Spheres of Influence: Arithmetic Geometry and Chromatic Homotopy Theory

首席研究员计划在数论和拓扑学这两个数学领域之间架起一座桥梁。数论是一个古老的数学分支，研究的是整数和素数。数论中的一些基本结果是素数的无穷性和给出毕达哥拉斯的所有三元组的公式。拓扑学是对形状的研究，但人们记不住长度和角度等细节；对于拓扑学家来说，甜甜圈和咖啡杯的表面是出了名的难以区分。拓扑学的一个重要主题是发明不变量来区分形状。例如，一条裤子不同于吸管，因为“洞的个数”是一个不变量，它为它们分配了不同的值(分别为2和1，但人们必须准确地知道洞是什么)。“洞”的概念可以推广到更高的维度：球体没有一维洞，但它确实有一个二维洞，甚至有一个三维洞(1931年发现的霍普夫纤维)。每个维度都有“球体”，确定每个球体有多少个洞是拓扑学中一大悬而未决的问题。最近，拓扑学家的方法已经侵入了数论的领域。特别值得一提的是，数论的一个分支，也就是所谓的p进几何，涉及允许小数位向左无限偏移的奇怪数字系统，已经出现了。首席研究人员将利用他在p进几何方面的专业知识来为球体中的孔洞计数问题做出贡献。他还将组织会议和研讨会，目的是将数字理论家和拓扑学家聚集在一起，因为目前这两个领域有点相互孤立。最后，首席研究人员计划对他的四名研究生进行与该项目相关的方法培训。计算一个形状上的孔洞数量的装置被称为“同伦群”。计算球面的同伦群是出了名的困难，但同时也很有趣。有一种分而治之的方法来做到这一点，称为色同伦理论，它用它的K(N)局部化版本取代了球面。这里K(N)是Morava K-理论谱。主要研究人员和合作者正在进行的工作已经确定了K(N)-局部球面到扭子群的同伦群。所使用的技术涉及形式群、p进几何，特别是完美拟态空间，这是菲尔兹奖牌获得者彼得·肖尔茨于2012年发明的某些类似于分数维的实体。项目的下一步是利用相关技术计算K(N)-局部范畴的Picard群。在此之后，首席研究员将把他的注意力转向被称为“色分裂猜想”的问题，该猜想与球体在不同K(N)处的迭代局部化有关。这是从球面的K(N)-局部类似物组装球面的同伦群所需的拼图中缺失的部分之一。该奖项由代数、数论和几何分析项目共同支持。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The principal investigator plans to build a bridge between two areas of mathematics: number theory and topology. Number theory is an ancient branch of mathematics concerned with the whole numbers and primes. Some basic results in number theory are the infinitude of primes and the formula which gives all the Pythagorean triples. Topology is the study of shapes, but one doesn't remember details like length and angles; the surfaces of a donut and a coffee mug are famously indistinguishable to a topologist. An overarching theme in topology is to invent invariants to distinguish among shapes. For instance, a pair of pants is different from a straw because "number of holes" is an invariant which assigns different values to them (2 and 1 respectively, but one has to be precise about what a hole is). The notion of "hole" can be generalized to higher dimensions: a sphere has no 1-dimensional hole, but it does have a 2-dimensional hole and even a 3-dimensional hole (known as the Hopf fibration, discovered in 1931). There are "spheres" in every dimension, and the determination of how many holes each one has is a major unsolved problem in topology. Lately, the topologists' methods have encroached into the domain of number theory. In particular the branch of number theory known as p-adic geometry, involving strange number systems allowing for decimal places going off infinitely far to the left, has made an appearance. The principal investigator will draw upon his expertise in p-adic geometry to make contributions to the counting-holes-in-spheres problem. He will also organize conferences and workshops with the intent of drawing together number theorists and topologists together, as currently these two realms are somewhat siloed from each other. Finally, the principal investigator plans to train his four graduate students in methods related to this project.The device which counts the number of holes in a shape is called the "homotopy group". Calculating the homotopy groups of the spheres is notoriously difficult and interesting at the same time. There is a divide-and-conquer approach to doing this known as chromatic homotopy theory, which replaces the sphere with its K(n)-localized version. Here K(n) is the Morava K-theory spectrum. Work in progress by the principal investigator and collaborators has identified the homotopy groups of the K(n)-local sphere up to a torsion subgroup. The techniques used involve formal groups, p-adic geometry, and especially perfectoid spaces, which are certain fractal-like entities invented in 2012 by Fields Medalist Peter Scholze. The next step in the project is to calculate the Picard group of the K(n)-local category, using related techniques. After this, the principal investigator will turn his attention to the problem known as the "chromatic splitting conjecture", which has to do with iterated localizations of the sphere at different K(n). This is one of the missing pieces of the puzzle required to assemble the homotopy groups of the spheres from their K(n)-local analogues. This award is jointly supported by the Algebra and Number Theory and Geometric Analysis programs.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.