CAREER: Randomness in Number Theory and Beyond

职业：数论及其他领域的随机性

• 批准号: 1652116
• 负责人: Melanie Wood
• 金额: $ 55万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-07-01 至 2019-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1652116&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.

该项目将加深对数字基本行为统计的理解，包括数字如何因式分解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。这项研究将研究神秘的微观数值结构如何聚合成普遍的宏观模式。这种从微观到宏观现象的一个基本例子是普遍存在的钟形曲线，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。该项目将帮助揭示数字基本行为的钟形曲线的模拟。具体来说，将使用概率、随机群、算术几何、代数几何、拓扑和数学等工具来研究数域和函数域的类群分布。数论。 该项目将开发随机群模型，这些模型是类群及其非交换类群的良好近似，并确定这些模型的基本概率结构。 来自代数几何和拓扑的工具将用于证明函数域类群的分布具有与模型预测一致的行为。 这将在数​​域及其非交换类似物的类组中展示新的结构。

项目成果

Cohen-Lenstra heuristics and local conditions

Cohen-Lenstra 启发法和当地条件

• DOI: 10.1007/s40993-018-0134-x
• 发表时间: 2018-12
• 期刊: Research in Number Theory
• 影响因子: 0.8
• 作者: Wood; Melanie Matchett
• 通讯作者: Melanie Matchett

Random integral matrices: universality of surjectivity and the cokernel

随机积分矩阵：满射性和 cokernel 的普遍性

• DOI: 10.1007/s00222-021-01082-w
• 发表时间: 2022-04
• 期刊: Inventiones mathematicae
• 影响因子: 3.1
• 作者: Nguyen, Hoi H.;Wood, Melanie Matchett
• 通讯作者: Wood, Melanie Matchett

Moments and interpretations of the Cohen–Lenstra–Martinet heuristics

Cohen-Lenstra-Martinet 启发法的时刻和解释

• DOI: 10.4171/cmh/514
• 发表时间: 2021-01
• 期刊: Commentarii Mathematici Helvetici
• 影响因子: 0.9
• 作者: Wang, Weitong;Wood, Melanie Matchett
• 通讯作者: Wood, Melanie Matchett