Motivic and Equivariant Stable Homotopy Groups

球体是所有几何对象的基本构建块。通过将这些球体拟合在一起可以构建更复杂的几何对象，但不同维度的球体只能以特定的组合组合在一起。列举这些球面组合是稳定同伦理论的基本问题之一。这个问题称为球面同伦群的计算。该项目使用谱序列来进行这些计算。它们是微妙的、错综复杂的、微妙的、复杂的，但如果有足够的洞察力和耐心也是可以理解的。每了解机械的一个新部分，就会有另一层复杂的东西可供进一步研究。计算机计算起到了很大的辅助作用。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给自足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，它们侵蚀了传统的进入壁垒。这对于偏远地理位置的人和那些非传统或非声望背景的人来说尤其有益，他们通常不能进入传统的数学系。该项目将计算经典的、C-Motivic、R-Motivic和C2-等变的稳定同伦群。关键工具是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关于稳定同伦群的计算数据的具体方法。其他问题涉及相关的结构性问题，如奇异周期和Mahowald不变量。一个关键的想法是使用C-Motivic计算来更深入地了解古典结构。初步结果表明，这是一种令人惊讶的强大技术，它允许在一定范围内计算新的稳定同伦群。该项目还包括一系列计算R-基元和C2-等变稳定同伦群的技术。这代表了第一次认真努力在计算水平上解决工具的等变版本，如亚当斯谱序列。关键是通过C-Motivic和R-Motivic中间步骤逐步建立C2-等变计算。这项工作的回报的一个例子是计算出出了名的困难的Mahowald不变量的新值。最后，该项目将研究有效的切片谱序列，特别是在R-动机同伦理论中，但也在任意域上。该谱序列是对Adams-Novikov谱序列的基序替代。R-Motivic计算更容易实现，而任意域涉及有趣的算术。在有效切片过滤的启发下，该项目还将尝试研究一种新的C2等变过滤，该过滤应该对研究C2等变稳定同伦群有用。这一奖项反映了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。

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.