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: 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
New directions in geometric evolution equations. Diffusion occurs in natural processes such as crystal growth and flame propagation and is also used as a technique in image processing. This project will allow Australian researchers to develop new methods for analysis of the mathematics underlying diffusion and to apply these methods to prove new theoretical results with broad applications.
Australian Laureate Fellowships - Grant ID: FL130100118
Funder
Australian Research Council
Funding Amount
$2,033,722.00
Summary
Nonlinear partial differential equations and applications. This project aims to confirm and enhance Australia as a world leader in this very active and highly significant area of nonlinear partial differential equations. We will develop new methods and techniques to solve challenging problems of immense international interest and continue building expertise and training in the area.
Discovery Early Career Researcher Award - Grant ID: DE190100147
Funder
Australian Research Council
Funding Amount
$336,174.00
Summary
Geometric flow of hypersurfaces and related problems. This project aims to address many of the important problems in the area of geometric flow of hypersurfaces. Geometric flow is the central direction in the field of geometric analysis and has proven to be powerful in understanding geometry and topology of the underlying manifolds. The project expects to improve our understanding of the flows and enable their application to unravel new results in geometry and topology through the development of ....Geometric flow of hypersurfaces and related problems. This project aims to address many of the important problems in the area of geometric flow of hypersurfaces. Geometric flow is the central direction in the field of geometric analysis and has proven to be powerful in understanding geometry and topology of the underlying manifolds. The project expects to improve our understanding of the flows and enable their application to unravel new results in geometry and topology through the development of new mathematical techniques. The project is expected to impact on a range of related fields, including image processing and materials science.Read moreRead less
Topological and analytic aspects of the Kaehler-Ricci flow. The project will make use of the Kaehler-Ricci flow in an original way to study algebraic geometry objects. It involves techniques from geometric analysis, algebraic geometry and several complex variables and is a powerful method to construct and analyse canonical singular metric, which is then applied for further understanding of the algebraic variety.
Geometric evolution problems in nonlinear partial differential equations. This project aims to address important problems key to the understanding of geometric evolution equations and certain other nonlinear partial differential equations. The problems to be tackled lie in a very active area of mathematics: harmonic maps, liquid crystals and Yang-Mills theory. Special aims are to exploit new methods to settle open problems in harmonic maps and Yang-Mills equations, and to improve understanding o ....Geometric evolution problems in nonlinear partial differential equations. This project aims to address important problems key to the understanding of geometric evolution equations and certain other nonlinear partial differential equations. The problems to be tackled lie in a very active area of mathematics: harmonic maps, liquid crystals and Yang-Mills theory. Special aims are to exploit new methods to settle open problems in harmonic maps and Yang-Mills equations, and to improve understanding of practical questions such as the mathematical modelling of liquid crystals via the celebrated Ericksen-Leslie and Landau-de Gennes theories. The expected outcomes are fundamental results in mathematics, with applications in other sciences.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE190101471
Funder
Australian Research Council
Funding Amount
$345,000.00
Summary
Singularity analysis for manifolds with Ricci curvature bounds. This project aims to investigate the central problem of singularity analysis in manifolds by profiling their topological structures and analysing their geometric structure. Understanding the geometric structure of manifolds is at the forefront of research in geometry and topology, with applications in disciplines such as physics and medical imaging. The project will undertake pressing research in this active and dynamic field and ex ....Singularity analysis for manifolds with Ricci curvature bounds. This project aims to investigate the central problem of singularity analysis in manifolds by profiling their topological structures and analysing their geometric structure. Understanding the geometric structure of manifolds is at the forefront of research in geometry and topology, with applications in disciplines such as physics and medical imaging. The project will undertake pressing research in this active and dynamic field and expects to generate greater understanding of limit spaces, deepening the theory of geometric measure theory, and enhancing their applications in mathematics and science.Read moreRead less