Equations of Monge-Ampere type and applications. Many fundamental problems in geometry, physics and applied sciences are related to equations of Monge-Ampere type. In recent years there have been rapid developments in the study of these equations with major breakthroughs made by the proposers. This project aims at new discoveries and findings in theory and applications by resolving outstanding open problems, and enhance Australian leadership, expertise, and training in key areas of mathematics a ....Equations of Monge-Ampere type and applications. Many fundamental problems in geometry, physics and applied sciences are related to equations of Monge-Ampere type. In recent years there have been rapid developments in the study of these equations with major breakthroughs made by the proposers. This project aims at new discoveries and findings in theory and applications by resolving outstanding open problems, and enhance Australian leadership, expertise, and training in key areas of mathematics and its applications.Read moreRead less
Geometric partial differential systems and their applications. This proposal addresses questions central to the understanding of nonlinear partial differential systems from classical, quantum field theory and liquid crystals. Applications to physical problems such as the Yang-Mills flow, Faddeev's model and liquid crystal systems are of great interest and importance in the broader scientific community. The project will yield internationally significant results in theoretical mathematics, with ....Geometric partial differential systems and their applications. This proposal addresses questions central to the understanding of nonlinear partial differential systems from classical, quantum field theory and liquid crystals. Applications to physical problems such as the Yang-Mills flow, Faddeev's model and liquid crystal systems are of great interest and importance in the broader scientific community. The project will yield internationally significant results in theoretical mathematics, with applications in physics and and other sciences. Specialist training will be provided for Australia's next generation of mathematicians. This project will enable Australian researchers to stay at the forefront of research in this area, strengthening links with a number of world-leading mathematicians.Read moreRead less
Nonlinear elliptic partial differential equations and applications. Many fundamental advances in modern technology, science and economics are driven by the analysis of nonlinear models based on nonlinear partial differential equations. In recent years there has been increasing use in applications of partial differential equations of elliptic type with major discoveries made and longstanding problems resolved by the two Chief Investigators, who have in return received many international accolades ....Nonlinear elliptic partial differential equations and applications. Many fundamental advances in modern technology, science and economics are driven by the analysis of nonlinear models based on nonlinear partial differential equations. In recent years there has been increasing use in applications of partial differential equations of elliptic type with major discoveries made and longstanding problems resolved by the two Chief Investigators, who have in return received many international accolades. This project provides for the continuation of Australian leadership in key strategic areas of international science, such as optimal transportation, as well as the continued building of related expertise and training.Read moreRead less
Nonlinear elliptic equations and applications. Many fundamental advances in modern technology, science and economics are driven through the analysis of nonlinear models based on nonlinear partial differential equations. In recent years there has been an explosion in applications of partial differential equations of elliptic type with major discoveries in underlying theory being made by the two Chief Investigators. This project provides for the continuation of Australian leadership in key st ....Nonlinear elliptic equations and applications. Many fundamental advances in modern technology, science and economics are driven through the analysis of nonlinear models based on nonlinear partial differential equations. In recent years there has been an explosion in applications of partial differential equations of elliptic type with major discoveries in underlying theory being made by the two Chief Investigators. This project provides for the continuation of Australian leadership in key strategic areas of international science, such as optimal transportation, as well as the continued building of related expertise and training.Read moreRead less
Variational problems of Monge-Ampere type. Nonlinear models dominate the frontline of modern theoretical and applied mathematics. This project concerns contemporary variational problems with analysis linked strongly to the Monge-Ampere equation, which is a fully nonlinear partial differential equation. Its study in recent years has generated complex and deep theoretical issues along with a diverse range of applications. The proposal is divided into two themes, affine maximal surfaces (involving ....Variational problems of Monge-Ampere type. Nonlinear models dominate the frontline of modern theoretical and applied mathematics. This project concerns contemporary variational problems with analysis linked strongly to the Monge-Ampere equation, which is a fully nonlinear partial differential equation. Its study in recent years has generated complex and deep theoretical issues along with a diverse range of applications. The proposal is divided into two themes, affine maximal surfaces (involving fourth order partial differential equations of Monge-Ampere type) and optimal transportation (where Monge-Ampere theory has been applied successfully in recent years). Each of these builds upon major recent research breakthroughs of the proposers.Read moreRead less
Geometric variational problems and nonlinear partial differential systems. We will investigate several important problems on non-linear partial differential systems, bridging analysis, differential geometry and mathematical physics. Harmonic maps are the prototype of maps minimizing the Dirichlet energy. The liquid crystal configuration generalizes the harmonic map with values into two dimensional spheres. The Yang-Mills equations originated from particle physics. We will make fundamental contri ....Geometric variational problems and nonlinear partial differential systems. We will investigate several important problems on non-linear partial differential systems, bridging analysis, differential geometry and mathematical physics. Harmonic maps are the prototype of maps minimizing the Dirichlet energy. The liquid crystal configuration generalizes the harmonic map with values into two dimensional spheres. The Yang-Mills equations originated from particle physics. We will make fundamental contributions to these topics: Regularity problem and energy minimality of weakly harmonic maps, Weak solutions of the liquid crystal equilibrium system, Yang-Mills heat flow and singular Yang-Mills connections.
