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
Quantum algebras with supersymmetries. The project aims to make fundamental advances in the theory of quantum algebras. It will develop explicit
structure and representation theory of major classes of quantum algebras which are of great importance to
quantum field theory and integrable models with supersymmetries. The intended outcomes include a solution of
the outstanding classification problem for representations of quantum algebras with supersymmetries, which has
remained open for the last tw ....Quantum algebras with supersymmetries. The project aims to make fundamental advances in the theory of quantum algebras. It will develop explicit
structure and representation theory of major classes of quantum algebras which are of great importance to
quantum field theory and integrable models with supersymmetries. The intended outcomes include a solution of
the outstanding classification problem for representations of quantum algebras with supersymmetries, which has
remained open for the last two decades. It will involve newly-developed methods within the theory of quantum
groups, and both the methods and classification will bring new mathematical instruments for the advance of
supesymmetric conformal field theory and soliton spin chain models.Read moreRead less
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
Deformation of singularities through Hodge theory and derived categories. Moduli theory, the modern classification theory of mathematical objects, is a branch of algebraic geometry with applications in wide-ranging areas from the theoretical high-energy physics (dark matter and Higgs boson) to data encryption and correction via cryptography. The aim of this project is to resolve central open problems in this theory. This will be achieved by developing new methods and establishing deeper connecti ....Deformation of singularities through Hodge theory and derived categories. Moduli theory, the modern classification theory of mathematical objects, is a branch of algebraic geometry with applications in wide-ranging areas from the theoretical high-energy physics (dark matter and Higgs boson) to data encryption and correction via cryptography. The aim of this project is to resolve central open problems in this theory. This will be achieved by developing new methods and establishing deeper connections between various dynamic branches of these fields. By undertaking research at the forefronts of these highly active areas, this project will both strengthen the current expertise within the Australian mathematical community and precipitate the advance of Australian high-tech industries. Read moreRead less
Class numbers and discriminants: algebraic and analytic number theory meet. This project aims to investigate connections between analytic and algebraic number theory utilising the theoretical and computational expertise of the research group in number theory at UNSW Canberra. The potential findings are highly significant since the innovative generation of new fundamental knowledge will expand the field, and have cryptographic applications.
The expected outcomes include increased capacity in fun ....Class numbers and discriminants: algebraic and analytic number theory meet. This project aims to investigate connections between analytic and algebraic number theory utilising the theoretical and computational expertise of the research group in number theory at UNSW Canberra. The potential findings are highly significant since the innovative generation of new fundamental knowledge will expand the field, and have cryptographic applications.
The expected outcomes include increased capacity in fundamental science and greater understanding of classical and quantum cryptographic protocols. This project will provide the additional, and substantial, benefit of generating research output, training HDR students, and contributions towards national security.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE230100054
Funder
Australian Research Council
Funding Amount
$432,000.00
Summary
Spectral estimates in the presence of a magnetic field. Estimates on eigenvalues of integral operators are at the core of numerous results in the study of quantum phenomena and in associated mathematical fields. This project aims to establish detailed spectral properties of the integral operators arising in quantum models incorporating magnetic fields. An anticipated goal is the generation of new and significant theoretical results in analysis that will open novel approaches to the use of magnet ....Spectral estimates in the presence of a magnetic field. Estimates on eigenvalues of integral operators are at the core of numerous results in the study of quantum phenomena and in associated mathematical fields. This project aims to establish detailed spectral properties of the integral operators arising in quantum models incorporating magnetic fields. An anticipated goal is the generation of new and significant theoretical results in analysis that will open novel approaches to the use of magnetic differential operators. This is expected to benefit Australian science by invigorating collaboration between mathematics and theoretical physics, by providing research training relevant to emerging quantum science based technology and strengthening research collaborations with world leading scientists.Read moreRead less
Geodesic arcs and surfaces for hyperbolic knots and 3-manifolds. This project aims to use recent breakthroughs in mathematics to determine explicit geometric information on mathematical spaces, namely knot complements and 3-manifolds. These spaces arise in applications across science and engineering. They break into pieces that admit geometry, where hyperbolic geometry is the most common. This project expects to generate new knowledge around a number of open questions and conjectures on the hype ....Geodesic arcs and surfaces for hyperbolic knots and 3-manifolds. This project aims to use recent breakthroughs in mathematics to determine explicit geometric information on mathematical spaces, namely knot complements and 3-manifolds. These spaces arise in applications across science and engineering. They break into pieces that admit geometry, where hyperbolic geometry is the most common. This project expects to generate new knowledge around a number of open questions and conjectures on the hyperbolic geometry of knots and 3-manifolds. Expected outcomes include development of theory, and improved geometric tools. It will benefit the mathematical community through new insights and improved methods, and possibly lead to downstream applications in other scientific fields that rely on geometry. Read moreRead less