Decoupling Theory and Exponential Sum Estimates

本项目是关于调和分析中解耦不等式的研究。这种不等式衡量了各种曲面(如抛物面、锥面或力矩曲线)在傅里叶变换中的振荡和抵消。这些不等式源于对偏微分方程式(如薛定谔方程或波动方程)的研究，以及从数论到指数和的研究，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相互独立地发展起来。这个项目的目标之一是将更多的数论工具引入傅里叶分析。将研究这两个领域之间的联系，希望证明更广泛和更一般类别的曲面的解耦合不等式，并改进这些不等式的量化版本。活动还将包括组织一个在线研讨会，几个本科生活动，甚至是对公众开放的研究激励的演讲。2015年，Bourain、Demeter和Guth证明了矩曲线的一个解耦定理，维诺格拉多夫中值定理(VMVT)中的主要猜想是作为推论得出的。维诺格拉多夫中值定理(VMVT)是1935年的一个长期悬而未决的问题。他们的方法是纯粹的傅立叶分析。大约在同一时间，Wooley用他的有效同余方法给出了VMVT的一个纯粹的数论证明。脱钩和有效的同余相互独立地发展。这个项目的一个目标是进一步研究它们之间的联系。以前试图从脱钩的角度来解释有效一致的思想，已经产生了新的见解和新的观点。该项目包括对VMVT进展的持续研究，希望发现调和分析中的新工具，以使用它们来证明对更一般类型的曲面的去耦合估计。该项目的其他目标是通过对具有应用的局部场的解耦来获得改进的VMVT的定量估计，例如，Riemann Zeta函数。此外，还将努力证明不同规范和表面的脱钩估计，以及比通常情况下已知更多信息的更精细的情况。该奖项反映了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。

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.