Equivariant Stable Stems

球体是拓扑学中简单而重要的研究对象。代数拓扑学的核心问题之一是高维球面到低维球面的所有可能映射的分类。事实证明，这种球体映射的分类在几何学和物理学中有着广泛的影响。最近，这个问题在其他情况下受到了关注：当球体在代数几何领域中被考虑时，或者当球体具有指定的对称性时，这些对称性必须由所讨论的映射保持。最近，人们对这些不同背景如何相互影响有了更深入的了解。该奖项支持的研究将利用这些新发现的联系来扩大对这些问题的理解范围，特别是在具有双重对称的球体的设置中。本项目为研究生提供科研培训并提供资金。首席研究员将继续与Dan Isaksen共同研究球的动力和c2等变稳定同伦群的计算。r动力计算更容易接近，这些计算决定了一部分c2等变稳定同伦群。主要的工具是rho-Bockstein谱序列和Adams谱序列。将采用各种技术来运行这些光谱序列，包括使用Massey产品。PI和合作者还将研究r -动力和c2等变环境下的v1-周期性，生成支持周期性算子的有限配合物。在这种情况下，这将导致稳定同伦群中元素的周期族。在另一个方向上，另一个合作将分析等变上同调理论的加性幂运算。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 simple yet important objects of study in topology. One of the central questions of algebraic topology is the classification of all possible mappings of a high-dimensional sphere onto a sphere of lower dimension. It turns out that this classification of mappings of spheres has wide-ranging repercussions in geometry and in physics. Recently, this question has received attention in other contexts: when the spheres are considered in the realm of algebraic geometry, or when the spheres have specified symmetries which must be preserved by the mappings in question. More recently, greater understanding of how these various contexts impact each other has emerged. The research supported by this award will employ these newfound connections to expand the range in which these questions are understood, especially in the setting of spheres with a twofold symmetry. This project provides and funds research training for graduate students.The principal investigator will continue joint work with Dan Isaksen on computations of the motivic and C2-equivariant stable homotopy groups of spheres. The R-motivic computations are more approachable, and these determine a portion of the C2-equivariant stable homotopy groups. The main tools will be the rho-Bockstein spectral sequence and the Adams spectral sequence. Various techniques will be employed to run these spectral sequences, including the use of Massey products. The PI and collaborators will also investigate v1-periodicity in the R-motivic and C2-equivariant settings, producing finite complexes that support periodicity operators. This will lead to periodic families of elements in the stable homotopy groups of spheres in these contexts. In another direction, another collaboration will analyze additive power operations for equivariant cohomology theories.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.