to progress by dramatically expanding the state of knowledge on fractional decomposition. Expected outcomes include major progress on Nash-Williams' conjecture and related graph decomposition problems. This should enhance Australia's research reputation in pure mathematics and provide benefits in downstream applications areas including statistics, data transmission, and fibre-optic networks.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE230101165
Funder
Australian Research Council
Funding Amount
$419,420.00
Summary
Geometric Scattering Theory, Resolvent Estimates, and Wave Asymptotics. This project aims to understand how fast the local energy of a wave decays when it propagates in a rough, open system. This projects will generate new knowledge in the mathematical subfields of microlocal analysis and partial differential equations by refining tools such as Carleman estimates, separation of variables, b-vector field analysis, and quasimode constructions. The expected outcome of this project is a novel and co ....Geometric Scattering Theory, Resolvent Estimates, and Wave Asymptotics. This project aims to understand how fast the local energy of a wave decays when it propagates in a rough, open system. This projects will generate new knowledge in the mathematical subfields of microlocal analysis and partial differential equations by refining tools such as Carleman estimates, separation of variables, b-vector field analysis, and quasimode constructions. The expected outcome of this project is a novel and comprehensive mathematical treatment of wave propagation in systems with weaker than Lipschitz regularity. This research should provide significant benefits such as informing predictions about waves in rough systems, including the propagation of seismic waves, and lead to advances in medical and geological imaging.Read moreRead less
Mathematics for future magnetic devices. The aim of this project is to develop a mathematical theory and numerical models of stochastic partial differential
equations for magnetic nano-structures. Such materials will yield next-generation magnetic memories with up to
three orders of magnitude faster switching speeds and dramatically increased data storage density. New
mathematical theories will help understand their sensitivity to small random fluctuations that can destroy stored
information. Th ....Mathematics for future magnetic devices. The aim of this project is to develop a mathematical theory and numerical models of stochastic partial differential
equations for magnetic nano-structures. Such materials will yield next-generation magnetic memories with up to
three orders of magnitude faster switching speeds and dramatically increased data storage density. New
mathematical theories will help understand their sensitivity to small random fluctuations that can destroy stored
information. This project aims to revolutionise mathematical modelling of magnetic memories and put Australia at
the forefront of international research. Technological advances to create much smaller and faster memory devices
are expected to enable groundbreaking ways of managing and mining big dataRead moreRead less
Geodesic arcs and surfaces for hyperbolic knots and 3-manifolds. This project aims to use recent breakthroughs in mathematics to determine explicit geometric information on mathematical spaces, namely knot complements and 3-manifolds. These spaces arise in applications across science and engineering. They break into pieces that admit geometry, where hyperbolic geometry is the most common. This project expects to generate new knowledge around a number of open questions and conjectures on the hype ....Geodesic arcs and surfaces for hyperbolic knots and 3-manifolds. This project aims to use recent breakthroughs in mathematics to determine explicit geometric information on mathematical spaces, namely knot complements and 3-manifolds. These spaces arise in applications across science and engineering. They break into pieces that admit geometry, where hyperbolic geometry is the most common. This project expects to generate new knowledge around a number of open questions and conjectures on the hyperbolic geometry of knots and 3-manifolds. Expected outcomes include development of theory, and improved geometric tools. It will benefit the mathematical community through new insights and improved methods, and possibly lead to downstream applications in other scientific fields that rely on geometry. Read moreRead less
The Global Structure of Sparse Networks. Graph theory (the mathematics of networks) models many real-world problems and is a major area of modern mathematics. This project aims to investigate the global structure of graphs using product structure theory, which is a recent breakthrough method that has been the key to solving several open problems. The goal is to extend the reach of product structure theory and to discover new fields of application, especially in theoretical computer science. It i ....The Global Structure of Sparse Networks. Graph theory (the mathematics of networks) models many real-world problems and is a major area of modern mathematics. This project aims to investigate the global structure of graphs using product structure theory, which is a recent breakthrough method that has been the key to solving several open problems. The goal is to extend the reach of product structure theory and to discover new fields of application, especially in theoretical computer science. It is expected that the tools developed will be widely applicable, for example, in network optimisation. The project aims to build collaborations between Australian researchers and world-leading international mathematicians, and provide advanced training for talented young researchers.Read moreRead less