Topics in Discrete Harmonic Analysis

Aesthetics提供了对各种对象的基本见解，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。中心主题将是一个平均数的研究，因为它们出现在一个重要的类的推广素数，抽象的号码字段设置。这无限类素数域揭示了支配素数的基本原理。事实证明，这些问题在数域设置中证明的共性和差异，为传统上回避的素数问题提供了新的见解。这些问题包括所谓的哥德巴赫定理，出现在素数中的模式，以及相关的和新的问题。这项研究将与下一代STEM学生和新兴研究人员的培训计划同时进行。例如，继续培训研究生，以及引导一个程序，指导来自格鲁吉亚州约70所高中的700名高中生在双招生班，如线性代数和向量微积分。数域设置中的一系列问题有望为经典问题（如数域设置中的哥德巴赫猜想）提供新的见解。这些见解来自于对素数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继续讨论。受研究将是新的问题，平均超过表面，结合联合收割机的重要元素的连续和离散的元素。这项研究将补充正在进行的制冷剂研究，并提供一系列关于被调查运营商的Schatten规范的新结果。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的知识价值和更广泛的影响审查标准进行评估来支持。

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.