Motivic Homotopy Theory, Group Actions, and K-theory

本项目研究动机同伦理论，这是一个相对较新的工具，使我们能够利用代数拓扑学的方法来理解代数几何中感兴趣的对象，即代数簇-多项式方程的解集。该理论在解决开放性问题方面取得了一些惊人的成功。在这个项目中，PI计划开发类似的工具，不仅适用于研究代数簇，而且适用于研究它们的对称性。PI与合作者将研究动机同伦理论的结构方面。这包括研究等变motivic球谱和代数结构的同伦层。他们将开发一个等变motivic切片谱序列，专注于感兴趣的例子，如埃尔米特K理论。他们将继续他们的研究张量三角几何motivic同伦和它的影响色motivic同伦理论。最后，他们将研究环谱的同伦代数K-理论方面。

This project studies motivic homotopy theory, which is a relatively new tool which allows us to utilize methods from algebraic topology to understand the objects of interest in algebraic geometry, namely algebraic varieties - sets of solutions of polynomial equations. The theory has had several spectacular successes in resolving open problems. In this project, the PI plans to develop analogous tools which are well suited for studying not just algebraic varieties, but also their symmetries. The PI, together with collaborators, will study structural aspects of motivic homotopy theory. This includes a study of the equivariant motivic sphere spectrum and the algebraic structure of its homotopy sheaves. They will develop an equivariant motivic slice spectral sequence, focusing on examples of interest such as Hermitian K-theory. They will continue their study of tensor triangular geometry of motivic homotopy and its implications for chromatic motivic homotopy theory. Lastly they will study aspects of homotopy algebraic K-theory of ring spectra.