EPSRC Reference: 
EP/T001364/1 
Title: 
Sharp Fourier Restriction Theory 
Principal Investigator: 
Oliveira e Silva, Dr D 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
School of Mathematics 
Organisation: 
University of Birmingham 
Scheme: 
New Investigator Award 
Starts: 
01 October 2019 
Ends: 
30 September 2021 
Value (£): 
246,055

EPSRC Research Topic Classifications: 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
Nature's efficiency is remarkable and everywhere to be seen: soap bubbles resemble perfect spheres, honeycombs are arranged in highly ordered hexagonal lattices, and light rays describe paths that minimise travel time. Observations like these have led scientists to formulate and successfully rely upon various extremal principles which have shaped the development of classical and modern physics. For instance, the principle of least action translates into the minimisation of a certain quantity  a functional  which can then be used to obtain the equations of motion of a given system.
Mathematical Analysis is a source of powerful tools to understand and classify the different ways in which various functionals can be minimised. Especially compelling examples come from Harmonic Analysis, which is the branch of mathematics concerned with the representation and reconstruction of signals (functions) as a superposition of basic harmonics  signals of wellspecified duration, intensity and frequency  as well as the study of how suitable operations (filtering, denoising, compression, etc.) affect the reconstructed signal. The Fourier Transform is a powerful tool that lies at the heart of Harmonic Analysis, and has been shaping the history of mathematics since it first appeared almost 200 years ago. Much more recently, it was understood that analysis and geometry can be linked via the Fourier Transform through the notion of curvature. The fertile research ground of Fourier Restriction Theory starts with the observation that curvature causes the Fourier Transform to decay. In turn, this leads to a number of surprising and deep applications. For instance, the Schrödinger equation describes the changes over time of a physical system in which quantum effects, such as waveparticle duality, are significant. Given its dispersive nature (i.e. different frequencies propagate in different directions), certain estimates quantifying the size of the solutions of the Schrödinger equati in terms of the size of the initial datum are a direct manifestation of Fourier Restriction Theory, and play a key role in quantum mechanics.
This project is concerned with the development of novel and robust methods to establish optimal (socalled sharp) control of the Fourier Transform in the presence of curvature. We aim to discover the sharp form of certain cornerstone inequalities in Fourier Restriction Theory, and to characterise the ways in which the corresponding functionals can be minimised. In particular, this will lead to a deeper understanding of the solutions of the Schrödinger equation. Multilinear analogues will be investigated, for two main reasons. Firstly, they will clarify some elusive measuretheoretical aspects of Fourier Restriction Theory which have hitherto remained unaccessible. Secondly, multilinear functionals lie at the frontier between linear and fully nonlinear phenomena. We further plan to develop and use appropriate multilinear tools in order to inaugurate a restriction theory for the Nonlinear Fourier Transform, in the version considered in recent influential work of Terence Tao and Christoph Thiele. Progress in this novel and exciting research area is expected to have a significant impact on theoretical foundations as well as in applications.

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Organisation Website: 
http://www.bham.ac.uk 