Topology in seven dimensions. Aims: The project aims to give a complete classification of a certain class of 7-dimensional spaces; namely simply-connected spin 7-manifolds. We also present related programs classify G_2-structures on 7-manifolds.
Significance: the proposed classification will be a signature achievement in the topology of manifolds, with applications likely in both geometry and mathematical physics.
Expected outcomes: The project will produce a series of papers published in high ....Topology in seven dimensions. Aims: The project aims to give a complete classification of a certain class of 7-dimensional spaces; namely simply-connected spin 7-manifolds. We also present related programs classify G_2-structures on 7-manifolds.
Significance: the proposed classification will be a signature achievement in the topology of manifolds, with applications likely in both geometry and mathematical physics.
Expected outcomes: The project will produce a series of papers published in high quality journals and enhanced scientific collaboration between Australia and the United Kingdom.
Benefits: The project will enhance Australia's research reputation by producing excellent research in a field not historically represented in the country.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
Triangulations in dimensions 3 and 4: discrete and geometric structures. Recently there have been spectacular advances in understanding 3-dimensional spaces and the interaction between ideas in mathematical physics (quantum invariants) and such spaces. This project aims at practical methods for finding geometric structures and advancing our understanding of the information that physics is providing about these spaces.
Interactions of geometry and knot theory. This project aims to use recent breakthroughs in hyperbolic geometry and Kleinian groups to relate geometry to knots which are mathematical objects arising in microbiology, chemistry, physics, and mathematics. Knots are often studied via the space around them known as the knot complement. Knot complements decompose into geometric pieces, and the most common geometry is hyperbolic, which completely determines a knot. However, how to obtain information on ....Interactions of geometry and knot theory. This project aims to use recent breakthroughs in hyperbolic geometry and Kleinian groups to relate geometry to knots which are mathematical objects arising in microbiology, chemistry, physics, and mathematics. Knots are often studied via the space around them known as the knot complement. Knot complements decompose into geometric pieces, and the most common geometry is hyperbolic, which completely determines a knot. However, how to obtain information on the hyperbolic geometry of a knot from a classical description is unknown. This project will obtain information by uncovering results that would enable classification of even extremely complicated knots, and could affect mathematics and other fields.Read moreRead less
Interactions between geometric and topological structures. This project aims to develop the necessary tools for a geometric imagination in higher dimensions and to bridge the gap between low and high dimensions. Topology is the mathematical study of the shape of spaces. Geometry endows spaces with additional structure such as distance, angle and curvature. Moduli space encodes the different ways in which a geometry can be assigned to a space. Expected outcomes of the project include construction ....Interactions between geometric and topological structures. This project aims to develop the necessary tools for a geometric imagination in higher dimensions and to bridge the gap between low and high dimensions. Topology is the mathematical study of the shape of spaces. Geometry endows spaces with additional structure such as distance, angle and curvature. Moduli space encodes the different ways in which a geometry can be assigned to a space. Expected outcomes of the project include construction of computable invariants, solution of realisation problems and understanding degeneration of geometries.Read moreRead less
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
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
Comprehensive Study of Kahler-Ricci Flows. The intended outcome of the project is to reveal the deep relation between geometry and topology of underlying spaces. Ricci flow has attracted major attention in pure mathematics over the past 30 years, including ground-breaking contributions by Perelman on Ricci flow regarding the famous Poincare and Thurston's Geometrisation Conjectures. The project focuses on the complex version of Ricci flow: Kahler-Ricci flow. The project plans to explore the Kahl ....Comprehensive Study of Kahler-Ricci Flows. The intended outcome of the project is to reveal the deep relation between geometry and topology of underlying spaces. Ricci flow has attracted major attention in pure mathematics over the past 30 years, including ground-breaking contributions by Perelman on Ricci flow regarding the famous Poincare and Thurston's Geometrisation Conjectures. The project focuses on the complex version of Ricci flow: Kahler-Ricci flow. The project plans to explore the Kahler-Ricci flow in the closed and complete non-compact settings and the corresponding versions of Geometric Minimal Model Program; and the Kahler-Ricci flow in the Fano manifold setting and stability conditions.Read moreRead less
Australian Laureate Fellowships - Grant ID: FL150100126
Funder
Australian Research Council
Funding Amount
$2,080,100.00
Summary
Geometric analysis of eigenvalues and heat flows. Geometric analysis of eigenvalues and heat flows: This fellowship project aims to build on Australia's leading position in the areas of nonlinear partial differential equations and geometric analysis to exploit new and highly innovative mathematical methods. It is expected that the methods will affect a range of related fields including stochastic modelling and finance, image processing, and the basic sciences. The project seeks to serve as a foc ....Geometric analysis of eigenvalues and heat flows. Geometric analysis of eigenvalues and heat flows: This fellowship project aims to build on Australia's leading position in the areas of nonlinear partial differential equations and geometric analysis to exploit new and highly innovative mathematical methods. It is expected that the methods will affect a range of related fields including stochastic modelling and finance, image processing, and the basic sciences. The project seeks to serve as a focal point for a developing community of Australian researchers in this field, providing a training ground for young researchers and students at the forefront of a vigorous and internationally active area of research, and bringing top international researchers to Australia to interact with the local research community.Read moreRead less