Computational Motivic and Equivariant Homotopy Theory

球面是拓扑学中简单而重要的研究对象。代数拓扑学的中心问题之一是对给定维数的球面到不同维数的球面上的所有可能映射进行分类。最近，这个问题在其他方面也受到了关注：当球被认为是代数几何领域，或者当球有特定的对称性，必须保持的映射问题。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。PI将奋进扩大理解这些问题的范围，特别是在具有双重对称性的球体的设置中。该项目的另一个组成部分集中在一个不变量称为K理论，它位于拓扑和代数的接口。PI将开发一个版本的K-理论，同时配备了对称性和乘法。PI将继续他的联合工作与丹Isaksen计算motivic和C2-等变稳定同伦群的领域。环C上运动稳定同伦群的计算与经典稳定同伦群的计算非常相似。从那里，Bockstein谱序列恢复了R上的动机稳定同伦群。最后，考虑点的C2-等变同调中的“负锥”，得到等变稳定同伦群。在另一个方向上，PI与May、Merling和Osorno一起，将继续研究G-谱作为谱麦基函子的表示。这个理论的核心是具有良好乘法性质的等变无限循环空间机。一些工具，图中心的应用，如等变交换乘法，范数映射，几何不动点，没有得到很好的理解，在这个模型的G-谱。

Spheres are simple yet important objects of study in topology. One of the central questions of algebraic topology is to classify all of the possible mappings of a sphere of a given dimension onto a sphere of different dimension. More 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. Even more recently, greater understanding of how these various contexts impact each other has emerged. The PI will endeavor to expand the range in which these questions are understood, especially in the setting of spheres with a twofold symmetry. Another component of the project focuses on an invariant known as K-theory, which lies at the interface of topology and algebra. The PI will develop a version of K-theory which is simultaneously equipped with symmetries and multiplications.The PI will continue his joint work with Dan Isaksen on computations in the motivic and C2-equivariant stable homotopy groups of spheres. The computation of motivic stable homotopy groups over C is fairly similar to that of classical stable homotopy groups. From there, the Bockstein spectral sequence recovers the motivic stable homotopy groups over R. Finally, accounting for the "negative cone" in the C2-equivariant homology of a point leads to the equivariant stable homotopy groups. In a different direction the PI, together with May, Merling, and Osorno, will continue to investigate the presentation of G-spectra as spectral Mackey functors. A centerpiece of this theory is an equivariant infinite loop space machine with good multiplicative properties. A number of tools that figure centrally in applications, such as equivariantly commutative multiplications, norm maps, and geometric fixed points, are not well understood in this model of G-spectra.