Fourier Analysis in Arithmetic Statistics

This award concerns research in arithmetic statistics which is the branch of number theory which considers "arithmetic objects" such as number fields and ideal class groups, and asks questions such as how many there are or how large they can be. The subject interfaces with a large number of areas of ongoing mathematical research, and this project will focus on connections to Fourier analysis, which appear particularly ripe for further study and development. These connections will be developed both for their intrinsic value, as well as for their utility as building blocks in other parts of arithmetic statistics. The PI will also continue his development of coursework (with an associated book project), his efforts to invite a variety of external speakers to Columbia, and his outreach activities to high school students. Part of the research will consist of finite Fourier transform computations, further developing lines where the PI and his collaborators have enjoyed success in the past. Another part will improve error terms in various number field counting results, bypassing a known obstacle. The research will also study a variety of associated Sato-Shintani zeta functions outside their regions of absolute convergence, where Fourier analysis is the key to proving that they are defined at all. 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.