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
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
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
Classical and quantum invariants of low-dimensional manifolds. This project aims to advance our understanding of knots and 3-dimensional spaces, which arise naturally in fields as diverse as physics, computer graphics, chemistry and biology. Recent ideas from quantum field theory link physics to topology in dimensions 3 and 4, leading to powerful invariants of knots and 3-dimensional manifolds that include the Jones polynomial and the 3D-index. This project aims to resolve key questions relating ....Classical and quantum invariants of low-dimensional manifolds. This project aims to advance our understanding of knots and 3-dimensional spaces, which arise naturally in fields as diverse as physics, computer graphics, chemistry and biology. Recent ideas from quantum field theory link physics to topology in dimensions 3 and 4, leading to powerful invariants of knots and 3-dimensional manifolds that include the Jones polynomial and the 3D-index. This project aims to resolve key questions relating these quantum invariants to classical topology and geometry. The project will have a major impact in low-dimensional topology, and lead to deep and unexpected connections between mathematics and mathematical physics.Read moreRead less
Connections in low-dimensional topology. This project aims to resolve important open questions in low-dimensional topology, by connecting hyperbolic geometry to invariants arising from quantum topology, cluster algebras, and spinors.
The spaces studied in this project, namely 3-manifolds and knots, arise in applications across engineering and science. The project expects to generate new insights into these spaces by applying tools connecting them to hyperbolic geometry.
Expected outcomes inclu ....Connections in low-dimensional topology. This project aims to resolve important open questions in low-dimensional topology, by connecting hyperbolic geometry to invariants arising from quantum topology, cluster algebras, and spinors.
The spaces studied in this project, namely 3-manifolds and knots, arise in applications across engineering and science. The project expects to generate new insights into these spaces by applying tools connecting them to hyperbolic geometry.
Expected outcomes include efficient techniques to compute important data about 3-manifolds and knots, particularly certain polynomials encoding geometry, and exact calculations of circle packings.
This should provide significant benefits, such as progress on difficult conjectures in hyperbolic geometry.
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Trisections, triangulations and the complexity of manifolds. This project aims at practical representations of 3-dimensional and 4-dimensional spaces as needed in applications. Topology is the mathematical study of the shapes of spaces. Geometry endows spaces with additional structure such as distance, angle and curvature. Special combinatorial structures, such as minimal triangulations, are often closely connected to geometric structures or topological properties. This project aims to construct ....Trisections, triangulations and the complexity of manifolds. This project aims at practical representations of 3-dimensional and 4-dimensional spaces as needed in applications. Topology is the mathematical study of the shapes of spaces. Geometry endows spaces with additional structure such as distance, angle and curvature. Special combinatorial structures, such as minimal triangulations, are often closely connected to geometric structures or topological properties. This project aims to construct computable invariants, connectivity results for triangulations, and algorithms to recognise fundamental topological properties and structures such as trisections and bundles.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
Quantum invariants and hyperbolic manifolds in three-dimensional topology. The project aims to broaden our understanding of three-dimensional (3-D) spaces, including spaces that arise in engineering, microbiology and physics. It is known that all 3-D spaces can be decomposed into geometric pieces. The most common type of geometry is hyperbolic. It is also known that such spaces have algebraic structures arising from quantum physics, known as quantum invariants. Several important conjectures, bas ....Quantum invariants and hyperbolic manifolds in three-dimensional topology. The project aims to broaden our understanding of three-dimensional (3-D) spaces, including spaces that arise in engineering, microbiology and physics. It is known that all 3-D spaces can be decomposed into geometric pieces. The most common type of geometry is hyperbolic. It is also known that such spaces have algebraic structures arising from quantum physics, known as quantum invariants. Several important conjectures, based on developments in physics, assert that hyperbolic geometry and quantum invariants are deeply related, but they remain unproved. The project aims to find new relationships between hyperbolic geometry and quantum invariants, advancing our understanding of both areas.Read moreRead less
Singular spaces in analysis and geometry. Singularities arise naturally in many areas of mathematics, as models of symmetry, degeneracy, and asymptotic collapse. The aim of this project is to provide powerful, generlisable tools to elucidate the interplay between modes of singularity formation and solutions to the important differential equations which arise in geometric analysis. The proposed framework builds upon the established success of microlocal analysis, initiated by Melrose in the 1980' ....Singular spaces in analysis and geometry. Singularities arise naturally in many areas of mathematics, as models of symmetry, degeneracy, and asymptotic collapse. The aim of this project is to provide powerful, generlisable tools to elucidate the interplay between modes of singularity formation and solutions to the important differential equations which arise in geometric analysis. The proposed framework builds upon the established success of microlocal analysis, initiated by Melrose in the 1980's, in the generalisation of landmark theorems like the Atiyah-Singer index theorem to more general Riemannian manifolds. This project will benefit Australia by increasing its capacity in pure mathematics in this highly active research area.Read moreRead less