Arithmetic Statistics: Geometry and Analysis

数论是数学中最古老的分支之一，但它在科学中有越来越多的应用。算术统计学是对数论对象的总体研究，而不是孤立的研究。这项研究有许多不同的形式，其中一些将在本项目中进行。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，对许多数论对象的行为的初步理解通常是由显式示例的研究驱动的；然而，产生足够的例子往往依赖于理论，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有更深的理解。本项目将采用的策略和技术源于广泛的其他数学学科，包括分析、几何、代数，在某些情况下还包括统计学。PI将研究与有限域上的代数变异、正特征函数域和数域相关的ζ函数和l函数的一系列问题。这包括关于点计数的统计问题，在算术方面（Q和其他数域上阿贝尔变量的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；以及函数域的枚举和制表。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。

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.