Oscillatory Integrals and Falconer's Conjecture

本课题是关于傅里叶分析中的限制理论。这个领域关注的是在一些弯曲的物体上具有傅里叶变换（频率）支持（最多非零）的函数，如球体或锥体。这样的函数自然地出现在科学和数学的几个领域：Schrödinger方程、波动方程和数论的研究中。例如，线性波动方程的解可以表示为在圆锥上支持傅里叶变换的函数。研究这些函数可以让我们理解波是如何随时间演化的。在数论中，人们可以通过估计丢芬图方程（具有整数系数的多项式方程）的函数来计算整数解的个数。也就是说，如果对应的函数是集中的，那么丢番图方程就会有很多整数解。解的个数的上限可以根据函数的展开程度给出。这个项目将集中在傅里叶支持的曲率如何防止函数集中。工作将集中在振荡积分和有关Falconer的猜想。后者是一个未解决的问题，涉及紧致d维空间中点之间的欧几里得距离集。振荡积分的课题涉及限制猜想、霍曼德算子和解耦问题。对于限制猜想，斯坦的限制猜想将在高维空间和三维空间进行研究。对于Hörmander算子，将Bochner-Riesz猜想视为不满足布尔甘“一般失效”条件的Hörmander算子来研究。工作将围绕Falconer猜想在径向投影的维度上进行。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.