Non-linear partial differential equations: Bubbles, layers and stability. This project aims to investigate non-linear elliptic partial differential equations in well-established models in applied sciences. The treatment of them challenges the existing mathematical theory. This project will enrich and expand the mathematical theory in semi-linear elliptic equations to understand the equations under investigation.
New perspectives on nonlocal equations. This project aims at tackling cutting-edge problems in the field of mathematical analysis, with specific focus on nonlocal equations, by introducing innovative approaches and a unified perspective. It focuses on the use of long-range interactions to deeply understand new effects arising in several mathematical problems of great impact.
The research will be performed through stimulating international collaborations, providing exchange opportunities and idea ....New perspectives on nonlocal equations. This project aims at tackling cutting-edge problems in the field of mathematical analysis, with specific focus on nonlocal equations, by introducing innovative approaches and a unified perspective. It focuses on the use of long-range interactions to deeply understand new effects arising in several mathematical problems of great impact.
The research will be performed through stimulating international collaborations, providing exchange opportunities and ideal conditions for students to complete their training.
The expected outcomes include new techniques to solve difficult problems, high impact international research collaborations, training of the next generation of mathematicians and top tier journal publications.Read moreRead less
Singularity and regularity for Monge-Ampere type equations. The Monge-Ampere equation, as a premier nonlinear partial differential equation, arises in several areas including geometry, physics, and optimal transportation. Many important problems and applications are related to the regularity of solutions, which are obstructed by singularities. This project aims to classify the geometry of the singular sets, and to establish a comprehensive regularity theory for general Monge-Ampere type equation ....Singularity and regularity for Monge-Ampere type equations. The Monge-Ampere equation, as a premier nonlinear partial differential equation, arises in several areas including geometry, physics, and optimal transportation. Many important problems and applications are related to the regularity of solutions, which are obstructed by singularities. This project aims to classify the geometry of the singular sets, and to establish a comprehensive regularity theory for general Monge-Ampere type equations by using innovative approaches and developing cutting-edge technologies in partial differential equations. Expected outcomes include the resolution of outstanding open problems. This project will significantly enhance Australia’s leadership and expertise in a major area of mathematics and applications.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE230101165
Funder
Australian Research Council
Funding Amount
$419,420.00
Summary
Geometric Scattering Theory, Resolvent Estimates, and Wave Asymptotics. This project aims to understand how fast the local energy of a wave decays when it propagates in a rough, open system. This projects will generate new knowledge in the mathematical subfields of microlocal analysis and partial differential equations by refining tools such as Carleman estimates, separation of variables, b-vector field analysis, and quasimode constructions. The expected outcome of this project is a novel and co ....Geometric Scattering Theory, Resolvent Estimates, and Wave Asymptotics. This project aims to understand how fast the local energy of a wave decays when it propagates in a rough, open system. This projects will generate new knowledge in the mathematical subfields of microlocal analysis and partial differential equations by refining tools such as Carleman estimates, separation of variables, b-vector field analysis, and quasimode constructions. The expected outcome of this project is a novel and comprehensive mathematical treatment of wave propagation in systems with weaker than Lipschitz regularity. This research should provide significant benefits such as informing predictions about waves in rough systems, including the propagation of seismic waves, and lead to advances in medical and geological imaging.Read moreRead less
Inverse problems with partial data. This project aims to use mathematics, in particular the theory of micro-local analysis, to determine the amount of measurements one needs in order to reconstruct an image by some of the tomography methods commonly used in medical imaging. Expected outcomes of this project include showing that an arbitrarily small set of boundary measurements is sufficient to reconstruct the coefficients of various important partial differential equations such as Schrodinger eq ....Inverse problems with partial data. This project aims to use mathematics, in particular the theory of micro-local analysis, to determine the amount of measurements one needs in order to reconstruct an image by some of the tomography methods commonly used in medical imaging. Expected outcomes of this project include showing that an arbitrarily small set of boundary measurements is sufficient to reconstruct the coefficients of various important partial differential equations such as Schrodinger equation, Dirac operators, and Maxwell equations. In addition to providing a theoretical foundation upon which one can build numerical algorithms, this project will also provide the missing link between inverse problems and unique continuation theory. The downstream impact of this research will lead to more efficient and accurate tomography methods which can be implemented in a range of imaging applications.Read moreRead less
