Discovery Early Career Researcher Award - Grant ID: DE160100024
Funder
Australian Research Council
Funding Amount
$391,509.00
Summary
Higgs bundle moduli spaces and spectral data. The aim of this Project is to advance the study of Higgs bundles using a construction known as spectral data. Higgs bundles are geometric structures bridging several branches of mathematics including differential geometry, representation theory and mathematical physics. This should lead to new results and solve some important open problems concerning the geometry of Higgs bundle moduli spaces and their symmetry groups. The results obtained in the Pro ....Higgs bundle moduli spaces and spectral data. The aim of this Project is to advance the study of Higgs bundles using a construction known as spectral data. Higgs bundles are geometric structures bridging several branches of mathematics including differential geometry, representation theory and mathematical physics. This should lead to new results and solve some important open problems concerning the geometry of Higgs bundle moduli spaces and their symmetry groups. The results obtained in the Project should benefit the many branches of mathematics interacting with Higgs bundles. Such theoretical underpinnings are the basis on which new innovations and technologies in science and engineering may be developed.Read moreRead less
Applications of generalised geometry to duality in quantum theory. This project will undertake research into mathematics at the forefront of modern physics. The aim of the project is to develop a mathematical theory of T-duality, a phenomenon in quantum physics, using generalised geometry.
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.
Parametrised gauge theory and positive scalar curvature. This project aims to study innovative extensions of Seiberg-Witten gauge theory with new applications to the topology of metrics of positive scalar curvature on four-dimensional manifolds. Since Atiyah-Bott, Donaldson, Hitchin, and Seiberg-Witten’s work on various equations in gauge theory, profound applications have changed the geometry and topology of low dimensional manifolds. Parametrised index theory has obtained deep results on the t ....Parametrised gauge theory and positive scalar curvature. This project aims to study innovative extensions of Seiberg-Witten gauge theory with new applications to the topology of metrics of positive scalar curvature on four-dimensional manifolds. Since Atiyah-Bott, Donaldson, Hitchin, and Seiberg-Witten’s work on various equations in gauge theory, profound applications have changed the geometry and topology of low dimensional manifolds. Parametrised index theory has obtained deep results on the topology of metrics of positive scalar curvature in higher dimensions, but these methods do not work in the case of the fourth dimension. This project will develop (parametrised) Seiberg-Witten gauge theory as a new approach to the study of the topology of metrics of positive scalar curvature in four dimensions. Expected outcomes include new invariants related to positive scalar curvature in four dimensions.Read moreRead less
New techniques and invariants in low-dimensional topology. The aim of this project is to introduce and apply new methods and invariants in the field of low-dimensional topology by developing parametrised and equivariant enhancements of Seiberg-Witten theory and Floer homology. These new refined invariants, made possible by recent advances in gauge theory, will be more powerful than existing ones, enabling the detection of new exotic phenomena. Expected outcomes include effective means for distin ....New techniques and invariants in low-dimensional topology. The aim of this project is to introduce and apply new methods and invariants in the field of low-dimensional topology by developing parametrised and equivariant enhancements of Seiberg-Witten theory and Floer homology. These new refined invariants, made possible by recent advances in gauge theory, will be more powerful than existing ones, enabling the detection of new exotic phenomena. Expected outcomes include effective means for distinguishing families of spaces, measuring their complexity and new obstructions for their existence. The new invariants and techniques will lead to the resolution of some open problems in low-dimensional topology and enhance Australia's reputation as a world leader in this field.Read moreRead less
Geometric transforms and duality. This Proposal 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.
Homotopical structures in algebraic, analytic, and equivariant geometry. This is a project for fundamental research in pure mathematics. It is focused on an emerging subfield of complex geometry concerned with spaces and maps that exhibit exceptional flexibility properties, which often go hand-in-hand with a high degree of symmetry. The project aims to develop the foundations of this new area, solve several open problems, and pursue interconnections with and applications to algebraic geometry, c ....Homotopical structures in algebraic, analytic, and equivariant geometry. This is a project for fundamental research in pure mathematics. It is focused on an emerging subfield of complex geometry concerned with spaces and maps that exhibit exceptional flexibility properties, which often go hand-in-hand with a high degree of symmetry. The project aims to develop the foundations of this new area, solve several open problems, and pursue interconnections with and applications to algebraic geometry, complex analysis, geometric invariant theory, and topology.Read moreRead less
Advances in index theory. The laws of nature are often expressed in terms of differential equations, which if elliptic, have an index being the number of solutions minus the number of constraints imposed. The Atiyah-Singer Index Theorem gives a striking calculation of this index and the projects involve innovative extensions of this theory with novel applications.
Advances in Index Theory. The laws of nature are often expressed in terms of differential equations which, if 'elliptic', have an 'index' being the number of solutions minus the number of constraints imposed. The Atiyah-Singer Index Theorem gives a striking calculation of this 'index', and this project involves innovative extensions of this theory with novel applications.
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.