Distance Questions, Fourier Restriction, and Beyond

距离问题是关联几何的驱动力，关联几何是研究基本几何对象(如点、直线或圆)的相交模式的数学领域。最著名的距离问题之一被称为埃德·S不同距离问题，该问题要求给定的一组点产生的不同距离的数目最少。另一个例子是单位距离问题，涉及在一组给定点之间可以出现固定距离的最大次数。这些问题推动了工具和思想的发展，这些工具和思想在数学以外的学科中有广泛的应用，如计算机科学、物理和工程。傅里叶限制与傅里叶变换有关，傅里叶变换将函数分解成具有不同振荡频率的片段。傅里叶限制研究了傅里叶变换的一个基本问题：函数的大小与其傅里叶变换的几何形状之间的关系。这种关系在偏微分方程组、数论等领域有着重要的应用。这项研究项目旨在加深对距离问题和傅立叶限制之间相互作用的理解，并开发现代工具，应用于不同的数学领域，包括调和分析、几何测度论和偏微分方程。该项目的主要方向之一是由Falconer猜想推动的，该猜想是对不同距离问题的连续模拟。我们猜想，如果紧集E的Hausdorff维度超过某一门限，则E的距离集必有正测度。通过傅里叶限制理论的新思想：入射几何和几何测度论，获得了解决猜想的最新结果。本项目将继续研究这些方法，并将它们应用于其他相关的距离问题，如一般几何构形、多参数距离和分形度的投影。这项工作旨在进一步研究加权傅里叶限制估计和解耦，并将它们应用于距离问题。该项目预计将揭示离散和连续环境之间更深层次的联系。关于高维圆锥的傅里叶限制的研究将继续使用与多项式方法相关的代数工具。此外，该项目还将探索傅里叶限制在色散偏微分方程式中的进一步应用。这一奖项反映了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。

Distance questions are the driving forces in incidence geometry, a field of mathematics studying intersection patterns of basic geometric objects (such as points, lines, or circles). One of the most famous distance questions is called the Erdös distinct distance problem, which asks for the least number of distinct distances generated by a given set of points. Another example is the unit distance problem, concerning the maximum number of times that a fixed distance can occur among a given set of points. These questions have motivated development of tools and ideas that have wide applications in disciplines beyond mathematics, such as computer sciences, physics, and engineering. Fourier restriction concerns the Fourier transform, which decomposes a function into pieces with different frequencies of oscillation. Fourier restriction studies a fundamental question about Fourier transform: the relation between the size of a function and the geometry of its Fourier transform. This relation has important applications in partial differential equations, number theory, and other areas. This research project aims to further the understanding of the interplay between distance questions and Fourier restriction, as well as to develop modern tools with applications in various areas of mathematics, including harmonic analysis, geometric measure theory, and partial differential equations.One of the main directions in the project is driven by Falconer's conjecture, a continuous analogue of the distinct distance problem. It is conjectured that the distance set of a compact set E must have positive measure if the Hausdorff dimension of E exceeds a certain threshold. Recent results towards resolving the conjecture were obtained via new ideas from Fourier restriction theory: incidence geometry and geometric measure theory. This project will continue investigation of these approaches and apply them to other related distance questions such as general geometric configurations, multiparameter distances, and projections of fractal measures. The work aims to further the study of weighted Fourier restriction estimates and decoupling and to apply them to distance questions. The project is expected to reveal deeper connections between the discrete and continuous setting. An investigation of Fourier restriction for the cone in high dimensions will be continued using algebraic tools in connection with polynomial methods. In addition, the project will explore further applications of Fourier restriction in dispersive partial differential equations.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.