Quantitative Aspects of Arithmetic Statistics

算术统计学关注数论的基本对象-例如数域和理想类群，并询问诸如它们可以有多大或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，包括代数数论，代数几何，表示论和傅立叶分析。这个项目的重点将是算术统计中的一系列问题的研究和算术统计中其他问题的应用。PI将继续他的课程开发，与相关的书籍项目，在研究生和本科生的水平，和资金也将加强在他的家庭部门的学术氛围，通过资助研究生，邀请客座演讲者，并协助哥伦比亚数学圈。技术上，研究将集中在算术统计中的上限和误差项的改进。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域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引入Bhargava的平均方法，再次导致更强的定量结果（与第一方面部分重叠）。第三个方面旨在改进最著名的计数数域的一般上限，这是PI与Robert Lemke奥利弗的联合工作。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

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.