Oscillatory Integrals and Falconer's Conjecture

振荡积分和福尔科纳猜想

• 批准号: 2141426
• 负责人: Hong Wang
• 金额: $ 17.93万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-08-15 至 2024-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2141426&HistoricalAwards=false
• 关键词: Oscillatory; Integrals; Falconer; Conjecture

The project is on the restriction theory in Fourier analysis. This field is concerns functions with Fourier transform (frequencies) supported (non-zero at most) on some curved objects such as a sphere or a cone. Such functions appear naturally in several areas of science and mathematics: in the study of Schrödinger equations, wave equations and number theory. For instance, a solution to the linear wave equation can be represented as a function with Fourier transform supported on a cone. Investigating these functions allows one to understand how waves evolve in time. In number theory, one can count the number of integer solutions to some Diophantine equations (polynomial equations with integer coefficients) by estimating such functions. Namely, if the corresponding functions are concentrated, then one expects the Diophantine equation to have many integer solutions. And an upper bound on the number of solutions can be given in terms of how spread out the functions are. This project will be focused on how the curvature of the Fourier support prevents the functions from being concentrated.The work will be concentrated on oscillatory integrals and related to Falconer's conjecture. The latter is an unsolved question concerning the sets of Euclidean distances between points in compact d-dimensional spaces. The projects on oscillatory integrals concern the restriction conjecture, the Hormander operator, and decoupling questions. For the restriction conjecture, Stein's restriction conjecture will be studied in higher dimensions and in dimension three. For the Hörmander operator the Bochner-Riesz conjecture will be investigated by considering it as a Hörmander operator not satisfying Bourgain's "generic failure" condition. Work will be done on the dimension of radial projections with applications surrounding Falconer's conjecture.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.

该项目在傅立叶分析中是限制理论。该字段是对某些弯曲对象（例如球体或锥体）上支持的傅立叶变换（频率）（最多为零）的函数。这种功能在科学和数学的几个领域中自然出现：在Schrödinger方程，波动方程和数理论的研究中。例如，对线性波方程的解决方案可以表示为在研究这些功能时支持的傅立叶变换的函数，因此可以理解波浪在时间上的发展。在数量理论中，可以通过估计此类功能来计算某些二磷酸方程（具有整数系数的多项式方程）的整数溶液数量。也就是说，如果相应的函数浓缩，那么人们期望双方方程具有许多整数溶液。并且可以根据函数的分散方式给出有关解决方案数量的上限。该项目将集中于傅立叶支持的货币如何阻止功能集中。该工作将集中在振荡性积分上，并与Falconer的概念有关。后者是一个未解决的问题，是关于紧凑的D维空间中点之间欧几里得距离集的集合。有关振荡积分的项目涉及限制概念，荷尔曼德操作员和解耦问题。对于限制的猜想，将在更高的维度和第三维中研究Stein的限制猜想。对于Hörmander运营商，Bochner-Riesz的猜想将通过将其视为不满足Bourgain的“通用失败”条件的Hörmander经营者。将在径向项目的维度上完成，并使用围绕Falconer猜想的应用进行工作。该奖项反映了NSF的法定任务，并通过使用基金会的知识分子优点和更广泛的影响评估标准来评估值得支持。