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Optimal shapes in geometry and physics: Isoperimetry in modern analysis. This project will find the best isoperimetric shapes in curved spaces: shapes that optimise geometric or analytic quantities, such as the volume enclosed by a surface of a given area, or the resonant frequency of a drum of given area. The optimal shapes lead to tools that are widely used in differential equations, geometric analysis, statistical physics, probability theory, and quantum computing. Through this work, we ....Optimal shapes in geometry and physics: Isoperimetry in modern analysis. This project will find the best isoperimetric shapes in curved spaces: shapes that optimise geometric or analytic quantities, such as the volume enclosed by a surface of a given area, or the resonant frequency of a drum of given area. The optimal shapes lead to tools that are widely used in differential equations, geometric analysis, statistical physics, probability theory, and quantum computing. Through this work, we will forge connections between the geometry of curved spaces, and the physics of operators therein. The significant benefits of this project include increasing fundamental mathematical knowledge, building capacity in Australia’s world-class geometric analysis community, and strong links with international partners.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE220100918
Funder
Australian Research Council
Funding Amount
$426,000.00
Summary
Teichmueller dynamics and the birational geometry of moduli space. The project aims to leverage recent cutting-edge advances in the area of Teichmueller dynamics to answer longstanding open questions from algebraic geometry on the moduli space of curves, an object with deep connections to many diverse areas of science including quantum gravity and theoretical physics. The project expects to generate new theories and increased understanding in both areas through the innovation of relating these p ....Teichmueller dynamics and the birational geometry of moduli space. The project aims to leverage recent cutting-edge advances in the area of Teichmueller dynamics to answer longstanding open questions from algebraic geometry on the moduli space of curves, an object with deep connections to many diverse areas of science including quantum gravity and theoretical physics. The project expects to generate new theories and increased understanding in both areas through the innovation of relating these perspectives, as well as uncovering new connections between the viewpoints. Further benefits should include building international collaborations and the contribution of this diverse perspective to the growing algebraic geometry community in Australia and to mathematics and related scientific fields more generally.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
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
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
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
Discovery Early Career Researcher Award - Grant ID: DE190101222
Funder
Australian Research Council
Funding Amount
$348,070.00
Summary
Elliptic representation theory: the study of symmetries across geometry, algebra and physics. This project aims to study symmetries which control geometry and physics of four-dimensional manifolds. Representation theory is the area in pure mathematics which studies symmetries coming from geometry, algebra, and physics. The expected outcome is new families of quantum algebra, their character formulas, and a solution to an infinite-dimensional moduli problem. This project will further deepen the m ....Elliptic representation theory: the study of symmetries across geometry, algebra and physics. This project aims to study symmetries which control geometry and physics of four-dimensional manifolds. Representation theory is the area in pure mathematics which studies symmetries coming from geometry, algebra, and physics. The expected outcome is new families of quantum algebra, their character formulas, and a solution to an infinite-dimensional moduli problem. This project will further deepen the mathematical knowledge of symmetries, and show unexpected new connections between different areas of pure mathematics and mathematical physics.Read moreRead less
Curvature flows and spectral estimates. Curvature flows are a class of geometrically motivated equations, modelled on the heat equation. Recently, researchers have developed new methods for studying the regularity of solutions to these equations, and applied them to a different problem, that of estimating quantities depending on the smaller eigenvalues of a Schroedinger operator. This project builds on the early success of this research and will produce a new understanding of the behaviour of ei ....Curvature flows and spectral estimates. Curvature flows are a class of geometrically motivated equations, modelled on the heat equation. Recently, researchers have developed new methods for studying the regularity of solutions to these equations, and applied them to a different problem, that of estimating quantities depending on the smaller eigenvalues of a Schroedinger operator. This project builds on the early success of this research and will produce a new understanding of the behaviour of eigenvalues, establish sharp estimates for spectral quantities, particularly on manifolds with curvature bounds, and find optimal conditions under which non-compact solutions to curvature flows are stable.Read moreRead less