Oscillatory Integrals and Falconer's Conjecture

该项目是关于傅立叶分析中的限制理论。这个领域是关于一些曲面物体(如球体或锥体)上支持(最多非零)傅立叶变换(频率)的函数。这样的函数自然出现在科学和数学的几个领域：研究薛定谔方程、波动方程和数论。例如，线性波动方程的解可以表示为锥面上支承傅里叶变换的函数。研究这些函数可以让我们了解波是如何随时间演变的。在数论中，人们可以通过估计某些丢番图方程(具有整系数的多项式方程)的函数来计算其整数解的个数。也就是说，如果相应的函数是集中的，那么人们预计丢番图方程会有许多整数解。根据函数的分布程度，可以给出解的数量的上限。这个项目将集中在傅里叶支撑曲率如何防止函数集中。工作将集中在振荡积分，并与法尔科纳猜想有关。后者是关于紧d维空间中点之间的欧几里得距离集的一个悬而未决的问题。关于振荡积分的投影涉及限制猜想、Hormander算子和解耦问题。对于限制猜想，Stein的限制猜想将在更高的维度和三维进行研究。对于Hörmander算子，Bochner-Riesz猜想将被视为不满足Bourain“一般失效”条件的Hörmander算子。将围绕福尔科纳猜想的应用在径向投影的维度上进行工作。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。

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.