RUI: Higher Structures in Stable, Equivariant, and Motivic Homotopy Theory

本研究探讨当代数学的三个分支：高等范畴、同伦理论和代数几何之间的接口。粗略地说，这些学科分别研究结构、变形和多项式。通过该奖项开展的项目以新颖的方式将这些概念联系起来，寻求揭示新的定理。首席研究员还将通过组织夏季研究小组和撰写与这些项目相关的高级论文来支持不同群体的本科生参与他们的研究。私人学院的研究项目有五个方面。首先，他们将共同研究绝对伽罗瓦群的谱Mackey函子映射到模拟动力谱的“谱Gysin函子”（由Garkusha-Panin-Voevodksy的框架对应理论构建）的方式。其次，他们将通过连接Witt - k理论研究eta周期动力球谱。第三，他们将通过Lazard Hopf代数群上的模进一步解释稳定动力同伦范畴的Balmer谱中的结构。第四，他们将构造处理真g置换范畴配对的等变无限循环空间机。最后，他们将研究Picard 2-群和稳定2型的同伦理论，以及两者之间的相互作用。

This research investigates the interface between three branches of contemporary mathematics: higher categories, homotopy theory, and algebraic geometry. Loosely speaking, these subjects study structure, deformations, and polynomials, respectively. The projects undertaken via this award link these concepts in novel ways, seeking to expose new theorems. The Principal Investigors will also support the participation of a diverse group of undergraduate students in their research by organizing summer research groups and leading senior theses related to these projects.The PIs' approach to their research program is five-fold. First, they will jointly study the fashion in which spectral Mackey functors of an absolute Galois group map to "spectral Gysin functors" (constructed from Garkusha-Panin-Voevodksy's theory of framed correspondences) which model motivic spectra. Second, they will study the eta-periodic motivic sphere spectrum via connective Witt K-theory. Third, they will explicate further structure in the Balmer spectrum of the stable motivic homotopy category via comodules over the Lazard Hopf algebroid. Fourth, they will construct equivariant infinite loop space machines which handle pairings of genuine G-permutative categories. And finally, they will study the homotopy theory of Picard 2-groupoids and stable 2-types, and the interaction between the two.