Quantitative Aspects of Arithmetic Statistics

算术统计涉及数论的基本对象，如数域和理想类群，并询问诸如它们可以有多大，或者它们有多常见等问题。它与正在进行的数学研究的大量领域相联系，包括代数论、代数几何、表示论和傅立叶分析。这个项目的重点将是研究算术统计中的一系列问题，并将其应用于算术统计中的其他问题。PI将继续发展他的课程工作，以及相关的研究生和本科水平的图书项目，资金还将通过资助研究生、邀请来访演讲者以及协助哥伦比亚数学圈来支持他所在部门的知识氛围。从技术上讲，这项研究将专注于改善算术统计中的上限和误差项。该项目的一个方面将利用Sato-Shintani Zeta函数的理论来改善各种数域计数结果中的误差项，绕过一个已知的障碍。第二个方面将把调和分析引入Bhargava的平均方法，再次导致更强的定量结果(与第一个方面部分重叠)。第三个方面旨在改进最著名的计数数字字段的一般上限，这是PI与Robert Lemke Oliver的联合工作。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。

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.