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
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
Partial differential equation: Schrodinger operator and long-time dynamics. This project aims to develop new analysis methods associated to the Schrodinger operator, and to solve several challenging problems regarding dispersive partial differential equations (PDE). Long-time dynamics of PDE solutions are a key goal in both pure and applied mathematics, and have been extensively studied by leading mathematicians and mathematical physicists. However, it is unknown how to investigate large soluti .... Partial differential equation: Schrodinger operator and long-time dynamics. This project aims to develop new analysis methods associated to the Schrodinger operator, and to solve several challenging problems regarding dispersive partial differential equations (PDE). Long-time dynamics of PDE solutions are a key goal in both pure and applied mathematics, and have been extensively studied by leading mathematicians and mathematical physicists. However, it is unknown how to investigate large solutions when the order of the PDE's nonlinearity is low. This project expects to develop new methods to attack such problems. The results of the project will be of great importance in mathematics and physics, as many fundamental physical models in areas such as optics, fluid mechanics and quantum mechanics fit the paradigm.Read moreRead less
Monge-Ampere type equations and their applications. The study of Monge-Ampere equations has attracted major attention in mathematics in recent years, due to many significant applications in geometry, physics and applied science. This project aims to resolve challenging problems involving Monge-Ampere type equations, by utilising new ideas and breakthroughs made by the proposer. A comprehensive regularity theory for Monge-Ampere type equations, particularly in the degenerate case, is expected to ....Monge-Ampere type equations and their applications. The study of Monge-Ampere equations has attracted major attention in mathematics in recent years, due to many significant applications in geometry, physics and applied science. This project aims to resolve challenging problems involving Monge-Ampere type equations, by utilising new ideas and breakthroughs made by the proposer. A comprehensive regularity theory for Monge-Ampere type equations, particularly in the degenerate case, is expected to be established. Innovative cutting-edge techniques and interdisciplinary approaches are expected to be developed. Anticipated outcomes of this project include the resolution of outstanding open problems and continuing enhancement of Australian leadership and expertise in a major area of mathematics.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
Discovery Early Career Researcher Award - Grant ID: DE230100415
Funder
Australian Research Council
Funding Amount
$422,154.00
Summary
Rigidity and boundary phenomena for geometric variational problems. The proposed project aims to investigate theoretical properties of thin films and fluid interfaces, which are modelled as surfaces driven by surface tension, possibly in an enclosing container. This project is expected to generate new knowledge in the area of geometric partial differential equations, by utilising new techniques in geometric flows, and by establishing novel methods for boundary value problems. The developed techn ....Rigidity and boundary phenomena for geometric variational problems. The proposed project aims to investigate theoretical properties of thin films and fluid interfaces, which are modelled as surfaces driven by surface tension, possibly in an enclosing container. This project is expected to generate new knowledge in the area of geometric partial differential equations, by utilising new techniques in geometric flows, and by establishing novel methods for boundary value problems. The developed techniques may have far-reaching applications in other areas of mathematical analysis, and the expected results would contribute greatly to the theory of surfaces governed by mean curvature, which arise in various real-world phenomena such as soap bubbles, black hole horizons and bushfire fronts. Read moreRead less
Australian Laureate Fellowships - Grant ID: FL220100072
Funder
Australian Research Council
Funding Amount
$2,490,704.00
Summary
Mathematical Breakthroughs in Wave Propagation. This Fellowship proposal in theoretical mathematics aims to solve three major open problems in wave propagation. These are the long-time behaviour of nonlinear waves, including the behaviour and interaction of solitary waves; the propagation of waves in rough media; and the small-scale behaviour of interacting waves under the assumption of chaotic ray dynamics. The research aims to analyse wave equations that model problems in optical media and wav ....Mathematical Breakthroughs in Wave Propagation. This Fellowship proposal in theoretical mathematics aims to solve three major open problems in wave propagation. These are the long-time behaviour of nonlinear waves, including the behaviour and interaction of solitary waves; the propagation of waves in rough media; and the small-scale behaviour of interacting waves under the assumption of chaotic ray dynamics. The research aims to analyse wave equations that model problems in optical media and waveguides, medical and seismic imaging, and nano-electronic devices. Outcomes and benefits are expected in new mathematical theory, Australian research capability, better algorithms for numerically computing waves, and technological advances in communications, medical imaging, and seismic imaging.Read moreRead less
Mathematics for future magnetic devices. The aim of this project is to develop a mathematical theory and numerical models of stochastic partial differential
equations for magnetic nano-structures. Such materials will yield next-generation magnetic memories with up to
three orders of magnitude faster switching speeds and dramatically increased data storage density. New
mathematical theories will help understand their sensitivity to small random fluctuations that can destroy stored
information. Th ....Mathematics for future magnetic devices. The aim of this project is to develop a mathematical theory and numerical models of stochastic partial differential
equations for magnetic nano-structures. Such materials will yield next-generation magnetic memories with up to
three orders of magnitude faster switching speeds and dramatically increased data storage density. New
mathematical theories will help understand their sensitivity to small random fluctuations that can destroy stored
information. This project aims to revolutionise mathematical modelling of magnetic memories and put Australia at
the forefront of international research. Technological advances to create much smaller and faster memory devices
are expected to enable groundbreaking ways of managing and mining big dataRead moreRead less
Discovery Early Career Researcher Award - Grant ID: DE230100954
Funder
Australian Research Council
Funding Amount
$354,968.00
Summary
Partial Differential Equations, geometric aspects and applications. The study of Partial Differential Equations (PDEs) is a classical and prolific field of research having a fundamental role in the development of mathematical analysis and motivated by important applications in natural and applied sciences.
This project aims to obtain substantial progress in the field of PDEs. The area of mathematical research covered is extremely broad, at the confluence of analysis and geometry, and with many a ....Partial Differential Equations, geometric aspects and applications. The study of Partial Differential Equations (PDEs) is a classical and prolific field of research having a fundamental role in the development of mathematical analysis and motivated by important applications in natural and applied sciences.
This project aims to obtain substantial progress in the field of PDEs. The area of mathematical research covered is extremely broad, at the confluence of analysis and geometry, and with many applications to other areas of mathematics and natural and applied sciences. The results that will be obtained will produce a significant amount of new knowledge in this extremely difficult, but rapidly growing, field, by exploiting international scientific collaborations and interdisciplinary methods.Read moreRead less