Decoupling Theory and Exponential Sum Estimates

解耦理论和指数和估计

• 批准号: 2311174
• 负责人: Zane Li
• 金额: $ 10.21万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-01-01 至 2024-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2311174&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。这种不等式测量各种弯曲几何表面（例如抛物面、圆锥体或力矩曲线）的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（例如薛定谔方程或波动方程）的研究，以及通过指数和的数论，可以将其视为编码某些算术数据的傅里叶级数。多年来，这两个领域的工具在某种程度上彼此独立地发展。该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的表面类别的解耦不等式，并改进这些不等式的定量版本。活动还包括组织一次在线研讨会、几项本科生活动，甚至是由研究推动的向公众开放的演讲。 2015 年，Bourgain、Demeter 和 Guth 证明了矩曲线的解耦定理，由此得出了维诺格拉多夫中值定理 (VMVT) 中的主要猜想（1935 年的一个长期悬而未决的问题）。他们的方法纯粹是傅立叶分析。大约在同一时间，伍利使用他的高效同余方法给出了 VMVT 的纯数论证明。解耦和高效一致性是分开且独立发展的。该项目的目标之一是进一步研究它们之间的联系。先前从解耦的角度解释有效一致性思想的尝试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，希望发现谐波分析中的新工具，以使用它们来证明更通用的表面类别的解耦估计。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，该局部场可应用于黎曼 zeta 函数等。此外，还将致力于证明不同规范和表面以及更精细情况下的解耦估计，在这些情况下，已知的信息比通常给出的信息更多。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。