Oscillatory Integrals and Falconer's Conjecture

振荡积分和福尔科纳猜想

• 批准号: 2424015
• 负责人: Hong Wang
• 金额: $ 17.93万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-03-01 至 2024-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2424015&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.

该项目涉及傅里叶分析中的限制理论，涉及某些弯曲物体（例如球体或圆锥体）上支持傅里叶变换（频率）的函数（最多非零）。科学和数学：例如，在薛定谔方程、波动方程和数论的研究中，线性波动方程的解可以表示为圆锥上支持傅里叶变换的函数。研究这些函数可以让人们了解波是如何产生的。在数论中，我们可以通过估计某些丢番图方程（具有整数系数的多项式方程）的整数解来计算这些函数，即，如果相应的函数是集中的，则可以预期丢番图方程有多个。整数解的数量可以根据函数的分布程度给出。该项目将重点研究傅立叶支持的曲率如何阻止函数。这项工作将集中于振荡积分并与福尔科纳猜想相关，后者是一个关于紧 d 维空间中点之间的欧几里得距离集的未解决问题。振荡积分的项目涉及限制猜想。 Hormander算子，以及解耦问题，对于限制猜想，将在高维和三维上研究Stein的限制猜想。 Bochner-Riesz 猜想将通过将其视为不满足 Bourgain 的“一般失效”条件的 Hörmander 算子来进行研究。将在径向投影的维度上开展工作，并围绕 Falconer 猜想进行应用。该奖项反映了 NSF 的法定使命，并已被视为。值得通过使用基金会的智力优点和更广泛的影响审查标准进行评估来支持。