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Motivic Homotopy Theory, Stable Homotopy Groups of Spheres and the Kervaire Invariant

动机同伦理论、球面稳定同伦群和 Kervaire 不变量

基本信息

批准号：
1810638
负责人：
Zhouli Xu
金额：
$ 16.1万
依托单位：
Massachusetts Institute of Technology
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-07-01 至 2020-09-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1810638&HistoricalAwards=false
关键词：
Motivic Homotopy Theory Stable Groups

项目摘要

Topology is a subject of mathematics that studies the shape of spaces, and homotopy theory studies how spaces can be deformed to each other without tearing things up or puncturing holes in the process. Among all spaces, the spheres are the most fundamental and beautiful objects. A central question in homotopy theory and algebraic topology - the computation of homotopy groups of spheres - is to understand how spheres can be mapped to other spheres, in different dimensions, and to understand how different maps can or cannot be deformed to each other. This topological question is not just a fundamental question on fundamental mathematical objects, but it also has deep connections and interactions to other subjects in mathematics. It has been a major research question since 1950's. For example, Kervaire and Milnor established a connection to the question on how many ways one could do calculus on spheres - smooth structures on spheres; Kervaire and Browder built a connection to the question on how one could or could not do surgeries to higher dimensional spaces to turn them into spheres - the Kervaire invariant problem. Moreover, a summand of the stable homotopy groups of spheres, the image of J, is closely related to the Bernoulli numbers; Quillen and Goerss, Hopkins, Miller, Lurie established a connection to the moduli stack of formal groups. More recently, Hill, Hopkins, Ravenel, Voevodsky, Morel, Isaksen and others established connections between equivariant homotopy theory and motivic homotopy theory. The goal of this project is to deepen the existing connections, as well as discovering new connections by pushing further the limit of existing ones.This research concentrates on computations of stable homotopy groups of spheres, with interactions among motivic, equivariant and chromatic homotopy theory, and applications to problems in differential topology, such as uniqueness of smooth structure on spheres and the Kervaire invariant problem. More specifically, in current and ongoing projects with Isaksen and Wang, the Principle Investigator (PI) develops new computational tools in motivic homotopy theory, with connection to chromatic homotopy theory, which computes 40 more new stems of classical stable homotopy groups of sphere within two years. The PI will deepen the new connection between motivic homotopy theory and chromatic homotopy theory, carry out more computations of stable stems in the next a few years, and use the computations to attack the last unsolved case of the Kervaire invariant problem in dimension 126. The PI will also explore connections between real motivic homotopy theory and C2 equivariant homotopy theory, following ongoing work of Behrens, Dugger, Guillou and Isaksen. The goal is to prove structural theorems in this direction, as well as providing concrete computational results. In ongoing projects with Hill, Shi and Wang, the PI will also apply equivariant techniques, such as the slice spectral sequences that are developed by Hill-Hopkins-Ravenel, to do computations in heights greater than 2 in chromatic homotopy theory and to understand its connection to stable homotopy groups of spheres.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.

拓扑学是一门研究空间形状的数学学科，而同伦理论研究的是空间如何在不撕裂或刺破洞的情况下相互变形。在所有的空间中，球体是最基本、最美丽的物体。同伦理论和代数拓扑的一个中心问题——球的同伦群的计算——是理解球如何在不同的维度上映射到其他的球，以及理解不同的映射如何能够或不能相互变形。拓扑问题不仅是一个关于基本数学对象的基本问题，而且与数学中的其他学科有着深刻的联系和相互作用。自20世纪50年代以来，这一直是一个主要的研究问题。例如，Kervaire和Milnor建立了一个问题的联系即有多少种方法可以在球面上做微积分球面上的光滑结构；Kervaire和Browder建立了一个关于如何能或不能对高维空间进行手术以将它们变成球体的问题——Kervaire不变量问题。此外，球的稳定同伦群的一个和，即J的象，与伯努利数密切相关；Quillen和Goerss， Hopkins, Miller， Lurie建立了与正式群的模堆栈的联系。最近，Hill、Hopkins、Ravenel、Voevodsky、Morel、Isaksen等人建立了等变同伦理论与动机同伦理论之间的联系。这个项目的目标是深化现有的联系，并通过进一步推动现有联系的限制来发现新的联系。本文主要研究了具有动机、等变和色同伦理论相互作用的球的稳定同伦群的计算，以及在球光滑结构唯一性和Kervaire不变量问题等微分拓扑问题中的应用。更具体地说，在与Isaksen和Wang的当前和正在进行的项目中，首席研究员（PI）开发了与色同伦理论相关的动力同伦理论的新计算工具，该工具在两年内计算了40多个经典稳定球同伦群的新系统。PI将深化动力同伦理论与色同伦理论之间的新联系，在未来几年内进行更多的稳定系统计算，并利用这些计算来解决126维Kervaire不变量问题的最后一个未解决的情况。继Behrens， Dugger， Guillou和Isaksen正在进行的工作之后，PI还将探索实动机同伦理论和C2等变同伦理论之间的联系。目标是在这个方向上证明结构定理，并提供具体的计算结果。在与Hill， Shi和Wang正在进行的项目中，PI还将应用等变技术，例如由Hill- hopkins - ravenel开发的切片光谱序列，在色同伦理论中进行大于2的高度计算，并了解其与稳定球同伦群的联系。该奖项反映了美国国家科学基金会的法定使命，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。