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.

该项目涉及调和分析中解耦不等式的研究，此类不等式测量抛物面、圆锥体或矩曲线等各种弯曲几何表面的傅立叶变换中的振荡和抵消。这些不等式源于研究偏微分方程。薛定谔方程或波动方程，以及从数论到指数和，可以被认为是编码某些算术数据的傅立叶级数多年来，这两个工具。该项目的一个目标是将更多的数论工具引入傅立叶分析中，以期证明更广泛和更通用的曲面类型的解耦不等式。活动将进一步包括组织在线研讨会、一些本科生活动，甚至是由研究推动的向公众开放的演讲。2015 年，布尔根、德米特和古斯能够做到这一点。为了证明矩曲线的解耦定理，维诺格拉多夫中值定理 (VMVT) 中的主要猜想是 1935 年的一个长期悬而未决的问题，他们的方法是纯粹的傅里叶分析，大约在同一时间。使用他的高效同余方法给出了 VMVT 的解耦和高效同余的纯数论证明，其中一个目标是独立发展的。该项目旨在进一步研究它们之间的联系，之前从解耦的角度解释有效一致性的想法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，希望能够发现新的工具。该项目的其他目标是通过对局部场进行解耦来获得改进的 VMVT 定量估计，该解耦可应用于黎曼方程等。此外，zeta 函数还反映出，我们将致力于证明不同规范和表面以及已知信息比通常给出的信息更多的更精细情况下的解耦估计。该奖项的法定使命是通过使用基金会的评估进行评估，并被认为值得支持。智力价值和更广泛的影响审查标准。