Motivic Homotopy Theory, Group Actions, and K-theory

动机同伦理论、群体行动和 K 理论

• 批准号: 1710966
• 负责人: Jeremiah Heller
• 金额: $ 18.68万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-07-01 至 2022-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1710966&HistoricalAwards=false
• 关键词: Motivic; Homotopy; Theory; Group; Actions

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.

本项目研究动机同伦理论，这是一个相对较新的工具，它允许我们利用代数拓扑的方法来理解代数几何中感兴趣的对象，即代数变异-多项式方程的解集。该理论在解决开放问题方面取得了几次惊人的成功。在这个项目中，PI计划开发类似的工具，不仅适合研究代数变量，而且适合研究它们的对称性。PI将与合作者一起研究动机同伦理论的结构方面。这包括对等变动力球谱及其同伦束的代数结构的研究。他们将开发一个等变的动机切片谱序列，重点关注像厄米k理论这样感兴趣的例子。他们将继续研究动同伦的张量三角几何及其对色动同伦理论的启示。最后，他们将研究环谱的同伦代数k理论。

项目成果

Rigidity for equivariant pseudo pretheories

等变伪前理论的刚性

• DOI: 10.1016/j.jalgebra.2018.09.027
• 发表时间: 2018
• 期刊: Journal of Algebra
• 影响因子: 0.9
• 作者: Heller, Jeremiah;Ravi, Charanya;Østvær, Paul Arne
• 通讯作者: Østvær, Paul Arne

Free (Z/p)n -complexes and p-DG modules

游离 (Z/p)n 配合物和 p-DG 模块

• DOI: 10.1016/j.jalgebra.2020.07.028
• 发表时间: 2021
• 期刊: Journal of Algebra
• 影响因子: 0.9
• 作者: Heller, Jeremiah;Stephan, Marc
• 通讯作者: Stephan, Marc

K-theoretic obstructions to bounded t-structures

• DOI: 10.1007/s00222-018-00847-0
• 发表时间: 2016-10
• 期刊: Inventiones mathematicae
• 影响因子: 3.1
• 作者: Benjamin Antieau;David Gepner;J. Heller

Some remarks on topological $K$-theory of dg categories

关于dg范畴的拓扑$K$理论的一些评论

• DOI: 10.1090/proc/14128
• 发表时间: 2018
• 期刊: Proceedings of the American Mathematical Society
• 影响因子: 1
• 作者: Antieau, Benjamin;Heller, Jeremiah
• 通讯作者: Heller, Jeremiah

Topological comparison theorems for Bredon motivic cohomology

Bredon 动机上同调的拓扑比较定理

• DOI: 10.1090/tran/7553
• 发表时间: 2018
• 期刊: Transactions of the American Mathematical Society
• 影响因子: 1.3
• 作者: Heller, J.;Voineagu, M.;Ostvær, P.
• 通讯作者: Ostvær, P.