Classical, Motivic and Equivariant Stable Homotopy Groups of Spheres.

拓扑学是一门研究空间形状及其相互作用的数学学科。拓扑学中的一个基本问题是拓扑空间及其映射的分类问题。在所有的空间中，球体是最重要的，我们可以通过一个单一的数字-维度-将它们分类为连续变形。一维球体，我们通常称之为圆，是我们可以在一张纸上画的东西，可以用我们熟悉的圆的方程的解来描述。二维球是我们可以在三维空间中看到的东西，比如篮球的表面，可以用类似的方程的解来描述。在拓扑学中，我们还研究了高维球面。虽然这些空间不能在三维中可视化，但它们确实存在于更高的维度，也可以研究。有些令人惊讶的是，球体之间的贴图更难分类，即使是连续变形。球面间映射的分类问题称为球面同伦群的计算问题。自20世纪50年代S以来，这一问题一直是一个重大而活跃的研究问题，通过与许多数学学科密切联系的各种方法，这一问题已经取得了很大进展。近年来，利用理据论和等变同伦论的方法已经取得了重大进展。这个项目的目的是进一步发展动机理论和等变同伦理论中的新技术，并研究球面同伦群的计算这一经典问题。研究生将参与这个项目。这个项目集中在计算稳定同伦群的球面，在经典，动机和等变背景下。更具体地说，主要研究人员和合作者将继续使用复数上的动机同伦理论中的技术，将经典计算推向126维的Kervaire不变问题。主要的研究人员和合作者将把定义在复数上的Chow t-结构技术推广到其他基场，并研究这些基场上的球面的动机稳定同伦群。此外，PI将与合作者研究亚当斯过滤3中的新末日猜想，并使用等变技术研究Hill-Hopkins-Ravenel类型光谱的同伦群。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.