Classical, Motivic and Equivariant Stable Homotopy Groups of Spheres.

拓扑学是一门数学学科，研究空间的形状和它们之间的相互作用。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，球面是最重要的，我们可以用一个数字--维数--把它们分类到连续变形。一个一维的球体，我们通常称之为圆，是我们可以在一张纸上画出来的东西，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看到的东西，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，我们也研究更高维的球体。虽然这些空间不能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。有些令人惊讶的是，球体之间的映射更难分类，即使是连续变形。球面之间映射的这种分类问题称为球面同伦群的计算问题。自20世纪50年代以来，这个问题一直是一个主要的和活跃的研究问题。通过各种与数学各学科密切相关的方法，对这一问题的研究取得了很大的进展。最近，利用动机同伦理论和等变同伦理论的技术已经取得了重大进展。这个项目的目标是进一步发展motivic和等变同伦理论的新技术，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领域，在经典，motivic和等变的背景下。更具体地说，主要研究者和合作者将继续使用复数上的motivic同伦理论技术来推动经典计算，走向126维的Kervaire不变量问题。 主要研究者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域，并研究这些基域上球面的motivic稳定同伦群。此外，PI将与合作者一起研究亚当斯过滤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。

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.