Stability Phenomena in Number Theory, Algebraic Geometry, and Topology

数论是最古老和最纯粹的数学领域之一，自欧几里得时代以来在许多方面都没有变化，但近年来它吸收了广泛的其他数学领域的思想和技术。这项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交汇点上。在一个主要项目中，PI和他的合作者展示了如何导致代数拓扑，研究高维形状及其之间的关系，转化为关于各种数字系统的算术的陈述。在另一篇文章中，他和另一个小组发展了一种新的表示理论形式(研究线性空间的对称性)，它揭示了数论、拓扑学和代数中的稳定现象。大致来说，这项工作回答了这样一个问题：一个无限大的物体什么时候可以用有限数量的数据来描述？PI还将继续他在数学推广方面的工作，包括将于2014年出版的一本面向普通读者的书。这个项目研究了关于数域的类群的p-部分的变化的Cohen-Lenstra猜想，更一般地，研究了关于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，Cohen-Lenstra猜想源于关于复射影直线的分支覆盖的某些模空间的上同调的断言，即众所周知的Hurwitz空间。这些空间可以纯粹用拓扑学来定义，事实上，这项工作的主旨是证明代数拓扑学中的新定理蕴含着许多关于函数域上的算术统计的流行猜想。更重要的是，拓扑结果作为一种机器，用于生成关于尚未被研究的关于Q上G-扩张的分布的问题的猜想，或者至少是启发式。例如，如果N是从大范围内均匀选择的随机无平方整数，且X是具有判别式N的全实五次扩张的个数，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-模理论，由PI与汤姆·丘奇和本森·法布合作开发。这一理论代表了同调稳定性的一种新方法，它的自然应用领域不是未加修饰的向量空间序列，而是其第n项是n字母上对称群的表示的向量空间序列。证明了存在一个自然的阿贝尔范畴，称为FI-模范畴，它涵盖了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的无平方多项式和环面的统计等广泛的现象。在这项研究项目中，除了继续调查FI-模块类别的内在结构外，还计划将这项工作与Venkatesh和Westland的其他工作联系起来。一个需要研究的典型问题是：有理函数域在判别式为素数(即有限域上的不可约多项式)的有限域上是否存在无穷多个三次扩张(或者，更好的是：三次扩张的期望数量渐近是多少)？Q上的相应问题是一个众所周知的公开问题。该项目还将解决一系列其他问题：调和分析中Kakeya问题的几何类比和方法、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.