Quantitative Aspects of Arithmetic Statistics

算术统计学关注的是数论的基本对象——比如数域和理想类群，并提出诸如它们有多大，或者它们有多普遍等问题。它与大量正在进行的数学研究领域相结合，包括代数数论、代数几何、表示理论和傅立叶分析。这个项目的重点将是研究算术统计中的一系列问题以及在算术统计中的其他问题的应用。PI将继续他在研究生和本科生阶段的课程作业和相关书籍项目的开发，这笔资金也将通过资助研究生、邀请客座演讲者和协助哥伦比亚数学圈来加强他所在部门的学术氛围。在技术上，重点研究算术统计中上界和误差项的改进。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字字段计数结果中的误差项，绕过已知的障碍。第二个方面将把谐波分析带入巴尔加瓦的平均方法，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域的一般上界，这是PI与Robert Lemke Oliver的联合工作。该奖项反映了美国国家科学基金会的法定使命，并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，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.