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
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
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.
Probing the earth and the universe with microlocal analysis. This project aims to use the theory of microlocal analysis to determine the amount of information one can recover about the earth and the universe by making observations on wave propagation. In addition to applications to seismic imaging and cosmology, this project will generate new knowledge in the field of differential geometry and dynamical systems. This will be accomplished by formulating the tomography problem in the language of d ....Probing the earth and the universe with microlocal analysis. This project aims to use the theory of microlocal analysis to determine the amount of information one can recover about the earth and the universe by making observations on wave propagation. In addition to applications to seismic imaging and cosmology, this project will generate new knowledge in the field of differential geometry and dynamical systems. This will be accomplished by formulating the tomography problem in the language of differential geometry and introduce new analysis techniques to study them. Expected outcome of this project will be new rigidity type results in Lorentzian and Riemannian geometry. There is also the potential for downstream impacts in seismic and cosmological imaging.Read moreRead less
Microlocal Analysis - A Unified Approach for Geometric Models in Biology . This project will use microlocal analysis to create a unified approach for predicting the outcome of a broad class of diffusion and reaction-diffusion models. This will replace the traditional theory which is no longer adequate for the level of geometric complexity demanded of current models arising in biology/ecology. This project will address the urgent need for a systematic theoretical underpinning of diffusion/reacti .... Microlocal Analysis - A Unified Approach for Geometric Models in Biology . This project will use microlocal analysis to create a unified approach for predicting the outcome of a broad class of diffusion and reaction-diffusion models. This will replace the traditional theory which is no longer adequate for the level of geometric complexity demanded of current models arising in biology/ecology. This project will address the urgent need for a systematic theoretical underpinning of diffusion/reaction-diffusion in geometric settings whose scope of application is broader than the the existing patchwork of methods.
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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