Cohen-Lenstra heuristics, Brauer relations, and low-dimensional manifolds

对称是数学中最重要的一个概念。人们曾经认为对称是几何形状的一种属性，但在19世纪，Evariste Galois将对称的概念扩展到代数对象，今天他的见解完全是纯数学的基础。这个提议的潜在目标是研究算术和几何对象的对称性，它介于代数、数论和拓扑之间，也依赖于概率论和可加组合学的技术。数论是一门古老的数学学科，有着2000多年的丰富历史，近年来也有了惊人的发展。最近一些最令人印象深刻的进展发生在被称为算术统计的数论领域：Manjul Bhargava的开创性贡献在2014年获得了菲尔兹奖（Fields Medal）。算术统计的目的是了解算术对象的行为，如（ray）类组，在家庭中。这个领域的诞生可以追溯到高斯，他提出了一些关于二次域类群行为的具体猜想。20世纪80年代，科恩和伦斯特拉提出了一个通用模型，该模型包含了高斯的所有猜想，甚至更多。粗略地说，他们假设虚二次域的类群服从一个概率分布，这个概率分配给一个有限阿贝尔群X，它的概率与X的对称数成反比。这实际上是随机代数对象的一个非常自然的模型。这一理论后来被Cohen和Martinet推广到其他数字领域，但在更一般的情况下，概率分布看起来更神秘。从那时起，Cohen-Lenstra-Martinet启发式被用作算术统计的指导原则，并在许多其他领域得到了应用，例如椭圆曲线理论、组合学和微分几何。这个项目将包括定理、猜想和计算的混合。我将：-证明原来的猜想是错误的，如前所述，-找到正确的公式，-把它们放在一个更概念化的基础上，通过使用我与亨德里克·伦斯特拉一起发展的代数对象的可通约性理论来解释科恩-马丁内的神秘概率权重，-扩展启发式的范围，例如射线类群。我们研究的另外两种非常基本的对称对象是有限集合和有限维向量空间。在数论和微分几何中有一个古老的问题，就是比较集合的对称性和向量空间的对称性，特别是当集合变成向量空间时，确定集合的哪些对称性是同构的（本质上是相同的）。这个问题有两种表现形式：一种是特征为0的域上的向量空间，例如实数上的向量空间，另一种是特征为正的域上的向量空间。在之前与Tim Dokchitser的合作中，我们已经解决了特征0的情况，从而解决了一个60多年的问题。利用我们开发的技术和新技术，本项目将解决正面特征的情况。最后，我也将研究低维流形的对称性。这些都是现代几何和拓扑学研究的基本对象，通过了解流形的对称性来确定流形的拓扑结构是一种古老而富有成果的研究方法。在最近与Aurel Page的联合工作中，我将一种新的表示理论工具引入了该领域，这是我在数论背景下工作过的。使用这些新技术，我计划更多地阐明对称和流形拓扑之间的联系。

A concept of central importance in mathematics is that of symmetry. One used to think of symmetry as a property of geometric shapes, but in the 19th century Evariste Galois extended the concept of symmetry to algebraic objects, and today his insights are completely fundamental to pure mathematics. The underlying goal of this proposal, which is situated between Algebra, Number Theory, and Topology, relying also on techniques from Probability Theory and Additive Combinatorics, is to study symmetries of arithmetic and geometric objects.Number Theory is an ancient mathematical discipline with a rich history of over 2000 years, but also with spectacular developments in recent years. Some of the most impressive recent advances have happened in the area of Number Theory called Arithmetic Statistics: the groundbreaking contributions of Manjul Bhargava have been rewarded with a Fields Medal in 2014. The aim of Arithmetic Statistics is to understand the behaviour of arithmetic objects, such as (ray) class groups, in families. The birth of this area goes back to Gauss, who formulated some concrete conjectures concerning the behaviour of class groups of quadratic fields. It was given a huge boost in the 1980s, when Cohen and Lenstra proposed a general model that implied all the conjectures of Gauss, and more. Roughly speaking, they postulated that class groups of imaginary quadratic fields obey a probability distribution that assigns to a finite abelian group X a probability that is inverse proportional to the number of symmetries of X. This is, in fact, a very natural model for random algebraic objects. This was later generalised to other number fields by Cohen and Martinet, but in more general cases the probability distributions looked more mysterious. The Cohen-Lenstra-Martinet Heuristics have been used as a guiding principle in Arithmetic Statistics since then, and have found applications in many other areas, such as the theory of Elliptic Curves, in Combinatorics, and in Differential Geometry. This project will consist of a blend of theorems, conjectures, and computations. I will:- show that the original conjectures are false, as stated,- find the correct formulations,- put them on a more conceptual footing, by explaining the mysterious looking probability weights of Cohen-Martinet using a theory of commensurability of algebraic objects that I have been developing together with Hendrik Lenstra,- extend the scope of the heuristics, e.g. to ray class groups.Two other kinds of very basic objects whose symmetries one studies are finite sets and finite dimensional vector spaces. An old problem in Representation Theory, with applications to Number Theory and Differential Geometry, is to compare symmetries of sets with symmetries of vector spaces, and in particular to determine which symmetries of sets become isomorphic (essentially the same) when the sets are turned into vector spaces. There are two incarnations of this problem: one where the vector spaces are over a field of characteristic 0, e.g. over the real numbers, and one where they are vector spaces over a field of positive characteristic. In previous joint work with Tim Dokchitser we have solved the case of characteristic 0, thereby settling an over 60 year old problem. Using the techniques that we developed, and new ones, this project will settle the case of positive characteristic.Finally, I will also investigate symmetries of low-dimensional manifolds. These are the basic objects studied by modern geometry and topology, and it is an old and fruitful line of investigation to determine what one can say about the topology of the manifold from knowing its symmetries. In recent joint work with Aurel Page, I have introduced a new representation theoretic tool into the area, which I had worked on in number theoretic contexts. Using these new techniques, I am planning to shed more light on the connection between symmetries and the topology of the manifold.