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.

拓扑是数学的主题，研究了空间的形状及其之间的相互作用。拓扑结构的一个基本问题是拓扑空间的分类问题以及其中的地图。在所有空间中，球体都是至关重要的，我们可以将它们分类为单个数字的连续变形 - 尺寸。我们通常称之为圆圈的一维球体是我们可以在纸上绘制的东西，可以用圆圈熟悉的方程式的解决方案来描述。我们可以在三维空间（例如篮球表面）中可视化二维球，并且可以通过类似方程式的解决方案来描述。在拓扑结构中，我们还研究了更高的维球。尽管这些空间无法在三个维度上可视化，但它们确实存在于更高的维度，也可以研究。令人惊讶的是，球之间的地图很难对其进行分类，甚至可以连续变形。球形之间地图的分类问题称为同型球体的计算问题。自1950年代以来，这个问题一直是一个重大而积极的研究问题。通过与许多数学主题有密切联系的各种方法，在这个问题上取得了很多进展。最近，使用动机和模棱两可的同义理论的技术取得了重大进展。该项目的目的是进一步开发动机和模棱两可的同型理论中的新技术，并研究这个球体同拷贝组的经典问题。研究生将参与该项目。该项目集中于在经典，动机和模棱两可的环境中稳定的球体稳定同型球体的计算。更具体地说，主要的研究者和合作者将继续使用动机同义理论中的技术来推动经典计算，将其推向Dimension 126中的Kervaire不变问题。主要的研究者和合作者将将CHOW T结构技术扩展到其他基础领域，并研究动机稳定的稳定稳定组的Sphers of Spheres of Spheres of Spheres of Spheres of Spheres of这些基础上，这些技术将其定义为这些基础。此外，PI将与合作者一起研究Adams Felttration 3中的新的世界末日猜想，并使用Equivariant技术来研究Hill-Hopkins-Ravenel型Spectra的同型群体组。该奖项反映了NSF的法定任务，并被认为是通过基金会的知识优点和广泛的criperia criperia criperia criperia criperia criperia criperia criperia criperia criperia criperia criperia criperia criperia criperia rection the Apportion。