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Discovery Early Career Researcher Award - Grant ID: DE240100447
Funder
Australian Research Council
Funding Amount
$438,847.00
Summary
The geometry of braids and triangulated categories. Triangulated categories play a central role in geometry, algebra, and topology. Their study can uncover deep structure connecting different areas of mathematics. This project aims to use novel approaches to answer fundamental questions about triangulated categories and their symmetries. These symmetries are encoded by braids, which are important objects with many applications across science. The project is expected to benefit Australia by stimu ....The geometry of braids and triangulated categories. Triangulated categories play a central role in geometry, algebra, and topology. Their study can uncover deep structure connecting different areas of mathematics. This project aims to use novel approaches to answer fundamental questions about triangulated categories and their symmetries. These symmetries are encoded by braids, which are important objects with many applications across science. The project is expected to benefit Australia by stimulating research in mathematics and computer science. It will invite connections with leading experts and students around the world and encourage overseas collaboration. There is a potential long-term benefit to cybersecurity, towards the development of new encryption schemes based on braids.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE230100579
Funder
Australian Research Council
Funding Amount
$445,754.00
Summary
The existence and abundance of small bases of permutation groups. This project aims to study bases for permutation groups, which are the mathematical formalisation of symmetry. Bases are crucial to encoding and computing with groups in diverse areas of science. Small bases are desirable for efficiency, but can be hard to find. This project expects to combine techniques from areas of algebra and probability to determine the existence and abundance of bases. Expected outcomes of this project inclu ....The existence and abundance of small bases of permutation groups. This project aims to study bases for permutation groups, which are the mathematical formalisation of symmetry. Bases are crucial to encoding and computing with groups in diverse areas of science. Small bases are desirable for efficiency, but can be hard to find. This project expects to combine techniques from areas of algebra and probability to determine the existence and abundance of bases. Expected outcomes of this project include new methods to address enduring open problems in the study of bases, as well as novel applications of existing techniques. This should provide significant benefits, such as creating and strengthening international collaborations, and building on Australia’s reputation as a powerhouse of finite group theory.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE230100303
Funder
Australian Research Council
Funding Amount
$352,200.00
Summary
New Foundations for Algebraic Geometry. Differential calculus is one of the most important and widely applied areas of mathematics. Differential categories are a modern foundational theory of differential calculus with applications throughout mathematics and computing. This project aims to use differential categories to create new foundations for algebraic geometry, and to generate new knowledge on the connection between algebraic and differential geometry. The generality of these foundations wi ....New Foundations for Algebraic Geometry. Differential calculus is one of the most important and widely applied areas of mathematics. Differential categories are a modern foundational theory of differential calculus with applications throughout mathematics and computing. This project aims to use differential categories to create new foundations for algebraic geometry, and to generate new knowledge on the connection between algebraic and differential geometry. The generality of these foundations will allow for novel applications of algebraic geometry with significant benefits to computer science, such as in machine learning and differentiable programming. We expect this to build Australia's profile in these important fields and help train the next generation of mathematicians.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
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
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