Applications of Analytic Uniformity in Arithmetic Statistics

这个项目将研究算术统计中的问题，这是数学的一个领域，与确定数论中对象的“典型”属性有关。在实践中，通过研究底层对象如何分解成更简单的对象，可以有效地理解算术统计中的许多问题。这个项目的大部分研究将致力于研究这些类型的分解，特别是这些简单物体的组合属性的过程，以提供复杂问题的答案。他还将继续担任本科生和研究生导师。他还将组织特别会议和研讨会，旨在汇集分析数论和代数数论的早期职业研究人员。从技术上讲，这个项目将侧重于开发数字字段和类组的统一边界（上和下）。它将考虑这些边界在算术统计中心问题上的应用，如关于给定伽罗瓦群的域数的Malle猜想和关于类群大小的ell-扭转猜想。另一个重点将是开发Artin l函数的零密度估计。这些密度估计将在一致界的证明中发挥作用，并将有其他应用，例如，PI将探索的某些随机多项式族的不可约性。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。

This project will investigate questions in arithmetic statistics, an area of mathematics concerned with determining the "typical" properties of objects in number theory. In practice, many questions in arithmetic statistics can be fruitfully understood by studying how the underlying objects decompose into simpler objects. Much of the research in this project will be aimed at studying these kinds of decompositions and, particularly, the process of assembling properties of these simpler objects to provide answers to complex questions. The PI will also continue his established record of undergraduate and graduate mentorship. He will also organize special sessions and workshops aimed at bringing together early career researchers in both analytic and algebraic number theory.More technically, this project will focus on the development of uniform bounds (both upper and lower) on number fields and class groups. It will consider the applications of these bounds to central problems in arithmetic statistics, like Malle's conjecture on the number of fields with a given Galois group and the ell-torsion conjecture on the size of the class group. Another focus will be the development of zero density estimates for Artin L-functions. These density estimates will play a role in the proofs of the uniform bounds and will have other applications, for example, to irreducibility in certain families of random polynomials, which the PI will explore.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.