Tomographic Fourier Analysis

这个项目在于欧几里德调和分析领域，特别是所谓的限制理论的傅里叶变换。这一数学理论关注的是欧几里德空间中不同方向传播的波族相互作用的方式，并建立了有效估计这种相互作用的深刻不等式。这一领域在过去十年中取得了显着的增长和影响，大大加深了与其他数学分支的联系，如微分方程，组合几何，代数几何和数论。该项目特别强调开发一种功能强大且广泛适用的新方法，以及其改变傅立叶限制理论的潜力。这种方法，自然称为层析傅立叶分析，旨在揭示在何种程度上叠加波在空间（称为傅立叶扩展）可以有效地研究通过他们的“部分”或“切片”。这个简单的想法打开了一个新的和直接的路线，经典的傅里叶分析方法可能适用于当代问题的谐波分析。具体目标是建立一系列重要的拓扑不等式，控制傅立叶扩展的经典层析变换，如X射线和氡变换（所谓的Mizohata-Takeuchi和Stein拓扑）。特别是，建立这样的控制将深刻地加强调和分析与组合几何的长期高调接口，并澄清著名的傅里叶限制和Kakeya定理之间的神秘关系。上述类型的复杂波状现象遍布数学科学，并且众所周知难以理解。在拟议的研究中开发的工具允许基本的振荡结构（所谓的振荡积分算子）被视为在纯粹的几何和组合的方式。从长远来看，这具有重大应用和效益的潜力。此外，该方法（层析傅立叶分析），正如其名称所示，有可能通过调和分析和反问题社区之间的新型双向互动来使数学受益。

This project lies in the field of euclidean harmonic analysis, and in particular the so-called restriction theory of the Fourier transform. This mathematical theory concerns the manner in which families of waves propagating in different directions in euclidean space can interact, and establishes deep inequalities that estimate this interaction effectively. This area has seen remarkable growth and impact over the last decade, considerably deepening its connections with other branches of mathematics, such as differential equations, combinatorial geometry, algebraic geometry and number theory. This project places particular emphasis on the development of a powerful and widely-applicable new methodology, and its potential to transform Fourier restriction theory. This methodology, naturally termed Tomographic Fourier Analysis, is designed to reveal the extent to which superpositions of waves in space (referred to as Fourier extensions) may be studied effectively via their "sections" or "slices". This simple idea opens a new and direct route by which classical methods of Fourier analysis may be applied to contemporary problems in harmonic analysis. The specific objectives are to establish a range of important conjectural inequalities that control Fourier extensions in terms of classical tomographic transforms, such as the X-ray and Radon transforms (the so-called Mizohata-Takeuchi and Stein conjectures). In particular, establishing such control would profoundly strengthen the longstanding high-profile interface of harmonic analysis with combinatorial geometry, and clarify the mysterious relationship between the celebrated Fourier restriction and Kakeya conjectures.Complex wave-like phenomena of the type described above pervade the mathematical sciences, and are notoriously difficult to understand. The tools developed in the proposed research allow the underlying oscillatory structures (so-called oscillatory integral operators) to be viewed in purely geometric and combinatorial ways. This has the potential for significant applications and benefits in the longer term. Furthermore, the methodology (Tomographic Fourier Analysis), as its name indicates, has the potential to benefit mathematics through novel two-way interactions between the harmonic analysis and inverse problems communities.