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

拓扑学是一门研究空间形状的数学学科，同伦理论研究空间如何在不撕裂物体或在此过程中刺穿孔的情况下相互变形。在所有空间中，球体是最基本、最美丽的物体。同伦理论和代数拓扑的一个核心问题——球体同伦群的计算——是理解球体如何在不同维度映射到其他球体，以及理解不同的映射如何能够或不能够相互变形。这个拓扑问题不仅是关于基本数学对象的基本问题，而且与数学中的其他学科也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 建立了与一个问题的联系：有多少种方法可以在球体上进行微积分——球体上的光滑结构；科维尔和布劳德与一个问题建立了联系，即人们如何能够或不能对高维空间进行手术以将其变成球体——科维尔不变问题。此外，球面稳定同伦群的被加数J的像与伯努利数密切相关； Quillen 和 Goerss、Hopkins、Miller、Lurie 建立了与正式群模数堆栈的连接。最近，Hill、Hopkins、Ravenel、Voevodsky、Morel、Isaksen 等人建立了等变同伦理论和动机同伦理论之间的联系。该项目的目标是加深现有的联系，并通过进一步突破现有联系的极限来发现新的联系。本研究集中于球体稳定同伦群的计算，以及动机、等变和色同伦理论之间的相互作用，以及在微分拓扑问题中的应用，例如球体上光滑结构的唯一性和Kervaire不变问题。更具体地说，在 Isaksen 和 Wang 当前和正在进行的项目中，首席研究员 (PI) 在动机同伦理论中开发了新的计算工具，并与色同伦理论相结合，该理论在两年内计算了球体经典稳定同伦群的 40 多个新主干。 PI 将深化动机同伦理论和色同伦理论之间的新联系，在未来几年内进行更多稳定茎的计算，并利用这些计算来解决 126 维 Kervaire 不变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 还将探索真实动机同伦理论和 C2 等变同伦理论之间的联系。目标是证明这个方向的结构定理，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 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，并了解其与球体稳定同伦群的联系。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 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.