CAREER: Randomness in Number Theory and Beyond

职业：数论及其他领域的随机性

• 批准号: 1952226
• 负责人: Melanie Wood
• 金额: $ 37.29万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-07-01 至 2020-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1952226&HistoricalAwards=false
• 关键词: CAREER; Randomness; Number; Theory; Beyond

This project will lead to a deeper understanding of the statistics of fundamental behaviors of numbers, including how numbers factor and how many solutions there are to given equations. The approach is to use random models to study these seemingly deterministic questions. This research will investigate how mysterious microlevel numerical structures aggregate into universal macrolevel patterns. A fundamental example of this micro-to-macro phenomenon is seen in the ubiquity of the bell curve, which describes many distributions seen in nature, even when they come from different sources. The project will help uncover the analog of the bell curve for fundamental behaviors of numbers.Specifically, the distribution of class groups of number fields and function fields will be studied using tools from probability, random groups, arithmetic geometry, algebraic geometry, topology, and number theory. The project will develop models of random groups that are good approximations for class groups and their non-abelian analogs, and determine the basic probabilistic structure of these models. Tools from algebraic geometry and topology will be used to prove that distributions of class groups of function fields have behavior that agrees with what is predicted by the models. This will exhibit new structure in the class groups of number fields and their non-abelian analogs.

这个项目将导致更深入地了解数字的基本行为的统计，包括数字如何因子以及给定方程有多少个解。方法是使用随机模型来研究这些看似确定性的问题。这项研究将探讨神秘的微观数值结构如何聚集成普遍的宏观模式。这种微观到宏观现象的一个基本例子是无处不在的钟形曲线，它描述了自然界中看到的许多分布，即使它们来自不同的来源。本课题将利用概率论、随机群、算术几何、代数几何、拓扑学、数论等工具，研究数域和函数域的类群分布，并通过数学形态学、数学形态 该项目将开发随机组模型，这些模型是类组及其非阿贝尔类似物的良好近似，并确定这些模型的基本概率结构。 代数几何和拓扑学的工具将被用来证明函数场的类群的分布具有与模型预测一致的行为。 这将在数域及其非阿贝尔类似物的类群中表现出新的结构。

项目成果

Random integral matrices and the Cohen-Lenstra heuristics

• DOI: 10.1353/ajm.2019.0008
• 发表时间: 2015-04
• 期刊: American Journal of Mathematics
• 影响因子: 1.7
• 作者: M. Wood

Nonabelian Cohen–Lenstra moments

• DOI: 10.1215/00127094-2018-0037
• 发表时间: 2017-02
• 期刊: Duke Mathematical Journal
• 影响因子: 2.5
• 作者: M. Wood;Philip Matchett Wood

A heuristic for boundedness of ranks of elliptic curves

• DOI: 10.4171/jems/893
• 发表时间: 2016-02
• 期刊: Journal of the European Mathematical Society
• 影响因子: 2.6
• 作者: Jennifer Park;B. Poonen;J. Voight;M. Wood