Arithmetic Statistics: Geometry and Analysis

算术统计：几何与分析

• 批准号: 2002716
• 负责人: Alina Bucur
• 金额: $ 16.5万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-07-15 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2002716&HistoricalAwards=false
• 关键词: Arithmetic; Statistics; Geometry; Analysis

Number theory is one of the oldest branches of mathematics, and yet it continues to have more and more applications within the sciences. Arithmetic statistics is the study of number-theoretic objects in aggregates, rather than in isolation. This study takes many different forms, several of which will be pursued in this project. The general philosophy behind the project is the interplay between theory and computation in number theory. On one hand, initial understanding of the behavior of many classes of number-theoretic objects is generally driven by the study of explicit examples; however, production of a sufficient supply of examples often depends on theory that will underpin the accuracy and range of computations. This project will lead to a deeper understanding of the statistics of fundamental behaviors of numbers. The strategies and techniques that will be utilized in this project originate in a broad range of other mathematical subjects, including analysis, geometry, algebra and in some cases, statistics. The PI will study a range of problems concerning zeta functions and L-functions associated to algebraic varieties over finite fields, function fields in positive characteristic, and number fields. These include statistical problems about point counts, in both the arithmetic aspect (Sato-Tate distributions for abelian varieties over Q and other number fields) and the geometric aspect (averages over geometric objects over a fixed finite field); algorithmic determination of L-functions; and enumeration and tabulation of function fields.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.

数论是数学中最古老的分支之一，但它在科学中的应用越来越多。算术统计学是对集合而不是孤立的数论对象的研究。这项研究采取了许多不同的形式，其中一些将在本项目中进行。该项目背后的一般哲学是数论中理论和计算之间的相互作用。 一方面，对许多类数论对象的行为的初步理解通常是由明确的例子的研究驱动的;然而，产生足够的例子通常取决于理论，这将支持计算的准确性和范围。这个项目将导致对数字基本行为的统计学有更深入的理解。将在这个项目中使用的策略和技术起源于广泛的其他数学科目，包括分析，几何，代数，在某些情况下，统计。PI将研究一系列关于zeta函数和L-函数的问题，这些函数与有限域上的代数簇，正特征函数域和数域有关。这些问题包括关于点数的统计问题，（Q和其他数域上阿贝尔簇的Sato-Tate分布）和几何方面（固定有限域上几何对象的平均值）; L函数的算法确定;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学术价值和更广泛的影响审查标准。

项目成果

Geometric generalizations of the square sieve, with an application to cyclic covers

方形筛的几何推广及其在循环覆盖中的应用

• DOI: 10.1112/mtk.12180
• 发表时间: 2022
• 期刊: Mathematika
• 影响因子: 0.8
• 作者: Bucur, Alina;Cojocaru, Alina Carmen;Lalín, Matilde N.;Pierce, Lillian B.
• 通讯作者: Pierce, Lillian B.