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，而同伦理论研究的是空间如何相互变形，而不会在变形过程中撕裂或刺穿物体。在所有的空间中，球体是最基本、最美丽的物体。同伦理论和代数拓扑学的一个中心问题--球面同伦群的计算--是理解球面如何映射到其他不同维度的球面，以及不同的映射如何相互变形。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有着深刻的联系和相互作用。自20世纪50年代以来，它一直是一个主要的研究问题。例如，Kervaire和Milnor建立了一个连接的问题，有多少种方法可以做微积分领域-光滑的结构领域; Kervaire和Browder建立了一个连接的问题，如何可以或不能做手术，以更高的维度空间，把它们变成球体-Kervaire不变的问题。此外，球面的稳定同伦群的一个被加数，J的像，与伯努利数密切相关;奎伦和戈斯、霍普金斯、米勒、卢里建立了与形式群的模栈的联系。最近，希尔，霍普金斯，拉文埃尔，Voevodsky，莫雷尔，Isaksen和其他人之间建立联系等变同伦理论和motivic同伦理论。本项目的目标是深化现有的联系，以及通过进一步推动现有联系的极限来发现新的联系。本研究集中于计算球面的稳定同伦群，以及motivic，等变和色同伦理论之间的相互作用，并应用于微分拓扑学中的问题，如球面上光滑结构的唯一性和Kervaire不变量问题。更具体地说，在目前和正在进行的项目与Isaksen和王，主要研究者（PI）开发新的计算工具，在motivic同伦理论，连接到色同伦理论，计算40个新的茎经典稳定同伦群的领域在两年内。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，并使用这些计算来攻击126维的Kervaire不变量问题的最后一个未解决的情况。PI还将探索真实的动机同伦理论和C2等变同伦理论之间的联系，以下正在进行的工作Behrens，Dugger，Guillou和Isaksen。我们的目标是证明在这个方向上的结构定理，以及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高度大于2的计算，并理解其与稳定同伦球群的联系。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的知识价值和更广泛的影响审查标准。

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.