Computational motivic homotopy theory

计算动机同伦理论

• 批准号: 0803997
• 负责人: Daniel Isaksen
• 金额: $ 9.92万
• 依托单位: Wayne State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-06-01 至 2012-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0803997&HistoricalAwards=false
• 关键词: Computational; motivic; homotopy; theory

My main goal in this project is to carry out computations in motivic homotopy theory, i.e., the homotopy theory of algebraic varieties.Voevodsky's now famous proof of the Milnor Conjecture is based primarily on techniques in motivic homotopy theory. Specifically, I am interested in the motivic version of the Adams spectral sequence, with base field R or C. One reason to focus only on R or C is that thorough computations are possible over these fields, but the computations still exhibit non-classical phenomena. The motivic Adams spectral sequence has a major deficiency in that it is not known to converge. Even if the spectral sequence does not converge, it is nonetheless possible to draw many conclusions about motivic stable homotopy groups. I believe that these and other further computations will be an important guide to further study of motivic homotopy theory. There is also a chance that motivic computations will reveal something new about classical stable homotopy groups.Homotopy theory is a technique for studying geometric objects up to certain kinds of deformations. Homotopical approaches often allow for concrete calculations that are otherwise inaccessible. Computations of stable homotopy groups have been a major topic of research in topology since the middle of the 20th century. Although tremendous progress has been made, much remains unknown. In the 1990's, Fabien Morel and Vladimir Voevodsky developed motivic homotopy theory. In analogy to classical homotopy theory, motivic homotopy theory allows us to study algebraic objects up to certain kinds of deformations. My goal is to carry out some fundamental computations in motivic homotopy theory. The goal is both a better understanding of the new algebraic theory as well as the classical geometric theory.

我在这个项目中的主要目标是在动机同伦理论中进行计算，即，同伦理论的代数簇。Voevodsky现在著名的证明米尔诺猜想主要是基于技术motivic同伦理论。 具体地说，我感兴趣的是亚当斯谱序列的动机版本，基场为R或C。 只关注R或C的一个原因是，在这些领域进行彻底的计算是可能的，但计算仍然表现出非经典现象。 动机亚当斯谱序列有一个主要的缺陷，它是不知道收敛。 即使谱序列不收敛，仍然可以得出许多关于动机稳定同伦群的结论。 我相信，这些和其他进一步的计算将是一个重要的指导，进一步研究动机同伦理论。 还有一个机会，motivic计算将揭示一些新的经典稳定同伦群。同伦理论是一种技术，用于研究几何物体的某些种类的变形。 同伦方法通常允许以其他方式无法实现的具体计算。 自世纪中期以来，稳定同伦群的计算一直是拓扑学研究的一个重要课题。 虽然已经取得了巨大进展，但仍有许多未知数。 在20世纪90年代，Fabien Morel和弗拉基米尔Voevodsky发展了motivic同伦理论。 类似于经典同伦理论，动机同伦理论允许我们研究代数对象的某些种类的变形。 我的目标是在动机同伦理论中进行一些基本的计算。 目标是更好地理解新的代数理论以及经典的几何理论。