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Sharp Fourier Restriction Theory

夏普傅里叶限制理论

基本信息

批准号：
EP/T001364/1
负责人：
Diogo Oliveira E Silva
金额：
$ 31.35万
依托单位：
University of Birmingham
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2020
资助国家：
英国
起止时间：
2020 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FT001364%2F1
关键词：
Sharp Fourier Restriction Theory

项目摘要

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.

大自然的效率是惊人的，随处可见：肥皂泡像完美的球体，蜂巢排列在高度有序的六边形晶格中，光线描绘的路径可以最大限度地减少旅行时间。像这样的观察使科学家们形成并成功地依赖于各种极端原理，这些原理塑造了古典和现代物理学的发展。例如，最小作用原理转化为一定数量的最小化-一个泛函-然后可以用来获得给定系统的运动方程。数学分析是一种强大的工具，可以理解和分类各种功能最小化的不同方式。特别引人注目的例子来自谐波分析，这是数学的一个分支，涉及信号（函数）的表示和重建，作为基本谐波的叠加-具有明确规定的持续时间，强度和频率的信号-以及研究合适的操作（滤波，去噪，压缩等）如何影响重建的信号。傅里叶变换是一个强大的工具，是谐波分析的核心，自200年前首次出现以来，一直在塑造数学的历史。最近，人们认识到分析和几何可以通过傅里叶变换通过曲率的概念联系起来。傅里叶限制理论的肥沃研究土壤始于对曲率导致傅里叶变换衰减的观察。反过来，这导致了许多令人惊讶和深入的应用。例如，Schrödinger方程描述了一个物理系统随时间的变化，其中量子效应，如波粒二象性，是重要的。鉴于其色散性质（即不同频率在不同方向传播），根据初始基准的大小量化Schrödinger方程解的大小的某些估计是傅里叶限制理论的直接表现，并在量子力学中发挥关键作用。这个项目是有关的新颖和鲁棒方法的发展，以建立最优（所谓的锐）控制的傅里叶变换在曲率的存在。我们的目标是发现傅里叶限制理论中某些基石不等式的尖锐形式，并描述相应泛函可以最小化的方法。特别是，这将导致对Schrödinger方程的解有更深的理解。我们将研究多线性类似物，主要有两个原因。首先，他们将澄清傅立叶限制理论的一些难以捉摸的测量理论方面，这些方面迄今为止仍然无法理解。其次，多线性泛函处于线性和全非线性现象之间的边界。我们进一步计划开发和使用适当的多线性工具，以便在Terence Tao和Christoph Thiele最近有影响力的工作中考虑的版本中开创非线性傅里叶变换的限制理论。这一新颖而令人兴奋的研究领域的进展预计将对理论基础和应用产生重大影响。