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
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
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
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: DE180101348
Funder
Australian Research Council
Funding Amount
$328,075.00
Summary
Singularity analysis for ricci flow and mean curvature flow. This project aims to investigate the central problem of singularity formation in Ricci flow and mean-curvature flow by profiling singular solutions and determining their stability and genericity. Geometric flows are powerful and successful ways of understanding classical problems in geometry and topology with applications in disciplines such as materials science and medical imaging. This project will generate significant results in sin ....Singularity analysis for ricci flow and mean curvature flow. This project aims to investigate the central problem of singularity formation in Ricci flow and mean-curvature flow by profiling singular solutions and determining their stability and genericity. Geometric flows are powerful and successful ways of understanding classical problems in geometry and topology with applications in disciplines such as materials science and medical imaging. This project will generate significant results in singularity analysis and will enrich understanding of geometric flows at and past singularities, deepen the theory of geometric flows, and enhance their applications in mathematics and science.Read moreRead less
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.
Discovery Early Career Researcher Award - Grant ID: DE210100535
Funder
Australian Research Council
Funding Amount
$340,548.00
Summary
Minimal surfaces and singularities of mean curvature flow. The project aims to characterise the geometric structure of minimal surfaces in the variational theory and classify singularities of mean curvature flow. Minimal surfaces are mathematical models of soap films, and their time-varying analogue is mean curvature flow, a dynamic process by which a surface flows to decrease its area as quickly as possible. As a central topic in geometric analysis, the theory of minimal surfaces and mean curv ....Minimal surfaces and singularities of mean curvature flow. The project aims to characterise the geometric structure of minimal surfaces in the variational theory and classify singularities of mean curvature flow. Minimal surfaces are mathematical models of soap films, and their time-varying analogue is mean curvature flow, a dynamic process by which a surface flows to decrease its area as quickly as possible. As a central topic in geometric analysis, the theory of minimal surfaces and mean curvature flow has proven to be a powerful and essential tool in mathematics. The project expects to generate new and significant results in minimal surfaces and singularity analysis of mean curvature flow and enhance potential applications in related disciplines such as computer vision and probability.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE180100110
Funder
Australian Research Council
Funding Amount
$343,450.00
Summary
Analysis of fully non-linear geometric problems and differential equations. This project aims to investigate non-linear geometric evolution equations that have received considerable attention in the past decades through their use in solving outstanding problems in mathematics, such as the Poincare conjecture. By developing innovative new techniques intertwining geometry and analysis, the project endeavours to make advances in non-linear problems modelling complex phenomena. The project addresses ....Analysis of fully non-linear geometric problems and differential equations. This project aims to investigate non-linear geometric evolution equations that have received considerable attention in the past decades through their use in solving outstanding problems in mathematics, such as the Poincare conjecture. By developing innovative new techniques intertwining geometry and analysis, the project endeavours to make advances in non-linear problems modelling complex phenomena. The project addresses topics as varied as hyperbolic geometry, and a geometric approach to irregularities forming in crystal growth in materials science, focusing on developing cutting-edge mathematical tools and connections to geometry.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE190100379
Funder
Australian Research Council
Funding Amount
$360,000.00
Summary
Curvature flow of clusters: optimal partitioning and merging fire fronts. This project aims to develop the curvature flow of clusters, a new mathematical innovation that builds on methods with proven success in making new progress on difficult problems in geometry and topology. The curvature flow of clusters will allow foams - partitions of space - to be viewed dynamically. This allows long-standing problems on their structure, a key mathematical challenge in material science, to be studied in a ....Curvature flow of clusters: optimal partitioning and merging fire fronts. This project aims to develop the curvature flow of clusters, a new mathematical innovation that builds on methods with proven success in making new progress on difficult problems in geometry and topology. The curvature flow of clusters will allow foams - partitions of space - to be viewed dynamically. This allows long-standing problems on their structure, a key mathematical challenge in material science, to be studied in a natural context. The project is expected to produce a software suite capable of simulating the movement of merging fire fronts with better accuracy than ever before. The mathematical tools developed by the project will have broad applicability, not only to space partitioning but also notably to bushfires, especially on the dynamics of merging fire fronts.Read moreRead less
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