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Analysis and applications of geometric evolution equations. This project will keep Australian research in geometric analysis at the leading edge of the field internationally. It will produce fundamental new insights in differential geometry and in the understanding of geometric partial differential equations, and will provide a rich and vigorous training ground for graduate and honours students.
Singular phenomena for nonlinear partial differential equations arising in applications. The development of nonlinear Partial Differential Equations (PDEs) in Australia is recognized worldwide through the outstanding contributions of mathematicians from the ANU, University of Sydney and other top Australian Universities. This project undertakes research in the PDEs field and follows directions of very current interest at an international level. Beyond the ANU, the project will enhance expertise ....Singular phenomena for nonlinear partial differential equations arising in applications. The development of nonlinear Partial Differential Equations (PDEs) in Australia is recognized worldwide through the outstanding contributions of mathematicians from the ANU, University of Sydney and other top Australian Universities. This project undertakes research in the PDEs field and follows directions of very current interest at an international level. Beyond the ANU, the project will enhance expertise in Australia in very active areas of mathematics research related to applications in physics, biology and other applied disciplines. Moreover, it will foster collaboration with mathematicians of international standing from Australia and abroad. Read moreRead less
A Dynamical Systems Approach to Mapping Southern Ocean Circulation Pathways. Climate change can be expected to have complex, long-term consequences for Australia's biodiversity, for our agricultural and marine production systems, and for regional communities. The Southern Ocean is a critical driver of global climate, connecting the three major oceanic basins. Using sophisticated mathematics to analyse state-of-the-art global ocean models, this project will create a detailed picture of hitherto i ....A Dynamical Systems Approach to Mapping Southern Ocean Circulation Pathways. Climate change can be expected to have complex, long-term consequences for Australia's biodiversity, for our agricultural and marine production systems, and for regional communities. The Southern Ocean is a critical driver of global climate, connecting the three major oceanic basins. Using sophisticated mathematics to analyse state-of-the-art global ocean models, this project will create a detailed picture of hitherto invisible Southern Ocean circulation 'pathways'. The newly discovered pathways will greatly enhance our understanding of the Southern Ocean circulation, lead to improved coupled climate models, and increase our ability to predict future climate change and threats to marine populations. Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE140101366
Funder
Australian Research Council
Funding Amount
$376,527.00
Summary
Fully nonlinear partial differential equations in optimisation and applications. Fully nonlinear partial differential equations of Monge-Ampere type and their applications in optimal transportation have been studied intensively in the past two decades. Optimal transportation is a subject in linear optimisation. This project will develop a new theory on Monge-Ampere type equations in nonlinear optimisation, which is a much broader area with many significant applications. This project will investi ....Fully nonlinear partial differential equations in optimisation and applications. Fully nonlinear partial differential equations of Monge-Ampere type and their applications in optimal transportation have been studied intensively in the past two decades. Optimal transportation is a subject in linear optimisation. This project will develop a new theory on Monge-Ampere type equations in nonlinear optimisation, which is a much broader area with many significant applications. This project will investigate fundamental properties of solutions to this problem and explore further real-world applications. The outcomes of this project will have a substantial impact on partial differential equations and related research subjects. Read moreRead less