Kakeya sets and rectifiability

本计画研究傅立叶分析与平面集合几何之间的相互影响。傅里叶变换将函数分解为不同频率的纯振荡，在纯数学之外有许多应用，包括信号处理，数据压缩和医学成像。尽管如此，关于傅里叶变换收敛的许多基本问题仍然不为人所知。这些问题与Kakeya集的几何性质有关，Kakeya集是包含多个方向的线但总面积很小的形状。本项目使用几何测度理论和调和分析等各个领域的技术分析挂谷型集。此外，为了使更多的低年级学生了解数学分析，并使更多的高年级学生了解该领域的最新研究，主要研究员将继续作为数学夏令营的讲师和导师，参加讲习班和会议。本项目的目的是研究挂谷型集合的性质以及投影和可求正性之间的定量关系。这些主题密切相关。例如，通过投影平面上的点线对偶，投影映射的几何性质可以用来证明平面Kakeya集的存在性。这个定理的定量类似物将被认为是使用工具，如多尺度分析。多尺度分解也将在度量空间的一般设置中进行研究;这些考虑因素与理论和算法计算机科学以及度量几何学的应用有关。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

This project studies the interplay between Fourier analysis and the geometry of planar sets. The Fourier transform, which decomposes a function into pure oscillations at various frequencies, has found many applications outside of pure mathematics, including to signal processing, data compression, and medical imaging. Despite this, many fundamental questions about the convergence of the Fourier transform are still not well-known. Such questions are connected to the geometric properties of Kakeya sets, which are shapes that contain lines in many directions but have small total area. This project analyzes Kakeya-type sets using techniques from various fields, including geometric measure theory and harmonic analysis. In addition, the Principal Investigator will continue to be involved as an instructor and mentor in summer math camps and to participate in workshops and conferences, in order to introduce more junior students to mathematical analysis, and to introduce more advanced students to current research in the area.The aim of this project is to study properties of Kakeya-type sets and quantitative relations between projections and rectifiability. These topics are closely related. For example, via point-line duality in the projective plane, geometric properties of projection mappings can be used to prove the existence of planar Kakeya sets. Quantitative analogues of this theorem will be considered using tools such as multiscale analysis. Multiscale decompositions will also be investigated in the general setting of metric spaces; such considerations are relevant for applications to theoretical and algorithmic computer science as well as to metric geometry.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.