New mathematics for understanding complex patterns in the natural sciences. This project aims to examine the interaction of fundamental two-dimensional patterns such as spots and stripes in reaction-diffusion equations, by developing and extending mathematical techniques. These fundamental planar structures form the backbone of more complex patterns and are, for example, observed in models that describe the propagation of impulses in nerve axons and the formation of vegetation patterns. The futu ....New mathematics for understanding complex patterns in the natural sciences. This project aims to examine the interaction of fundamental two-dimensional patterns such as spots and stripes in reaction-diffusion equations, by developing and extending mathematical techniques. These fundamental planar structures form the backbone of more complex patterns and are, for example, observed in models that describe the propagation of impulses in nerve axons and the formation of vegetation patterns. The future impact of this research will have economic and environmental benefits. For example, the project will develop a deeper understanding of interacting patterns that will provide insights into the role of vegetation in ecosystems that are undergoing desertification.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE180101348
Funder
Australian Research Council
Funding Amount
$328,075.00
Summary
Singularity analysis for ricci flow and mean curvature flow. This project aims to investigate the central problem of singularity formation in Ricci flow and mean-curvature flow by profiling singular solutions and determining their stability and genericity. Geometric flows are powerful and successful ways of understanding classical problems in geometry and topology with applications in disciplines such as materials science and medical imaging. This project will generate significant results in sin ....Singularity analysis for ricci flow and mean curvature flow. This project aims to investigate the central problem of singularity formation in Ricci flow and mean-curvature flow by profiling singular solutions and determining their stability and genericity. Geometric flows are powerful and successful ways of understanding classical problems in geometry and topology with applications in disciplines such as materials science and medical imaging. This project will generate significant results in singularity analysis and will enrich understanding of geometric flows at and past singularities, deepen the theory of geometric flows, and enhance their applications in mathematics and science.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE200101834
Funder
Australian Research Council
Funding Amount
$418,410.00
Summary
The structure of singularities in geometric flows. The proposed research aims to develop our understanding of the structure of singularities in mean curvature and related flows, with certain applications in mind.
Geometric evolution of spaces with symmetries. Symmetries underpin numerous laws of nature and mathematical constructions. This project aims to develop a comprehensive theory of the famous Ricci flow equation in the presence of symmetries. Previous study of this equation has led to many ground-breaking results, such as Perelman's celebrated proof of the century-old Poincaré conjecture. Outcomes are expected to fill major knowledge gaps in mathematics, opening doors to applications in quantum fie ....Geometric evolution of spaces with symmetries. Symmetries underpin numerous laws of nature and mathematical constructions. This project aims to develop a comprehensive theory of the famous Ricci flow equation in the presence of symmetries. Previous study of this equation has led to many ground-breaking results, such as Perelman's celebrated proof of the century-old Poincaré conjecture. Outcomes are expected to fill major knowledge gaps in mathematics, opening doors to applications in quantum field theory, relativity and other fields. Anticipated benefits include strengthening Australia’s leadership in mathematical innovation, advancing the internationalisation of the Australian research scene, and increasing the involvement of women in STEM.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE210100535
Funder
Australian Research Council
Funding Amount
$340,548.00
Summary
Minimal surfaces and singularities of mean curvature flow. The project aims to characterise the geometric structure of minimal surfaces in the variational theory and classify singularities of mean curvature flow. Minimal surfaces are mathematical models of soap films, and their time-varying analogue is mean curvature flow, a dynamic process by which a surface flows to decrease its area as quickly as possible. As a central topic in geometric analysis, the theory of minimal surfaces and mean curv ....Minimal surfaces and singularities of mean curvature flow. The project aims to characterise the geometric structure of minimal surfaces in the variational theory and classify singularities of mean curvature flow. Minimal surfaces are mathematical models of soap films, and their time-varying analogue is mean curvature flow, a dynamic process by which a surface flows to decrease its area as quickly as possible. As a central topic in geometric analysis, the theory of minimal surfaces and mean curvature flow has proven to be a powerful and essential tool in mathematics. The project expects to generate new and significant results in minimal surfaces and singularity analysis of mean curvature flow and enhance potential applications in related disciplines such as computer vision and probability.Read moreRead less