Harmonic analysis and dispersive partial differential equations. This project aims to develop theoretical results and practical techniques in the study of Partial Differential Equations. Harmonic analysis is used to study these equations; in which a system’s local behaviour is used to analyse global properties, using techniques such as the Fourier transform. The project will investigate central problems in the area, revealing deep connections between analysis and geometry, and apply these to stu ....Harmonic analysis and dispersive partial differential equations. This project aims to develop theoretical results and practical techniques in the study of Partial Differential Equations. Harmonic analysis is used to study these equations; in which a system’s local behaviour is used to analyse global properties, using techniques such as the Fourier transform. The project will investigate central problems in the area, revealing deep connections between analysis and geometry, and apply these to study the solutions’ long-term behaviour to non-linear equations. Expected outcomes include theoretical results and practical techniques to solve non-linear dispersive equations, which arise in quantum and fluid mechanics.Read moreRead less
Nonlinear harmonic analysis and dispersive partial differential equations. This proposal is devoted to linear and nonlinear harmonic analysis. It aims to unify the most significant attributes of harmonic analysis such as restriction estimates, dispersive properties of differential operators, spectral multipliers, uniform Sobolev estimates and sharp Weyl formula. Such unification will strongly improve tools for mathematical modelling in all areas of technology and science. Notable applications in ....Nonlinear harmonic analysis and dispersive partial differential equations. This proposal is devoted to linear and nonlinear harmonic analysis. It aims to unify the most significant attributes of harmonic analysis such as restriction estimates, dispersive properties of differential operators, spectral multipliers, uniform Sobolev estimates and sharp Weyl formula. Such unification will strongly improve tools for mathematical modelling in all areas of technology and science. Notable applications include medical imaging, fluid dynamics and subatomic modelling using quantum interpretation.
It will solve several important open problems in spectral analysis of partial differential operators and develop new cutting-edge techniques in harmonic analysis with application to nonlinear partial differential equations.Read moreRead less
Finite dimensional integrable systems and differential geometry. Mathematical models of many processes in science (physics, engineering) and in the real world (nature, economics) are governed by complicated systems of differential equations. An important, distinguished class of such models is described by integrable systems, the systems for which one can provide a comprehensive qualitative picture, and in many cases, a complete solution. Using recently developed, powerful methods of integrable s ....Finite dimensional integrable systems and differential geometry. Mathematical models of many processes in science (physics, engineering) and in the real world (nature, economics) are governed by complicated systems of differential equations. An important, distinguished class of such models is described by integrable systems, the systems for which one can provide a comprehensive qualitative picture, and in many cases, a complete solution. Using recently developed, powerful methods of integrable systems and differential geometry, this project will focus on a range of important, interconnected theoretical problems in both disciplines. The expected outcomes will provide new, deep, mathematically and physically significant results which will lead to applications and developments across a range of fields.Read moreRead less
Theory and applications of symmetries of relational structures. Relational structures underpin information science, and symmetries of the structures determine their behaviour. This project exploits the interplay between several distinct branches of mathematics concerned with symmetries of relational structures, and will lead to new understandings and breakthroughs of high theoretical interest.
Symmetries in real and complex geometry. This project concerns an important area of abstract modern geometry. The results and techniques of the project will lead to significant progress in this area. It will benefit the national scientific reputation, strengthen the research profile of the home institutions, and provide training to young researchers.