Local and global methods in homotopy theory

同伦理论中的局部和全局方法

• 批准号: 0605100
• 负责人: Mark Behrens
• 金额: $ 13.94万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0605100&HistoricalAwards=false
• 关键词: Local; global; methods; homotopy; theory

ABSTRACTThe PI will investigate how systematic phenomena in the homotopygroups of spheres is reflected in arithmetic phenomena occurring inmodular and automorphic forms. The stable homotopy groups exhibitchromatic layers of periodic behavior. The PI will continue his workstudying how the moduli space of elliptic curves, and the theory ofmodular forms, detects the second chromatic layer. The PI, with hiscollaborators, will investigate generalizations that relate the higherchromatic layers to Shimura varieties, and their automorphic forms. The unstable homotopy groups of spheres will be investigated within asimilar algebro-geometric framework using the Goodwillie tower. ThePI will study the relationships between ramification phenomena innumber theory and unstable phenomena in homotopy theory that arise.The PI's work studies higher dimensional geometry using the theory ofnumbers. The spaces of elliptic curves and their generalizations thatthe PI is studying are central objects in number theory, playing apivotal role both in Wiles' proof of Fermat's last theorem and inHarris and Taylor's proof of the local Langlands correspondence. Geometry in higher dimensions occurs in economics and physics. Thestudy of number theory gives rise to the encryption systems that allowfor the secure transmission of data, such as those used by secure webpages. The links between the seemingly disparate fields of geometryand number theory that the PI proposes to investigate will encouragedialog that breeds new science, produce problems suitable for engagingstudents in research activities, and stir public interest in themathematical sciences.

摘要PI将研究球体同质组中的系统现象如何反映在发生的算术现象中发生的，并发生了自型形式。 稳定的同型组表现出周期性行为的层层。 PI将继续他的工作研究，以研究椭圆曲线的模量空间以及模化形式的理论如何检测第二个色层。 带有Hiscollaborator的PI将调查将较高颜色层与Shimura品种及其自动形式相关的概括。将使用Goodwillie Tower在不稳定的球体组合框架内研究不稳定的球体。 ThePI将研究出现的同型理论中的分支现象无数理论与不稳定的现象之间的关系。PI的工作研究了使用Numbers理论研究更高维几何形状。 椭圆曲线的空间及其概括的PI正在研究是数量理论的中心对象，在威尔斯的Fermat的最后定理和Inharris和Taylor证明了当地的Langlands对应方面，都扮演了惯用的作用。在经济学和物理学中，几何形状发生在更高维度。 数字理论的概念产生了允许数据安全传输的加密系统，例如安全网页使用的数据。 PI提出的研究的几何学和数字理论的看似不同的领域之间的联系将鼓励繁殖新科学，产生适合于参与研究活动的问题，并激发公众对Themathalationalationalcation Science的兴趣。