Motivic and Equivariant Stable Homotopy Groups

球体是所有几何物体的基本组成部分。更复杂的几何物体可以通过将这些球体组合在一起来构造，但是不同维度的球体只能以特定的组合方式组合在一起。列举这些球的组合是稳定同伦理论的基本问题之一。这个问题被称为球的同伦群的计算。该项目使用光谱序列来进行这些计算。它们是微妙的、错综的、微妙的、复杂的，但只要有足够的洞察力和耐心，它们也是可以理解的。每当我们理解了机器的一个新部分，就可以进一步研究另一层复杂性。计算机计算起着很大的辅助作用。该项目促进使用视频会议与同行合作，为研究生提供建议，并举办在线研讨会。这些努力建立了一个自我维持的虚拟数学研究社区。这些新的互动模式的一个主要好处是，它们打破了传统的进入壁垒。这对于那些在偏远地区的人，以及那些具有非传统或非声望背景的人来说尤其有益，这些人通常无法进入传统的数学系。该项目将计算经典、c -动机、r -动机和c2 -等变稳定同伦群。主要工具有Adams谱序列、Adams- novikov谱序列和有效切片谱序列。该项目包括一系列相互关联的问题，既有代数问题，也有同伦问题。许多问题提出了获得稳定同伦群计算数据的具体方法。其他问题涉及相关的结构问题，如奇异周期性和Mahowald不变量。一个关键的想法是使用c动机计算来更深入地了解经典结构。初步结果表明，这是一种令人惊讶的强大技术，允许在一个范围内计算新的稳定同伦群。该项目还包括一系列计算r -动机和c2等变稳定同伦群的技术。这代表了在计算层面上首次认真努力解决Adams光谱序列等工具的等变版本。关键是通过c动力和r动力的中间步骤，逐步建立c2等变计算。这项工作的回报之一是计算出了非常困难的马霍瓦尔德不变量的新值。最后，本项目将研究有效切片谱序列，特别是在r -动机同伦理论中，以及在任意场上。这个谱序列是亚当斯-诺维科夫谱序列的动力替代。r动力计算更容易理解，而任意场涉及有趣的算术。受动力有效切片过滤的启发，本项目还将尝试研究一种新的c2等变过滤，这将有助于研究c2等变稳定同伦群。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Spheres are the basic building blocks of all geometric objects. More complicated geometric objects can be constructed by fitting these spheres together, but spheres of different dimensions can fit together in only certain combinations. Enumerating these combinations of spheres is one of the fundamental questions of stable homotopy theory. This problem is known as the computation of homotopy groups of spheres. The project uses spectral sequences to carry out these computations. They are delicate, intricate, subtle, and complicated, but they are also understandable with enough insight and patience. Each time a new part of the machinery is understood, another layer of complexities becomes accessible for further study. Computer calculations play a large supporting role. The project promotes the use of videoconferencing to collaborate with peers, to advise graduate students, and to host online seminars. These efforts build towards a self-sustaining virtual mathematical research community. One major benefit of these new modes of interaction is that they erode traditional barriers to entry. This is especially beneficial for people in remote geographical locations and for those with non-traditional or non-prestigious backgrounds who are not typically afforded access to traditional departments of mathematics.The project will compute classical, C-motivic, R-motivic, and C2-equivariant stable homotopy groups. The key tools are the Adams spectral sequence, the Adams-Novikov spectral sequence, and the effective slice spectral sequence. The project consists of a series of interlocking problems, both algebraic and homotopical. Many of the problems suggest specific methods for obtaining calculational data about stable homotopy groups. Other problems address related structural issues, such as exotic periodicity and Mahowald invariants. A key idea is to use C-motivic calculations to see deeper into classical structure. Preliminary results suggest that this is a surprisingly powerful technique that allows for the computation of new stable homotopy groups in a range. The project also includes a series of techniques for computing R-motivic and C2-equivariant stable homotopy groups. This represents the first serious effort to grapple with equivariant versions of tools like the Adams spectral sequence at a computational level. The key point is to build up to the C2-equivariant computations gradually, through C-motivic and R-motivic intermediate steps. One example of the kind of payoff for this work are computations of new values of notoriously difficult Mahowald invariants. Finally, the project will study the effective slice spectral sequence, especially in R-motivic homotopy theory but also over arbitrary fields. This spectral sequence is a motivic replacement for the Adams-Novikov spectral sequence. The R-motivic calculation is more accessible, while the arbitrary fields involve interesting arithmetic. Inspired by the motivic effective slice filtration, the project will also attempt to study a new C2-equivariant filtration that ought to be useful for studying C2-equivariant stable homotopy groups.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.