Motivic and Equivariant Stable Homotopy Groups

球体是所有几何对象的基本构建块。 更复杂的几何物体可以通过将这些球体拟合在一起来构造，但是不同尺寸的球体只能以特定的组合来拟合。 枚举这些球的组合是稳定同伦理论的基本问题之一。 这个问题被称为球面同伦群的计算。该项目使用光谱序列来执行这些计算。 它们是微妙的，错综复杂的，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。 每当机器的一个新部分被理解时，另一层复杂性就可以进一步研究。 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，为研究生提供咨询，并举办在线研讨会。 这些努力建立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，它们消除了传统的进入壁垒。 这对于那些生活在偏远地区的人以及那些通常无法接触到传统数学部门的非传统或非声望背景的人来说尤其有益。该项目将计算经典，C-motivic，R-motivic和C2-等变稳定同伦群。 主要工具是亚当斯谱序列、亚当斯-诺维科夫谱序列和有效切片谱序列。该项目包括一系列的连锁问题，代数和同伦。 许多问题提出了获得稳定同伦群的计算数据的具体方法。 其他问题解决相关的结构问题，如奇异周期性和Mahowald不变量。一个关键的想法是使用C-motivic计算来深入了解经典结构。 初步结果表明，这是一个令人惊讶的强大的技术，允许计算新的稳定同伦群的范围。该项目还包括一系列用于计算R-motivic和C2-等变稳定同伦群的技术。 这代表了第一次认真努力，以解决同变版本的工具，如亚当斯光谱序列在计算水平。 关键点是通过C-动机和R-动机的中间步骤逐渐建立到C2-等变计算。 这项工作的一个例子是计算出臭名昭著的困难Mahowald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-motivic同伦理论，但也在任意领域。 这个谱序列是亚当斯-诺维科夫谱序列的动机替代。 R-motivic计算更容易理解，而任意字段涉及有趣的算术。 受motivic有效切片过滤的启发，该项目还将尝试研究一种新的C2-等变过滤，这种过滤应该对研究C2-等变稳定同伦群有用。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智力价值和更广泛的影响审查标准进行评估来支持。

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.