A new approach to compressed sensing. Compressed sensing is an exciting new paradigm promising vastly improved signal sampling and reconstruction in a wide variety of applications including digital cameras, mobile phones and MRI machines. This project will explore a newly discovered approach to compressed sensing which uses mathematical arrays known as hash families.
Parametrised gauge theory and positive scalar curvature. This project aims to study innovative extensions of Seiberg-Witten gauge theory with new applications to the topology of metrics of positive scalar curvature on four-dimensional manifolds. Since Atiyah-Bott, Donaldson, Hitchin, and Seiberg-Witten’s work on various equations in gauge theory, profound applications have changed the geometry and topology of low dimensional manifolds. Parametrised index theory has obtained deep results on the t ....Parametrised gauge theory and positive scalar curvature. This project aims to study innovative extensions of Seiberg-Witten gauge theory with new applications to the topology of metrics of positive scalar curvature on four-dimensional manifolds. Since Atiyah-Bott, Donaldson, Hitchin, and Seiberg-Witten’s work on various equations in gauge theory, profound applications have changed the geometry and topology of low dimensional manifolds. Parametrised index theory has obtained deep results on the topology of metrics of positive scalar curvature in higher dimensions, but these methods do not work in the case of the fourth dimension. This project will develop (parametrised) Seiberg-Witten gauge theory as a new approach to the study of the topology of metrics of positive scalar curvature in four dimensions. Expected outcomes include new invariants related to positive scalar curvature in four dimensions.Read moreRead less
Propagation via nonlinear partial differential equations. This project aims to develop new theories in nonlinear partial differential equations to better understand propagation phenomena. Propagation occurs in various forms, such as the spreading of invasive species, infectious diseases or cancer cells, or the progression of the healing front of a wound. This project aims to understand propagation speed and profile, criteria for spreading and vanishing, and other qualitative properties of the eq ....Propagation via nonlinear partial differential equations. This project aims to develop new theories in nonlinear partial differential equations to better understand propagation phenomena. Propagation occurs in various forms, such as the spreading of invasive species, infectious diseases or cancer cells, or the progression of the healing front of a wound. This project aims to understand propagation speed and profile, criteria for spreading and vanishing, and other qualitative properties of the equations. The project will develop new mathematical theories, and build bridges between the theories and applications.Read moreRead less
Multiplicative structure of rational functions. This project aims to develop new methods of investigating fundamental number theoretic notions of torsion and multiplicative dependence between objects of great interest such as rational functions and their values. This includes investigating such celebrated objects as torsion points on elliptic curves and torsion subgroups on algebraic varieties. The goal is to develop new methods and make pivotal advances towards solving several fundamental probl ....Multiplicative structure of rational functions. This project aims to develop new methods of investigating fundamental number theoretic notions of torsion and multiplicative dependence between objects of great interest such as rational functions and their values. This includes investigating such celebrated objects as torsion points on elliptic curves and torsion subgroups on algebraic varieties. The goal is to develop new methods and make pivotal advances towards solving several fundamental problems where multiplicative dependence plays a crucial role. The expected outcome is to provide deeper understanding of the intriguing nature of torsion and multiplicative dependence and thus open new perspectives for their applications in number theory and beyond.Read moreRead less
Probing the earth and the universe with microlocal analysis. This project aims to use the theory of microlocal analysis to determine the amount of information one can recover about the earth and the universe by making observations on wave propagation. In addition to applications to seismic imaging and cosmology, this project will generate new knowledge in the field of differential geometry and dynamical systems. This will be accomplished by formulating the tomography problem in the language of d ....Probing the earth and the universe with microlocal analysis. This project aims to use the theory of microlocal analysis to determine the amount of information one can recover about the earth and the universe by making observations on wave propagation. In addition to applications to seismic imaging and cosmology, this project will generate new knowledge in the field of differential geometry and dynamical systems. This will be accomplished by formulating the tomography problem in the language of differential geometry and introduce new analysis techniques to study them. Expected outcome of this project will be new rigidity type results in Lorentzian and Riemannian geometry. There is also the potential for downstream impacts in seismic and cosmological imaging.Read moreRead less
Discrete Projective Differential Geometry: Comprehensive Theory and Integrable Structure. Differential geometry has been developed over centuries by the most distinguished of mathematicians and its applicability in the mathematical and physical sciences is beyond doubt. However, both natural and man-made structures are inherently discrete. Discrete differential geometry constitutes a relatively new and active research area located between pure and applied mathematics which is more fundamental th ....Discrete Projective Differential Geometry: Comprehensive Theory and Integrable Structure. Differential geometry has been developed over centuries by the most distinguished of mathematicians and its applicability in the mathematical and physical sciences is beyond doubt. However, both natural and man-made structures are inherently discrete. Discrete differential geometry constitutes a relatively new and active research area located between pure and applied mathematics which is more fundamental than differential geometry in that it aims to establish an autonomous discrete analogue from which differential geometry may be derived via an appropriate continuum limit. Even though discrete differential geometry has reached a high degree of sophistication, this project seeks to deliver the first comprehensive theory in this area. Read moreRead less
Connections in low-dimensional topology. This project aims to resolve important open questions in low-dimensional topology, by connecting hyperbolic geometry to invariants arising from quantum topology, cluster algebras, and spinors.
The spaces studied in this project, namely 3-manifolds and knots, arise in applications across engineering and science. The project expects to generate new insights into these spaces by applying tools connecting them to hyperbolic geometry.
Expected outcomes inclu ....Connections in low-dimensional topology. This project aims to resolve important open questions in low-dimensional topology, by connecting hyperbolic geometry to invariants arising from quantum topology, cluster algebras, and spinors.
The spaces studied in this project, namely 3-manifolds and knots, arise in applications across engineering and science. The project expects to generate new insights into these spaces by applying tools connecting them to hyperbolic geometry.
Expected outcomes include efficient techniques to compute important data about 3-manifolds and knots, particularly certain polynomials encoding geometry, and exact calculations of circle packings.
This should provide significant benefits, such as progress on difficult conjectures in hyperbolic geometry.
Read moreRead less
Advances in index theory. The laws of nature are often expressed in terms of differential equations, which if elliptic, have an index being the number of solutions minus the number of constraints imposed. The Atiyah-Singer Index Theorem gives a striking calculation of this index and the projects involve innovative extensions of this theory with novel applications.
Advances in Index Theory. The laws of nature are often expressed in terms of differential equations which, if 'elliptic', have an 'index' being the number of solutions minus the number of constraints imposed. The Atiyah-Singer Index Theorem gives a striking calculation of this 'index', and this project involves innovative extensions of this theory with novel applications.
Geodetic groups: foundational problems in algebra and computer science. The project aims to resolve important and longstanding open problems in Geometric Group Theory and Theoretical Computer Science. Since the 1980s researchers have conjectured that the geometric property of being geodetic is equivalent to several purely algebraic, algorithmic, and language-theoretic characterisations.
The project team's expertise in geodesic properties of groups, the interaction between formal languages and g ....Geodetic groups: foundational problems in algebra and computer science. The project aims to resolve important and longstanding open problems in Geometric Group Theory and Theoretical Computer Science. Since the 1980s researchers have conjectured that the geometric property of being geodetic is equivalent to several purely algebraic, algorithmic, and language-theoretic characterisations.
The project team's expertise in geodesic properties of groups, the interaction between formal languages and groups, and the theory of rewriting systems, together with recent breakthroughs by the team ensures that significant results can be expected.
Benefits include training research students and postdoctoral researchers in cutting-edge techniques, and advancing fundamental knowledge in mathematics and computer science.Read moreRead less