Decoupling Theory and Exponential Sum Estimates

本课题研究谐波分析中的解耦不等式。这些不等式测量了各种几何曲面（如抛物面、锥面或力矩曲线）的傅里叶变换中的振荡和抵消。这些不等式来自于研究偏微分方程，如Schrödinger方程或波动方程，以及数论中的指数和，指数和可以被认为是编码某些算术数据的傅立叶级数。多年来，这两个领域的工具在某种程度上是相互独立的。这个项目的目的之一是将更多的工具从数论引入傅里叶分析。这两个领域之间的联系将被研究，希望证明解耦不等式适用于更广泛和更一般的曲面类别，并改进这些不等式的定量版本。活动将进一步包括组织一个在线研讨会，几个本科生活动，甚至是由公众可以访问的研究激发的演讲。2015年，Bourgain、Demeter和Guth证明了力矩曲线的解耦定理，由此得出了维诺格拉多夫中值定理（Vinogradov’s Mean Value theorem， VMVT）的主要猜想，这是1935年以来一个长期悬而未决的问题。他们的方法是纯傅立叶分析法。大约在同一时间，Wooley用他的有效同余方法给出了VMVT的纯数论证明。解耦和有效同余是各自独立发展的。该项目的目的之一是进一步研究它们之间的联系。以往从解耦的角度对有效同余进行解释的尝试，产生了新的见解和新的观点。该项目包括继续研究VMVT的进展，希望在谐波分析中发现新的工具，用它们来证明更一般类型表面的解耦估计。该项目的其他目标是通过对局部域的解耦来获得VMVT的改进的定量估计，例如对Riemann zeta函数的应用。此外，工作将在证明不同规范和曲面的解耦估计以及已知信息多于通常给定信息的更精细的情况下完成。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。

This project concerns the study of decoupling inequalities in harmonic analysis. Such inequalities measure the oscillation and cancellation in the Fourier transform of various curved geometric surfaces such as the paraboloid, cone, or moment curve. These inequalities arise from studying partial differential equations such as the Schrödinger equation or wave equation and also from number theory through exponential sums, which can be thought of as Fourier series that encode certain arithmetic data. Over the years, tools in these two areas have developed somewhat independently from each other. One aim of this project is to bring more tools from number theory into Fourier analysis. The connections between these two areas will be studied with hopes of proving decoupling inequalities for a wider and more general class of surfaces and improving upon quantitative versions of these inequalities. Activities will further include organizing an online seminar, several undergraduate activities, and even presentations motivated by the research that are accessible to the public.In 2015, Bourgain, Demeter, and Guth were able to prove a decoupling theorem for the moment curve from which the Main Conjecture in Vinogradov's Mean Value Theorem (VMVT), a longstanding open question from 1935, followed as a corollary. Their method was purely Fourier analytic. At roughly at the same time, Wooley used his method of efficient congruencing to give a purely number theoretic proof of the VMVT. Decoupling and efficient congruencing developed separately and independently of each other. One objective of this project is to further study connections between them. Previous attempts at interpreting ideas from efficient congruencing from the perspective of decoupling have yielded fresh insights and new points of views. The project includes a continued study of progress on VMVT in hopes of uncovering new tools in harmonic analysis to use them to prove decoupling estimates for a more general class of surfaces. Other goals of this project are to obtain improved quantitative estimates for VMVT via decoupling over local fields which has applications, for example, to the Riemann zeta function. Additionally, work will be done on proving decoupling estimates for different norms and surfaces and more refined situations where more information is known than what is typically given.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.