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.

该项目将使对数字基本行为的统计数据有更深入的了解，包括数字因素和多少解决方案。该方法是使用随机模型来研究这些看似确定的问题。这项研究将调查神秘的微观数值结构如何汇总为通用的宏观模式。这种微观到麦克罗现象的基本例子在钟形曲线的无处不在，它描述了自然界中许多分布，即使它们来自不同的来源。该项目将有助于揭示钟形曲线的类似物，以实现数字的基本行为。特别是，将使用概率，随机组，算术几何形状，代数几何学，拓扑，拓扑，拓扑，拓扑，兼容性几何形状和数字字段和功能字段的分布进行研究。数字理论。 该项目将开发随机组的模型，这些模型是班级及其非亚伯类似物的良好近似值，并确定这些模型的基本概率结构。 代数几何学和拓扑的工具将用于证明班级函数字段的分布具有与模型预测的行为相符的行为。 这将在数​​字字段及其非亚洲类似物的阶级组中表现出新的结构。

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