Spheres of Influence: Arithmetic Geometry and Chromatic Homotopy Theory

影响范围：算术几何和色同伦理论

• 批准号: 2401472
• 负责人: Jared Weinstein
• 金额: $ 34万
• 依托单位: Trustees of Boston University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-09-01 至 2027-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2401472&HistoricalAwards=false
• 关键词: Spheres; Influence; Arithmetic; Geometry; Chromatic

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.

首席研究员计划在数学的两个领域之间建立一座桥梁：数论和拓扑学。 数论是数学的一个古老的分支，研究的是整数和素数。 数论中的一些基本结果是素数的无限性和给出所有毕达哥拉斯三元组的公式。 拓扑学是研究形状的，但人们不记得长度和角度等细节;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。 拓扑学中的一个首要主题是发明不变量来区分形状。 例如，一条裤子与一根吸管不同，因为“洞数”是一个不变量，它赋予它们不同的值（分别为2和1，但必须精确地说明洞是什么）。 “洞”的概念可以推广到更高的维度：一个球体没有一维洞，但它有一个二维洞，甚至一个三维洞（称为霍普夫纤维化，发现于1931年）。 在每一个维度上都有“球体”，确定每个球体有多少个洞是拓扑学中一个未解决的主要问题。 最近，拓扑学家的方法已经侵入数论领域。 特别是被称为p进几何的数论的分支，它涉及到允许小数位向左无限远的奇怪的数字系统，已经出现了。 首席研究员将利用他的专业知识在p-adic几何作出贡献的计数孔的领域的问题。 他还将组织会议和研讨会，目的是将数论家和拓扑学家聚集在一起，因为目前这两个领域彼此有些孤立。 最后，首席研究员计划训练他的四个研究生与这个项目有关的方法。计算形状中孔的数量的装置被称为“同伦组”。 计算球面的同伦群是出了名的困难，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）-局部化版本代替球面。 这里K（n）是Morava K理论谱。 主要研究者和合作者正在进行的工作已经确定了K（n）-局部球面的同伦群，直到挠子群。 所使用的技术涉及形式群，p-adic几何，特别是perfectoid空间，这是由菲尔兹奖得主Peter Scholze在2012年发明的某些类似分形的实体。 该项目的下一步是使用相关技术计算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，这与在不同K（n）的球体的迭代局部化有关。 这是从球面的K（n）-局部类似物中组装球面的同伦群所需的难题中缺失的部分之一。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知识价值和更广泛的影响审查标准进行评估，被认为值得支持。