Topics in Discrete Harmonic Analysis

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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.