Discovery Early Career Researcher Award - Grant ID: DE120102657
Funder
Australian Research Council
Funding Amount
$375,000.00
Summary
Group actions and K-theory: a new direction. This project investigates cutting-edge research in the mathematics of symmetries arising in nature. The aim is to significantly advance the frontiers of our knowledge by introducing new examples, original methods and a modern perspective.
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
Moduli spaces of geometric structures. One of the most spectacular recent advances in mathematics was Perelman's resolution of Thurston's geometrisation program for three-dimensional spaces. This makes it very important to understand sets of geometric structures on such spaces, called moduli spaces. The challenge of this project is to develop practical methods to compute moduli spaces, understand their global properties, and use them to define new invariants giving insights about shape, geometry ....Moduli spaces of geometric structures. One of the most spectacular recent advances in mathematics was Perelman's resolution of Thurston's geometrisation program for three-dimensional spaces. This makes it very important to understand sets of geometric structures on such spaces, called moduli spaces. The challenge of this project is to develop practical methods to compute moduli spaces, understand their global properties, and use them to define new invariants giving insights about shape, geometry and algebra of a space. This is a key problem as moduli spaces describe whether a space is rigid or can be deformed. They are a central object in several fields of mathematics, including geometry and topology, gauge theory, dynamical systems, mathematical physics and invariant theory.Read moreRead less
Invariants, geometric and discrete structures on manifolds. This project aims to develop practical methods for finding geometric and discrete structures on manifolds in both low and high dimensions and advancing our understanding of the information that physics is providing about these spaces. Recently there have been spectacular advances in understanding 3-D spaces and the interaction between ideas in mathematical physics (quantum invariants, string theory) and such spaces. In this project, the ....Invariants, geometric and discrete structures on manifolds. This project aims to develop practical methods for finding geometric and discrete structures on manifolds in both low and high dimensions and advancing our understanding of the information that physics is providing about these spaces. Recently there have been spectacular advances in understanding 3-D spaces and the interaction between ideas in mathematical physics (quantum invariants, string theory) and such spaces. In this project, the first aim is to construct structures with good geometric properties on 3- and 4-manifolds, using triangulations. The second aim is to study combinatorial decompositions of n-manifolds, using our new technique of multisections and also searching for polyhedral metrics of non-positive curvature. The third aim is to connect quantum invariants and geometric structures, again using triangulations.Read moreRead less
Flag varieties and configuration spaces in algebra. School students learn that curves may be described by means of equations, which may therefore be solved geometrically; this is an example of the interaction of algebra and geometry. In this project geometric ideas such as simplicial geometry and cohomological representation theory will be developed, which address deep questions in modern algebra.
Bundle gerbes: generalisations and applications. This project is fundamental, basic research at the forefront of modern differential geometry and its application to physics. It will ensure that Australia is involved in today's mathematical and physical advances and that we have Australian mathematicians trained to take advantage of the future benefits of these advances.
Discovery Early Career Researcher Award - Grant ID: DE120100232
Funder
Australian Research Council
Funding Amount
$375,000.00
Summary
Fusion categories and topological quantum field theory. This project will involve mathematical research of the highest international calibre on fusion categories and topological field theory. Progress in these fields will lead to advances in computing (for example substrates for quantum computers), condensed matter physics, and the mathematical fields of operator algebra, quantum algebra, and quantum topology.
Generalised topological spaces. Pure mathematics creates abstractions of real-world entities; one such is the idea of a 'topological space', which abstracts from geometric forms like cubes and toruses. But topological spaces fail to capture geometric structures arising in areas like quantum physics; and this project seeks to rectify this, by developing a new more general notion.
Australian Laureate Fellowships - Grant ID: FL100100137
Funder
Australian Research Council
Funding Amount
$1,868,132.00
Summary
Derived categories and applications. This project will deepen our understanding of homological algebra, a mathematical tool that has proved useful in areas ranging from physics to the coding of information for computer transmission. Also, having a thriving research presence in Australia, of this vibrant, modern field, should inspire more students to seek a career in mathematics; this would help relieve the acute, well-documented shortage of mathematicians in Australia. It has been established th ....Derived categories and applications. This project will deepen our understanding of homological algebra, a mathematical tool that has proved useful in areas ranging from physics to the coding of information for computer transmission. Also, having a thriving research presence in Australia, of this vibrant, modern field, should inspire more students to seek a career in mathematics; this would help relieve the acute, well-documented shortage of mathematicians in Australia. It has been established that Australia is not producing enough mathematicians to meet the needs of industry; a lively centre, full of young, productive mathematicians, will go a long way towards correcting this problem.Read moreRead less
Enriched higher category theory. At the beginning of 21st century it became clear that further progress in mathematics and physics required a development of a new powerful language, which received the name Higher Category Theory. This project is devoted to the development of this new exciting theory.