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Variational theory for fully nonlinear elliptic equations. This project aims to develop new methods and techniques to solve challenging mathematical problems in fully nonlinear partial differential equations arising in important applications. The project will develop methods and techniques to study these equations’ regularity and variational properties. This project is expected to establish comprehensive theories and enhance and promote Australian participation and leadership in this area of mat ....Variational theory for fully nonlinear elliptic equations. This project aims to develop new methods and techniques to solve challenging mathematical problems in fully nonlinear partial differential equations arising in important applications. The project will develop methods and techniques to study these equations’ regularity and variational properties. This project is expected to establish comprehensive theories and enhance and promote Australian participation and leadership in this area of mathematics.Read moreRead less
Propagation described by partial differential equations with free boundary. Cutting edge nonlinear mathematics is required to understand many important propagation phenomena in nature, such as the spreading of invasive species or nerve signals. This project aims to systematically investigate nonlinear partial differential equation models that govern the dynamics of such propagations, with emphasis on the development of new approaches that enable deeper insights on the evolution of the propagatin ....Propagation described by partial differential equations with free boundary. Cutting edge nonlinear mathematics is required to understand many important propagation phenomena in nature, such as the spreading of invasive species or nerve signals. This project aims to systematically investigate nonlinear partial differential equation models that govern the dynamics of such propagations, with emphasis on the development of new approaches that enable deeper insights on the evolution of the propagating fronts. The project aims to develop new mathematics for new applications of lasting values.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE170101410
Funder
Australian Research Council
Funding Amount
$330,324.00
Summary
Nonlinear free boundary problems: Propagation and regularity. This project aims to understand the propagation profile and regularity of two important classes of free boundary problems. Nonlinear free boundary problems arise from many applied fields, and pose great challenges to the theory of nonlinear partial differential equations, as the underlying domain of the solution to such problems has to be solved together with the solution itself. This research is expected to enhance the existing theor ....Nonlinear free boundary problems: Propagation and regularity. This project aims to understand the propagation profile and regularity of two important classes of free boundary problems. Nonlinear free boundary problems arise from many applied fields, and pose great challenges to the theory of nonlinear partial differential equations, as the underlying domain of the solution to such problems has to be solved together with the solution itself. This research is expected to enhance the existing theory of partial differential equations, and extend its applications to new situations such as flow through porous media and spreading of invasive species.Read moreRead less
Propagation and free boundary problems in nonlinear partial differential equations. Understanding the propagation of invasive species, flames and disadvantageous genes is a challenging problem in many areas of modern science. This project develops a new mathematical approach to better understand such propagation problems, where the mathematical model predicts a precise location of the propagating front for future time.
Stable and Finite Morse index solutions and peak solutions of nonlinear elliptic equations. The project aims to produce new results of mathematical interest which are also useful in the applications of mathematics. These should be of use in the study of industrial processes and in the study of the environment.
Singularity, degeneracy and related problems in nonlinear partial differential equations. The aim of this project is to solve some long standing open problems in nonlinear partial differential equations, modeling the processes in various applied sciences. New ideas and techniques will be developed to explain novel phenomena observed in the applied areas.
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