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Classical, Motivic and Equivariant Stable Homotopy Groups of Spheres.

球面的经典、动机和等变稳定同伦群。

基本信息

批准号：
2105462
负责人：
Zhouli Xu
金额：
$ 26.41万
依托单位：
University of California-San Diego
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-08-15 至 2024-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2105462&HistoricalAwards=false
关键词：
Classical Motivic Equivariant Stable Homotopy

项目摘要

Topology is a subject of mathematics that studies the shapes of spaces and interactions among them. A fundamental problem in topology is the classification problem of topological spaces and maps among them. Among all spaces, the spheres are of central importance, and we can classify them up to continuous deformation by a single number - the dimension. A one dimensional sphere, which we usually call a circle, is something that we can draw on a piece of paper, and can be described by the solutions of the familiar equation for a circle. A two dimensional sphere is something we can visualize in three dimensional space, such as the surface of a basketball, and can be described by the solutions of a similar equation. In topology, we also study higher dimensional spheres. Although these spaces cannot be visualized in three dimensions, they do exist in higher dimensions, and can also be studied. Somewhat surprisingly, maps between spheres are much harder to classify, even up to continuous deformations. This classification problem of maps between spheres is called the problem of computations of homotopy groups of spheres. This problem has been a major and active research problem since the 1950's. Much progress has been made on this problem, through all kinds of methods that have close connections to many subjects of mathematics. Recently, significant progress has been made using techniques from motivic and equivariant homotopy theory. The goal of this project is to further develop new techniques in motivic and equivariant homotopy theory, and to study this classical problem of computations of homotopy groups of spheres. Graduate students will be involved in this project.This project concentrates on computations of stable homotopy groups of spheres, in the classical, motivic and equivariant context. More specifically, the principal investigator and collaborators will continue using techniques in motivic homotopy theory over the complex numbers to push classical computations, towards the Kervaire invariant problem in dimension 126. The principal investigator and collaborators will extend the Chow t-structure technique defined over the complex numbers to other base fields, and study motivic stable homotopy groups of spheres over these base fields. Moreover, the PI will study with collaborators the New Doomsday Conjecture in Adams filtration 3, and use equivariant techniques to study homotopy groups of Hill-Hopkins-Ravenel type spectra.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及其相互作用的数学学科。拓扑学的一个基本问题是拓扑空间及其映射的分类问题。在所有空间中，球体是最重要的，我们可以通过一个数字（维度）将它们分类为连续变形。一维球体，我们通常称为圆，是我们可以在一张纸上绘制的东西，并且可以通过熟悉的圆方程的解来描述。二维球体是我们可以在三维空间中可视化的东西，例如篮球的表面，并且可以通过类似方程的解来描述。在拓扑学中，我们还研究更高维的球体。尽管这些空间无法在三维空间中可视化，但它们确实存在于更高的维度中，并且也可以进行研究。有点令人惊讶的是，球体之间的地图更难分类，甚至连续变形也是如此。这种球体之间映射的分类问题称为球体同伦群的计算问题。自 20 世纪 50 年代以来，这个问题一直是一个主要且活跃的研究问题。通过与许多数学学科密切相关的各种方法，这个问题已经取得了很大进展。最近，利用动机和等变同伦理论的技术已经取得了重大进展。该项目的目标是进一步开发动机和等变同伦理论的新技术，并研究球面同伦群计算的经典问题。研究生将参与该项目。该项目专注于在经典、动机和等变背景下计算球面的稳定同伦群。更具体地说，首席研究员和合作者将继续使用复数上的动机同伦理论技术来推动经典计算，以解决维度 126 中的 Kervaire 不变问题。首席研究员和合作者将把在复数上定义的 Chow t 结构技术扩展到其他基域，并研究这些基域上球体的动机稳定同伦群。此外，PI将与合作者一起研究Adams过滤3中的新末日猜想，并使用等变技术来研究Hill-Hopkins-Ravenel型光谱的同伦群。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。