Unlocking the potential for linear and discrete optimisation in knot theory and computational topology. Computational topology is a young, energetic field that uses computers to solve complex geometric problems, such as whether a loop of string is tangled. Such computations are becoming increasingly important in mathematics, and applications span biology, physics and information sciences, however many core problems in the field remain intractable for all but the simplest cases. This project unit ....Unlocking the potential for linear and discrete optimisation in knot theory and computational topology. Computational topology is a young, energetic field that uses computers to solve complex geometric problems, such as whether a loop of string is tangled. Such computations are becoming increasingly important in mathematics, and applications span biology, physics and information sciences, however many core problems in the field remain intractable for all but the simplest cases. This project unites geometric techniques with powerful methods from operations research, such as linear and discrete optimisation, to build fast, powerful tools that can for the first time systematically solve large topological problems. Theoretically, this project has significant impact on the famous open problem of detecting knottedness in fast polynomial time.Read moreRead less
Matchings in Combinatorial Structures. The theory of matching in graphs concerns the problem of pairing up objects, subject to constraints on which objects may be paired. It is a well-developed theory that is not only of tremendous mathematical importance, but is also widely applied to efficiently deal with allocation and scheduling problems. Much less is known, however, about the equally important but harder problem of dividing objects into collections of three or more. This project aims to add ....Matchings in Combinatorial Structures. The theory of matching in graphs concerns the problem of pairing up objects, subject to constraints on which objects may be paired. It is a well-developed theory that is not only of tremendous mathematical importance, but is also widely applied to efficiently deal with allocation and scheduling problems. Much less is known, however, about the equally important but harder problem of dividing objects into collections of three or more. This project aims to address this deficiency by developing the theory of matching in important combinatorial objects. The problems it expects to solve are of great significance in their own right, and when considered together may help to lay a foundation for a more general theory of matching.Read moreRead less
Categorical symmetries in representation theory. This project aims to develop categorical symmetries of central objects in mathematics such as braid groups, the Hilbert scheme of points, and the Virasoro algebra. The concept of symmetry is an important organising principle in science. Representation theory is the field of mathematics concerned with studying symmetries. The problems proposed have connections to many different areas including algebra, geometry, topology, and mathematical physics. ....Categorical symmetries in representation theory. This project aims to develop categorical symmetries of central objects in mathematics such as braid groups, the Hilbert scheme of points, and the Virasoro algebra. The concept of symmetry is an important organising principle in science. Representation theory is the field of mathematics concerned with studying symmetries. The problems proposed have connections to many different areas including algebra, geometry, topology, and mathematical physics. This project expects to advance pure mathematics and provide potential benefit in many related fields.Read moreRead less
Special Research Initiatives - Grant ID: SR0354716
Funder
Australian Research Council
Funding Amount
$10,000.00
Summary
Energetically Open Systems Research Network Study. Conceptual frameworks arising in the physical sciences, such as non-equilibrium statistical mechanics and thermodynamics, synergetics, chaos and dynamical systems theory, are seminal in the emerging science of complexity. This study will lay the groundwork for a network to link Australian and overseas research on these fundamental concepts, and their application within the context of entropy-producing systems vital to the long-term sustainabilit ....Energetically Open Systems Research Network Study. Conceptual frameworks arising in the physical sciences, such as non-equilibrium statistical mechanics and thermodynamics, synergetics, chaos and dynamical systems theory, are seminal in the emerging science of complexity. This study will lay the groundwork for a network to link Australian and overseas research on these fundamental concepts, and their application within the context of entropy-producing systems vital to the long-term sustainability of the earth - oceans, atmosphere, biosphere, CO2-free energy production, space and solar environment. The network would facilitate the development of young investigators and be linked into wider complex systems networks such as the CSIRO Centre for Complex Systems Science.Read moreRead less
Algorithms and computation in four-dimensional topology. This project will establish Australia as a world leader in computational topology, particularly in the all-important areas of topology in three and four dimensions. In four dimensions this work will be truly groundbreaking; until now the field has seen little development due to the complexity of the algorithms and computations required, and the applicant is in the unique position of having the necessary tools to make significant progress ....Algorithms and computation in four-dimensional topology. This project will establish Australia as a world leader in computational topology, particularly in the all-important areas of topology in three and four dimensions. In four dimensions this work will be truly groundbreaking; until now the field has seen little development due to the complexity of the algorithms and computations required, and the applicant is in the unique position of having the necessary tools to make significant progress in a feasible time frame. In three dimensions this project will strengthen the distinguished computational topology community in Melbourne, led by pioneers such as Rubinstein, Goodman, Hodgson as well as the applicant himself.Read moreRead less
A new approach to compressed sensing. Compressed sensing is an exciting new paradigm promising vastly improved signal sampling and reconstruction in a wide variety of applications including digital cameras, mobile phones and MRI machines. This project will explore a newly discovered approach to compressed sensing which uses mathematical arrays known as hash families.
Macdonald polynomials: Combinatorics and representations. This proposal is part of the aim to build a world class research team in algebraic combinatorics and combinatorial representation theory at the University of Melbourne, led by the two CI.
These fields are currently experiencing very rapid growth and development, and a strong Australia based team will further enhance the country's strong reputation in combinatorics and algebra.
The project will also provide a perfect training ground fo ....Macdonald polynomials: Combinatorics and representations. This proposal is part of the aim to build a world class research team in algebraic combinatorics and combinatorial representation theory at the University of Melbourne, led by the two CI.
These fields are currently experiencing very rapid growth and development, and a strong Australia based team will further enhance the country's strong reputation in combinatorics and algebra.
The project will also provide a perfect training ground for Higher Degree Students with interests in pure mathematics as well as computer
algebra and symbolic computation.Read moreRead less
Towards the prime power conjecture. This project attacks a famous and long standing conjecture in pure mathematics that has important ramifications in many applied areas. The project aims to determine when it is possible to produce more efficient codes for electronic communication and statistically balanced designs for experiments in areas as diverse as agriculture and psychology.