Sharp Fourier Restriction Theory

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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 well-specified 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 wave-particle 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 equation 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 (so-called 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 measure-theoretical 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.