CAREER: Randomness in Number Theory and Beyond

这个项目将导致对数字基本行为的统计的更深入的理解，包括数字是如何影响的，以及给定的方程有多少解。方法是使用随机模型来研究这些看似确定性的问题。这项研究将调查神秘的微观数字结构如何聚合成普遍的宏观模式。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。该项目将帮助揭示数的基本行为的钟形曲线的模拟。具体地说，将使用概率、随机群、算术几何、代数几何、拓扑学和数论的工具来研究数字域和函数域的类群的分布。该项目将开发随机群的模型，这些模型是类群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。来自代数几何和拓扑学的工具将被用来证明函数域的类组的分布具有与模型所预测的一致的行为。这将在数域及其非阿贝尔类似的类群中显示出新的结构。

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.