Stability Phenomena in Number Theory, Algebraic Geometry, and Topology

数论、代数几何和拓扑中的稳定性现象

• 批准号: 1402620
• 负责人: Jordan Ellenberg
• 金额: $ 27.8万
• 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: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1402620&HistoricalAwards=false
• 关键词: Stability; Phenomena; Number; Theory; Algebraic

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.

数论是最古老、最纯粹的数学领域之一，自欧几里得时代以来，它在许多方面都没有改变，但近年来，它已经从广泛的其他数学领域吸收了思想和技术。这个研究项目站在关于整数的经典问题和其他学科的思想之间的接口。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，如何转化为各种数字系统的算术陈述。另一方面，他和另一个小组发展了一种新的表示理论（对线性空间对称性的研究），它揭示了数论、拓扑学和代数中的稳定现象。近似地说，这项工作回答了这个问题：什么时候一个无限的物体可以用有限的数据来描述？PI还将继续他的数学推广工作，包括将于2014年出版的一本面向大众的书。本文研究了关于数域类群的p部变分的Cohen-Lenstra猜想，以及关于任意有限群G-扩展的判词的分布问题。所使用的方法是新颖的——PI和合作者表明，Cohen-Lenstra猜想遵循复射影线分支覆盖的某些模空间（称为Hurwitz空间）的上同调断言。这些空间可以纯拓扑学地定义，事实上，这项工作的主旨是表明代数拓扑学中的新定理隐含了许多关于函数域上算术统计的流行猜想。更重要的是，拓扑结果作为一种机器，用于产生关于尚未研究的Q上g扩展分布问题的猜想，或至少是启发式。例如，结果表明，如果N是在大范围内均匀选择的随机无平方整数，X是具有判别N的全实数五次扩展的个数，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。这一理论代表了同调稳定性的一种新方法，它的自然应用领域不是无修饰的向量空间序列，而是第n项是n个字母上对称群的表示的向量空间序列。事实证明，有一个自然的阿贝尔范畴，称为fi -模范畴，它捕获了广泛的现象，从模空间的上同调到代数组合中的协不变代数，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，除了继续研究fi模块类别的内在结构外，还计划将这项工作与Venkatesh和Westerland的其他工作联系起来。要研究的一个典型问题是：在有限域上是否存在无穷多个有理函数域的三次扩展（或者，更好的是：三次扩展的期望值是多少，渐近地），其判别式是素数（即有限域上的不可约多项式）？Q对应的问题是一个众所周知的开放问题。该项目还将解决一系列其他问题；调和分析、随机矩阵和有限域上普通曲线的比例中Kakeya问题的几何类似物和方法，以及在应用方面，关于几何在数据科学问题中的应用的一些问题。