Quantitative Aspects of Arithmetic Statistics

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Arithmetic statistics is concerned with the fundamental objects of number theory – such as number fields and ideal class groups, and asks questions such as how large they can be, or how common they are. It interfaces with a large number of areas of ongoing mathematical research, including algebraic number theory, algebraic geometry, representation theory, and Fourier analysis. The focus of this project will be the study of a family of problems in arithmetic statistics and applications to other problems in arithmetic statistics. The PI will continue his development of coursework, with associated book projects, at both the graduate and undergraduate levels, and the funding will also bolster the intellectual atmosphere in his home department, by funding graduate students, inviting visiting speakers, and assisting with the Columbia Math Circle.Technically, the research will focus on the improvement of upper bounds and error terms in arithmetic statistics. One aspect of the project will leverage the theory of Sato-Shintani zeta functions to improve error terms in various number field counting results, bypassing a known obstacle. A second aspect will bring harmonic analysis into Bhargava's averaging method, again leading to stronger quantitative results (with partial overlap with the first aspect). A third aspect aims to improve upon the best known general upper bound for counting number fields, which is joint work of the PI with Robert Lemke Oliver.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.