Tomographic Fourier Analysis

这个项目属于欧几里得谐波分析领域，特别是所谓的傅立叶变换的限制理论。这个数学理论关注的是在欧几里得空间中以不同方向传播的波族可以相互作用的方式，并建立了有效估计这种相互作用的深度不等式。在过去的十年里，这个领域有了显著的增长和影响，大大加深了它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。这个项目特别强调发展一种强大的、广泛适用的新方法，以及它对傅立叶限制理论的潜力。这种方法，自然地被称为层析傅立叶分析，旨在揭示空间中波的叠加（称为傅立叶扩展）可以通过它们的“截面”或“切片”进行有效研究的程度。这一简单的思想为傅立叶分析的经典方法应用于当代谐波分析问题开辟了一条新的、直接的途径。具体目标是建立一系列重要的推测不等式，这些推测不等式控制了经典层析变换中的傅里叶扩展，例如x射线和Radon变换（即所谓的Mizohata-Takeuchi和Stein猜想）。特别是，建立这样的控制将深刻地加强长期以来备受瞩目的谐波分析与组合几何的接口，并澄清著名的傅立叶限制和Kakeya猜想之间的神秘关系。上述类型的复杂波状现象遍及数学科学，并且众所周知难以理解。在提出的研究中开发的工具允许以纯几何和组合的方式来观察潜在的振荡结构（所谓的振荡积分算子）。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，有潜力通过谐波分析和反问题社区之间的新型双向交互使数学受益。

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.