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
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.
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.
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
Heat kernel and Riesz transform on non-compact metric measure spaces. This project will develop new techniques in heat kernel theory, with applications to such important topics as Schrodinger model for quantum mechanics. The proposed research is at the forefront of research in harmonic analysis and partial differential equations and will further enhance Australia's high international standing in these research fields.
Inverse Problems For Partial Differential Equations - A Geometric Analysis Perspective. This project will study mathematical models of various medical imaging techniques. These problems are formulated as inverse problems in partial differential equations (PDE) where one wishes to obtain information about a differential equation from data about its solutions. This problem is not well understood in the geometric setting where the PDE is taking place on a manifold and the goal of this research is t ....Inverse Problems For Partial Differential Equations - A Geometric Analysis Perspective. This project will study mathematical models of various medical imaging techniques. These problems are formulated as inverse problems in partial differential equations (PDE) where one wishes to obtain information about a differential equation from data about its solutions. This problem is not well understood in the geometric setting where the PDE is taking place on a manifold and the goal of this research is to advance the field in this direction. This project will introduce novel and innovative ideas from geometry and topology to overcome some of these difficulties. This project will enrich mathematics by providing links between different fields. Furthermore, it will enable the application of imaging techniques in a broader geometric setting to provide more efficient and accurate non-invasive detection techniques.Read moreRead less
Dynamic Equations on Measure Chains. Boundary value problems (BVPs) on ``measure chains''are new and useful mathematical equations that describe the world around. This project aims to answer some imporotant and fundamental mathematical questions such as
(i) Under what conditions do BVPs on measure chains actually have solutions?
(ii) If solutions do exist, then what are their properties?
The approach is to use modern tools from mathematical analysis, including topological transversality ....Dynamic Equations on Measure Chains. Boundary value problems (BVPs) on ``measure chains''are new and useful mathematical equations that describe the world around. This project aims to answer some imporotant and fundamental mathematical questions such as
(i) Under what conditions do BVPs on measure chains actually have solutions?
(ii) If solutions do exist, then what are their properties?
The approach is to use modern tools from mathematical analysis, including topological transversality and Leray-Schauder degree.
The project outcomes will
(a) significantly advance current mathematical theory for BVPs on measure chains
(b) unify the theory of BVPs for differential and difference equations
(c) potentially apply to many real-world phenomena.Read moreRead less
Singularities And Classifications Of Integrable Systems. What mathematical models of engineering and nature exclude chaos and have globally predictable solutions? What models occur ubquitously in fields as diverse as photonics and quantum gravity? The answers lie in the theory of integrable systems. We aim to develop powerful new algorithms for identifying integrable models and for deducing their remarkable properties. These algorithms are expected to answer fundamental questions of contempora ....Singularities And Classifications Of Integrable Systems. What mathematical models of engineering and nature exclude chaos and have globally predictable solutions? What models occur ubquitously in fields as diverse as photonics and quantum gravity? The answers lie in the theory of integrable systems. We aim to develop powerful new algorithms for identifying integrable models and for deducing their remarkable properties. These algorithms are expected to answer fundamental questions of contemporary importance. Longer term possible outcomes include applications to nonlinear optics and quantum computing.Read moreRead less
Finite Morse index solutions of nonlinear partial differential equations. We aim to produce mathematics which is of not only of interest to mathematicians but is useful in the study of many physical and biological processes. They occur in the study of processes in industry and the study of the environment.