Stability Phenomena in Number Theory, Algebraic Geometry, and Topology

数论是数学中最古老、最纯粹的领域之一，自欧几里得时代以来，它在许多方面都没有改变，但近年来，它吸收了许多其他数学领域的思想和技术。 这个研究项目站在关于整数的经典问题和其他学科的想法之间的界面。 在一个主要项目中，PI和他的合作者展示了代数拓扑的结果，高维形状及其之间关系的研究，如何转化为关于各种数字系统的算术的陈述。 在另一个，他和另一个小组发展了一种新形式的表示论（线性空间对称性的研究），揭示了数论，拓扑学和代数中的稳定现象。第一近似，工作回答了这个问题：当一个无限的对象可以描述的有限数量的数据？ PI还将继续他在数学推广方面的工作，包括将于2014年发布的普通读者书籍。本项目研究了关于数域类群p-部分的变化的Cohen-Lenstra定理，更一般地，研究了关于任意有限群G的G-扩张的判别式的分布问题。 所使用的方法是新颖的-PI和合作者表明，科恩-伦斯特拉代数遵循关于复射影线的分支覆盖的某些模空间（称为Hurwitz空间）的上同调的断言。 这些空间可以定义纯粹的拓扑，而事实上的重点工作已表明，新的定理代数拓扑意味着许多流行的approachtures算术统计功能领域。 更重要的是，拓扑结果作为一种机器，用于产生的示意图，或至少是示意图，有关的问题，分布的G-扩张在Q尚未被调查。 例如，结果表明，如果N是从大范围内均匀选择的随机无平方因子整数，X是具有判别式N的全真实的五次扩张的个数，则X具有均值为1/120的泊松分布。 该项目的一个新方面是FI模块理论，由PI与Tom Church和Benson Farb合作开发。 这一理论代表了一种新的方法同调稳定性，其自然域的应用不是序列的unadorned向量空间，而是序列的向量空间，其第n项是一个代表的对称群的n个字母。 事实证明，有一个自然的阿贝尔范畴，称为FI-模范畴，它捕捉了广泛的现象，从模空间的上同调到代数组合学中的上不变代数，再到有限域上李群中无平方多项式和环面的统计。 在这个研究项目中，除了继续调查的固有结构的类别的FI-模，计划使这项工作与其他工作与Venkatesh和韦斯特兰。 要研究的一个典型问题是：在有限域上是否有无穷多个有理函数域的三次扩张（或者，更好的是：三次扩张的期望数量是多少，渐近），其判别式是素数（即有限域上的不可约多项式）？ Q上的相应问题是一个著名的公开问题。 该项目还将解决一系列其他问题;调和分析中Kakeya问题的几何类似物和方法，随机矩阵和有限域上普通曲线的比例，以及在应用方面，关于几何应用于数据科学问题的一些问题。

Number theory is one of the oldest and purest areas of mathematics, unchanged in many ways since the time of Euclid, but in recent years it has incorporated ideas and techniques from a wide range of other mathematical areas. This research project stands at the interface between classical questions about whole numbers and ideas from other subjects. In one main project, the PI and his collaborators show how results in algebraic topology, the study of high-dimensional shapes and the relations between them, translate into statements about the arithmetic of various number systems. In another, he and another group develop a new form of representation theory (the study of symmetries of linear spaces), which sheds light on phenomena of stabilization in number theory, topology, and algebra. To a first approximation, the work answers the question: when can an infinite object be described by a finite amount of data? The PI will also continue his work in mathematical outreach, including a general-audience book to be released in 2014. This project investigates the Cohen-Lenstra conjectures concerning the variation of the p-part of the class group of number fields, and, more generally, distributional questions about the discriminants of G-extensions for G an arbitrary finite group. The methods used are novel -- the PI and collaborators show that the Cohen-Lenstra conjectures follow from assertions about the cohomology of certain moduli spaces of branched covers of the complex projective line, known as Hurwitz spaces. These spaces can be defined purely topologically, and in fact the thrust of the work has been to show that new theorems in algebraic topology imply many popular conjectures about arithmetic statistics over function fields. What's more, the topological results serve as a kind of machine for generating conjectures, or at least heuristics, about questions concerning the distribution of G-extensions over Q which have not yet been investigated. For instance, the results suggest that if N is a random squarefree integer chosen uniformly from a large range, and X is the number of totally real quintic extensions with discriminant N, then X has the Poisson distribution with mean 1/120. A new aspect of the project is the theory of FI-modules, developed by the PI in collaboration with Tom Church and Benson Farb. This theory represents a new approach to homological stability, whose natural domain of application is not sequences of unadorned vector spaces but rather sequences of vector spaces whose nth term is a representation of the symmetric group on n letters. It turns out that there is a natural abelian category, called the category of FI-modules, which captures a broad spectrum of phenomena ranging from cohomology of moduli spaces to the coinvariant algebras arising in algebraic combinatorics to the statistics of squarefree polynomials and tori in Lie groups over finite fields. In this research project, besides continuing investigation of the inherent structure of the category of FI-modules, it is planned to bring this work into contact with other work with Venkatesh and Westerland. A typical question to be investigated is: are there infinitely many cubic extensions of a rational function field over a finite field (or, better: what is the expected number of cubic extensions, asymptotically) whose discriminant is prime (i.e. an irreducible polynomial over the finite field)? The corresponding question over Q is a well-known open problem. The project will also address a suite of other problems; geometric analogues of and approaches to the Kakeya problem in harmonic analysis, random matrices and the proportion of ordinary curves over finite fields, and, on the applied side, some questions about the application of geometry to problems in data science.