Applications of Analytic Uniformity in Arithmetic Statistics

分析均匀性在算术统计中的应用

• 批准号: 2200760
• 负责人: Robert Lemke Oliver
• 金额: $ 14.53万
• 依托单位: Tufts University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-09-01 至 2025-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2200760&HistoricalAwards=false
• 关键词: Applications; Analytic; Uniformity; Arithmetic; Statistics

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.

该项目将研究算术统计中的问题，算术统计是一个数学领域，涉及确定数论中对象的“典型”属性。 在实践中，通过研究底层对象如何分解为更简单的对象，可以有效地理解算术统计中的许多问题。 该项目的大部分研究将旨在研究这些类型的分解，特别是组装这些更简单对象的属性以提供复杂问题的答案的过程。 PI 还将继续他在本科生和研究生指导方面的既定记录。 他还将组织特别会议和研讨会，旨在将分析和代数数论领域的早期职业研究人员聚集在一起。从技术上讲，该项目将重点关注数域和类组的统一界限（上限和下限）的开发。 它将考虑这些界限在算术统计中的核心问题上的应用，例如马勒关于给定伽罗瓦群的域数的猜想以及关于类群大小的埃尔扭转猜想。 另一个重点是 Artin L 函数的零密度估计的开发。 这些密度估计将在统一界限的证明中发挥作用，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 NSF 的法定使命，并被认为是值得的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来提供支持。