Kakeya sets and rectifiability

本项目研究傅里叶分析与平面集合几何之间的相互作用。傅里叶变换将一个函数分解成不同频率的纯振荡，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性的许多基本问题仍然不为人所知。这些问题与Kakeya集合的几何性质有关，Kakeya集合是一种形状，它包含许多方向的线，但总面积很小。本项目利用几何测量理论和谐波分析等不同领域的技术分析kakya型集合。此外，首席研究员将继续在夏季数学夏令营中担任讲师和导师，并参加研讨会和会议，以向更多的初级学生介绍数学分析，并向更多的高级学生介绍该领域的最新研究。本课题的目的是研究kakeya型集合的性质以及投影与可纠偏性之间的定量关系。这些话题是密切相关的。例如，通过投影平面上的点线对偶，可以利用投影映射的几何性质来证明平面Kakeya集合的存在性。将使用诸如多尺度分析之类的工具来考虑这个定理的定量类比。多尺度分解也将在度量空间的一般设置中进行研究；这些考虑与理论和算法计算机科学以及度量几何的应用相关。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.