Topics in Discrete Harmonic Analysis

离散谐波分析主题

• 批准号: 2247254
• 负责人: Michael Lacey
• 金额: $ 40.01万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-06-01 至 2026-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2247254&HistoricalAwards=false
• 关键词: Topics; Discrete; Harmonic; Analysis

Averages give basic insights into a wide range of objects, since they aggregate complex data into a single number. They appear, in different forms, in many areas of mathematics. The central topic will be a study of averages over prime numbers as they appear in an important class of generalizations of prime numbers, the abstract number field settings. This infinite class of prime number fields sheds light on the underlying principles that govern prime numbers. It turns out that the commonalities and differences in proofs about these questions in the number field settings give new insights into traditionally evasive questions about prime numbers. These questions include the so-called Goldbach conjectures, patterns that emerge in prime numbers, as well as related and new questions. The research will be conducted in parallel with a training program for a next generation of STEM students, and emerging researchers. For example, continued training of graduate students will take place, and as well lead a program instructing 700 High School students from about 70 High Schools across the state of Georgia in dual enrollment classes such as Linear Algebra and Vector Calculus.A vigorous program of study in discrete harmonic analysis will be carried out. A range of questions in the number field setting promise new insights into even classical questions like the Goldbach conjecture in the number field setting. These insights come about by a delicate analysis of the Fourier transforms of averages over prime numbers, as the most accessible example. So, questions about Goldbach conjecture are then related to the second and third order convolutions of the averages. Further questions about multilinear averages in the number field setting will be pursued. Subject to research will be novel questions concerning averages over surfaces that combine important elements of continuous and discrete elements. This research will complement the continuing study of commutators, with a range of new results about the Schatten norms of the operators under investigation.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.

平均值为广泛的对象提供了基本的见解，因为它们将复杂数据汇总到一个数字中。它们以不同形式出现在数学的许多领域中。中心主题将是对质数的平均值的研究，因为它们出现在质数的一系列重要类别（抽象数字字段设置）中。这个无限的质数字段阐明了控制质数的基本原则。事实证明，在数量字段设置中，有关这些问题的证明的共同点和差异为传统上关于质数的传统回避问题提供了新的见解。这些问题包括所谓的Goldbach猜想，质量数量出现的模式以及相关和新问题。该研究将与针对下一代STEM学生和新兴研究人员的培训计划并行进行。例如，将继续对研究生进行培训，并领导一项计划，指导佐治亚州约70名高中的700名高中学生参加双重入学课程，例如线性代数和矢量计算。数字字段设置中的一系列问题有助于对诸如数字字段设置中的Goldbach猜想之类的经典问题的新见解。这些见解是通过对最易于访问的示例对平均值的傅立叶变换的微妙分析来实现的。因此，有关戈德巴赫猜想的问题与平均值的第二和三阶卷积有关。将在数字字段设置中有关多线性平均值的进一步问题。受研究的前提将是关于结合了连续和离散元素的重要要素的表面上的平均值的新问题。这项研究将补充对换向因子的持续研究，并提供有关正在调查的运营商的沙特登规范的一系列新结果。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和更广泛影响的审查标准的评估来通过评估来获得支持的。

项目成果

Endpoint $\ell^r$ improving estimates for prime averages

端点 $ell^r$ 改进了素数平均值的估计

• DOI: 10.4310/mrl.2022.v29.n6.a6
• 发表时间: 2022
• 期刊: Mathematical Research Letters
• 影响因子: 1
• 作者: Lacey, Michael T.;Mousavi, Hamed;Rahimi, Yaghoub
• 通讯作者: Rahimi, Yaghoub