Discovery Early Career Researcher Award - Grant ID: DE140100259
Funder
Australian Research Council
Funding Amount
$379,549.00
Summary
New Compactifications of Moduli Spaces of Curves. A moduli space is a geometric representation of a system of algebraic equations. Many fundamental problems in algebra, dynamics and physics can be reformulated as questions concerning the geometry of moduli spaces, in particular the moduli space of curves. This project will produce a systematic classification of compactifications of the moduli space of curves, and develop new tools for investigating the geometry of these compactifications. One of ....New Compactifications of Moduli Spaces of Curves. A moduli space is a geometric representation of a system of algebraic equations. Many fundamental problems in algebra, dynamics and physics can be reformulated as questions concerning the geometry of moduli spaces, in particular the moduli space of curves. This project will produce a systematic classification of compactifications of the moduli space of curves, and develop new tools for investigating the geometry of these compactifications. One of these compactifications already lies at the centre of an extraordinary web of connections linking together topology, combinatorics and quantum field theory, and there is a strong possibility that this web fits into a more comprehensive picture involving all compactifications on an equal footing.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE180101360
Funder
Australian Research Council
Funding Amount
$328,075.00
Summary
The geometry and cohomology of moduli spaces of curves. This project aims to develop new insights on moduli spaces in algebraic geometry. Algebraic geometry is the field of mathematics that uses geometric methods to analyse algebraic equations, with wide applications ranging from cryptography to genetics. Moduli spaces in algebraic geometry provide powerful methods to geometrically analyse collections of related equations. Using innovative new techniques, the project aims to generate new knowled ....The geometry and cohomology of moduli spaces of curves. This project aims to develop new insights on moduli spaces in algebraic geometry. Algebraic geometry is the field of mathematics that uses geometric methods to analyse algebraic equations, with wide applications ranging from cryptography to genetics. Moduli spaces in algebraic geometry provide powerful methods to geometrically analyse collections of related equations. Using innovative new techniques, the project aims to generate new knowledge about fundamental moduli spaces. Expected outcomes include the establishment of an active community of algebraic geometers in Australia. These outcomes should provide significant benefits to pure mathematics and related scientific fields.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.
Noncommutative analysis and geometry in interaction with quantum physics. Quantum theory has produced many advances in our understanding of the physical world for the last hundred years while mathematical breakthroughs have been made through exploiting innovative ideas from quantum physics. This project continues in this highly successful framework and will lead to advances in geometry both classical and noncommutative.
Topological stability from spectral analysis. The aim is to use mathematical scattering theory to find and study new topological features of the spectra of linear transformations on Hilbert space. The significance derives from mathematical models of low temperature conducting quantum materials. These have revealed `topological phases of matter' that are stable with respect to a range of variations in the parameters that determine the system. The stability is desired for applications to quantum ....Topological stability from spectral analysis. The aim is to use mathematical scattering theory to find and study new topological features of the spectra of linear transformations on Hilbert space. The significance derives from mathematical models of low temperature conducting quantum materials. These have revealed `topological phases of matter' that are stable with respect to a range of variations in the parameters that determine the system. The stability is desired for applications to quantum devices. Our results will give topological stability from the scattering spectrum, a feature not previously seen. The benefits stem from new results in mathematical scattering theory with a primary novelty being the analysis of ``zero energy resonances'' in mathematical models of graphene.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE120101167
Funder
Australian Research Council
Funding Amount
$375,000.00
Summary
Canonical metrics on Kahler manifolds and Monge-Ampere equations. This project will introduce new ideas and techniques to study the existence of canonical metrics on Kahler manifolds, which is a fundamental problem in geometry. Advances in this research will have influence on other areas of science such as mechanics, string theory and mathematical physics.
Higher order curvature flow of curves and hypersurfaces. This project aims to analyse higher order geometric partial differential equations that have important mathematical applications in differential geometry of submanifolds as well as practical applications in physics and mathematical biology. The project aims to prove new general principles that reveal properties of these higher order elliptic and parabolic partial differential equations, producing a unified framework with applications to va ....Higher order curvature flow of curves and hypersurfaces. This project aims to analyse higher order geometric partial differential equations that have important mathematical applications in differential geometry of submanifolds as well as practical applications in physics and mathematical biology. The project aims to prove new general principles that reveal properties of these higher order elliptic and parabolic partial differential equations, producing a unified framework with applications to various specific problems. This project aims to increase Australia's research capacity in geometric evolution problems, provide training for some of Australia's next generation of mathematicians and build Australia's international reputation for significant research in geometric analysis.Read moreRead less
Fully nonlinear elliptic equations and applications. This project aims to develop new methods to solve challenging problems in fully nonlinear elliptic equations, and to confirm and enhance Australia as a world leader in this very active area. In addition to high impact publications, this highly innovative research also provides continued building of expertise and training in the area.
Discovery Early Career Researcher Award - Grant ID: DE200101834
Funder
Australian Research Council
Funding Amount
$418,410.00
Summary
The structure of singularities in geometric flows. The proposed research aims to develop our understanding of the structure of singularities in mean curvature and related flows, with certain applications in mind.