Kakeya sets and 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.