Computational Motivic and Equivariant Homotopy Theory

球面是拓扑学中简单而又重要的研究对象。代数拓扑学的核心问题之一是将给定维度的球面到不同维度的球面上的所有可能映射分类。最近，这个问题在其他情况下受到了关注：当球体被考虑在代数几何的领域中时，或者当球体指定了必须由所讨论的映射保持的对称性时。甚至在最近，对这些不同背景如何相互影响的理解也得到了更多的理解。PI将努力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专注于一个被称为K-理论的不变量，它位于拓扑学和代数的交界处。PI将发展一种同时配备对称和乘法的K-理论。PI将继续与Dan Isaksen在球面的Motivic和C2-等变稳定同伦群中的计算方面的联合工作。C上的动机稳定同伦群的计算与经典稳定同伦群的计算非常相似。从这里，Bockstein谱序列恢复了R上的动机稳定同伦群。最后，考虑了点的C2-等变同调中的“负锥”，得到了等变稳定同伦群。在一个不同的方向上，PI将与May、Merling和Osorno一起，继续研究G谱作为光谱Mackey函子的呈现。该理论的核心是一种具有良好乘性的等变无限循环空间机。在这个G谱模型中，一些在应用中居于中心位置的工具，如等变交换乘法、范数映射和几何不动点，并没有得到很好的理解。

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.