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.

数字理论是数学中最古老，最纯净的领域之一，自欧几里得时代以来就以多种方式保持不变，但是近年来，它融合了来自其他许多数学领域的思想和技术。 该研究项目位于有关其他主题的整数和想法的经典问题之间的界面。 在一个主要项目中，PI及其合作者表明如何导致代数拓扑，对高维形状及其之间的关系的研究如何转化为有关各种数字系统算术的陈述。 在另一个人中，他和另一个群体发展了一种新的表示理论（线性空间对称性的研究），该理论阐明了数字理论，拓扑和代数中稳定现象的阐明。在第一个近似过程中，工作回答了一个问题：什么时候可以通过有限的数据来描述无限的对象？ PI还将继续他的数学外展工作，包括将于2014年发行的一般性声明书。该项目调查了有关Cohen-Lenstra关于班级数字群体P-PART的构想的猜想，还有更多信息，以及更多信息。通常，关于G一个任意有限组的G延期判别的分布问题。 所使用的方法是新颖的 - PI和合作者表明，Cohen-Lenstra的猜想源于对复杂投影线的某些模量覆盖物的某些模量空间的主张，称为Hurwitz Spaces。 这些空间可以纯粹拓扑定义，实际上，工作的目的是表明代数拓扑中的新定理暗示了许多关于功能领域算术统计的流行猜想。 更重要的是，拓扑结果是产生猜想或至少启发式方法的机器，涉及有关Q extensions在Q上的分布的问题，但尚未研究。 例如，结果表明，如果n是一个从较大范围统一选择的随机无平方整数，而x是具有判别n的完全真实的五重奏扩展的数量，则x具有平均1/120的泊松分布。 该项目的一个新方面是由PI与Tom Church和Benson Farb合作开发的FI模型理论。 该理论代表了一种新的同源稳定性方法，其自然应用不是未经修饰的向量空间的序列，而是向量空间的序列，其n个项是对称组的表示对称群的。 事实证明，有一个天然的阿贝尔类别，称为fi模型类别，它捕获了从模量空间的共同学到在代数组合中引起的共同变体的广泛现象，以及在代数组合中产生的代数到谎言的统计数据，小组在有限的字段上。 在该研究项目中，除了继续研究FI模型类别的固有结构外，还计划将这项工作与Venkatesh和Westerland接触。 要研究的一个典型问题是：有限领域的理性功能领域的许多立方扩展是否无限地存在（或者更好的：什么是偶然的立方扩展的预期数），它们的判别是质量的（即，在不可避免的多项官方中，有限领域）？ Q上的相应问题是一个众所周知的开放问题。 该项目还将解决其他一系列问题；在谐波分析，随机矩阵以及普通曲线比有限场上的普通曲线的比例，以及在应用方面，有关几何学在数据科学中的问题上应用于问题的一些问题。