Decoupling Theory and Exponential Sum Estimates

解耦理论和指数和估计

• 批准号: 2154531
• 负责人: Zane Li
• 金额: $ 10.21万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-08-01 至 2023-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2154531&HistoricalAwards=false
• 关键词: Decoupling; Theory; Exponential; Sum; Estimates

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.

该项目涉及对谐波分析中解耦不平等的研究。这种不平等衡量了各种弯曲几何表面（例如抛物线，锥形或力矩曲线）的傅立叶变换中的振荡和取消。这些不平等是由研究部分微分方程（例如schrödinger方程或波动方程）以及数字理论通过指数总和来引起的，可以将其视为编码某些算术数据的傅立叶序列。多年来，这两个领域的工具彼此独立发展。该项目的一个目的是将更多的工具从数字理论带入傅立叶分析。这两个区域之间的连接将进行研究，希望为更广泛，更一般的表面提供脱钩不平等，并改善这些不平等的定量版本。活动将进一步包括组织在线开创性，几项本科活动，甚至是由公众可以访问的研究所激发的。他们的方法纯粹是傅立叶分析。大致同时，伍利使用了他的有效一致性方法给出了VMVT的纯粹数字理论证明。脱钩和有效的一致性彼此独立发展。该项目的一个目的是进一步研究它们之间的联系。以前的尝试从脱钩的角度从有效的一致性来解释思想已经产生了新的见解和新的观点。该项目包括对VMVT的进度进行的持续研究，希望在谐波分析中发现新工具，以使用它们证明对更一般类别的表面进行解耦估计。该项目的其他目标是通过将其分解为具有应用程序的本地字段，例如在Riemann Zeta函数上，获得了VMVT的改进定量估计值。此外，将在为不同的规范和表面提供脱钩估计值以及比通常给出的信息更多的信息的情况下进行的脱钩估计。该奖项反映了NSF的法定任务，并被认为是通过基金会的知识分子优点和更广泛的影响审查标准通过评估来评估的。