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
Parabolic methods for elliptic boundary value problems. This project aims to uncover new results for second order nonlinear elliptic partial differential equations via the use of uniqueness properties of solutions for related nonlinear parabolic partial differential equations. This will build on theory for fully nonlinear equations developed over the last 30 years. The project expects to generate new knowledge in the theory that will guide future research and have direct impact to applications ....Parabolic methods for elliptic boundary value problems. This project aims to uncover new results for second order nonlinear elliptic partial differential equations via the use of uniqueness properties of solutions for related nonlinear parabolic partial differential equations. This will build on theory for fully nonlinear equations developed over the last 30 years. The project expects to generate new knowledge in the theory that will guide future research and have direct impact to applications in optimal transport, geometric problems and more applied areas including image analysis and mathematical finance. The project will enhance Australia's international reputation for research in the field and train some of the next generation of mathematical analysts.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
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