Topics in Discrete Harmonic Analysis

平均值提供了对广泛对象的基本洞察，因为它们将复杂的数据聚合到一个数字中。它们以不同的形式出现在数学的许多领域。中心主题将是研究素数上的平均值，因为它们出现在素数的一类重要推广中，即抽象数域设置。这类无限的素数字段揭示了支配素数的基本原理。事实证明，在数域设置中，关于这些问题的证明的共性和差异为传统上回避质数的问题提供了新的见解。这些问题包括所谓的哥德巴赫猜想，素数中出现的模式，以及相关的和新的问题。这项研究将与针对下一代STEM学生和新兴研究人员的培训计划同时进行。例如，将继续培训研究生，并领导一个项目，指导来自佐治亚州约70所高中的700名高中生参加双重招生课程，如线性代数和矢量微积分。将实施一个积极的离散调和分析研究项目。数域环境中的一系列问题有望为甚至像数域环境中的哥德巴赫猜想这样的经典问题提供新的见解。这些洞察力来自于对素数上平均值的傅里叶变换的精细分析，作为最容易理解的例子。因此，关于哥德巴赫猜想的问题就与平均值的二阶和三阶卷积有关。关于数域设置中的多线性平均值的进一步问题将被继续探讨。需要研究的是关于曲面上的平均值的新问题，这些曲面结合了连续和离散元素的重要元素。这项研究将补充对换向器的持续研究，以及关于被调查操作员的Schatten规范的一系列新结果。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。

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.