Kakeya sets and rectifiability

挂屋组和可校正性

• 批准号: 2247233
• 负责人: Alan Chang
• 金额: $ 18.05万
• 依托单位: Washington University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-08-15 至 2026-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2247233&HistoricalAwards=false
• 关键词: Kakeya; sets; rectifiability

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.

这个项目研究傅立叶分析和平面集合几何之间的相互作用。傅里叶变换将函数分解成各种频率的纯振荡，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，关于傅里叶变换的收敛的许多基本问题仍然不为人所知。这样的问题与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。这个项目使用不同领域的技术来分析Kakeya类型的集合，包括几何测度论和调和分析。此外，首席调查员将继续担任暑期数学夏令营的导师和导师，并参加工作坊和会议，以向更多的低年级学生介绍数学分析，并向更多的高级学生介绍该领域的当前研究。本项目的目的是研究Kakeya类型集合的性质以及投影和可纠正性之间的定量关系。这些话题是密切相关的。例如，通过射影平面上的点-线对偶，可以利用射影映射的几何性质证明平面Kakeya集的存在性。这一定理的定量类比将使用多尺度分析等工具加以考虑。多尺度分解也将在公制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公制几何的应用相关。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。