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
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.
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.
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.
Transitions and singular behaviour in nonlinear partial differential equations. This research produces crucial mathematical methods and results for the understanding of several important nonlinear problems in applied sciences. The mathematical theory itself is of significant value in nonlinear partial differential equations. The project strengthens Australian research in nonlinear science, a critical area of modern international research, and provides high level training for the next generation ....Transitions and singular behaviour in nonlinear partial differential equations. This research produces crucial mathematical methods and results for the understanding of several important nonlinear problems in applied sciences. The mathematical theory itself is of significant value in nonlinear partial differential equations. The project strengthens Australian research in nonlinear science, a critical area of modern international research, and provides high level training for the next generation of Australian mathematicians.Read moreRead less
Integrable Lattice Equations. When mathematical models are simulated on a computer, the result is a system of partial difference equations, whose solutions evolve with discrete steps on a lattice in space and time. While many tools have been developed to study continuous equations, very few mathematical techniques are available for analysing non-linear lattice equations. We aim to develop techniques of solving the initial-value problem for a class of such equations. Our examples include integrab ....Integrable Lattice Equations. When mathematical models are simulated on a computer, the result is a system of partial difference equations, whose solutions evolve with discrete steps on a lattice in space and time. While many tools have been developed to study continuous equations, very few mathematical techniques are available for analysing non-linear lattice equations. We aim to develop techniques of solving the initial-value problem for a class of such equations. Our examples include integrable lattice equations that arise in the simulation of many physical problems ranging from the progression of shallow water waves to signals in an optical fibre.Read moreRead less
Abstract methods for nonlinear partial differential equations. To use abstract methods to study nonlinear partial differential equations where nonlinear effects dominate and where the diffusion is possibly small. These equations arise in many applications of mathematics such as population models and catalysis theory.