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.
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
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.
Geometric evolution of spaces with symmetries. Symmetries underpin numerous laws of nature and mathematical constructions. This project aims to develop a comprehensive theory of the famous Ricci flow equation in the presence of symmetries. Previous study of this equation has led to many ground-breaking results, such as Perelman's celebrated proof of the century-old Poincaré conjecture. Outcomes are expected to fill major knowledge gaps in mathematics, opening doors to applications in quantum fie ....Geometric evolution of spaces with symmetries. Symmetries underpin numerous laws of nature and mathematical constructions. This project aims to develop a comprehensive theory of the famous Ricci flow equation in the presence of symmetries. Previous study of this equation has led to many ground-breaking results, such as Perelman's celebrated proof of the century-old Poincaré conjecture. Outcomes are expected to fill major knowledge gaps in mathematics, opening doors to applications in quantum field theory, relativity and other fields. Anticipated benefits include strengthening Australia’s leadership in mathematical innovation, advancing the internationalisation of the Australian research scene, and increasing the involvement of women in STEM.Read moreRead less
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: DE190101241
Funder
Australian Research Council
Funding Amount
$350,000.00
Summary
Gauged sigma model, mirror symmetry, and related topics. This project aims to lay down a rigorous mathematical foundation of the gauged linear sigma model and seek its mathematical applications. The gauged linear sigma model is an important theory introduced by the great physicist Edward Witten. It is a fundamental framework in studying superstring theory, which is one of the most promising candidates for the unification of all aspects of physics. This project will generate new and significant r ....Gauged sigma model, mirror symmetry, and related topics. This project aims to lay down a rigorous mathematical foundation of the gauged linear sigma model and seek its mathematical applications. The gauged linear sigma model is an important theory introduced by the great physicist Edward Witten. It is a fundamental framework in studying superstring theory, which is one of the most promising candidates for the unification of all aspects of physics. This project will generate new and significant results in geometry, with many benefits, providing solid mathematical foundations of the gauged linear sigma model, deepening the understanding of this theory, and providing new methods for solving classical problems.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