Local and global methods in homotopy theory

摘要本文将研究球同伦群中的系统现象如何反映在非模自同构形式的算术现象中。稳定同伦群表现出周期行为的色层。PI将继续他的工作，研究椭圆曲线的模空间，以及模形式理论，如何检测第二色层。PI和他的合作者将研究高色层与志村变体及其自同构形式之间的关系。利用古德威利塔在类似的代数-几何框架内研究了球的不稳定同伦群。本项目将研究数论中的分支现象与同伦理论中出现的不稳定现象之间的关系。PI的工作是用数论研究高维几何。PI所研究的椭圆曲线空间及其推广是数论的核心对象，在怀尔斯对费马大定理的证明以及哈里斯和泰勒对局部朗兰兹对应的证明中都起着至关重要的作用。高维几何出现在经济学和物理学中。对数论的研究产生了允许数据安全传输的加密系统，例如安全网页所使用的加密系统。PI提议调查的几何和数论这两个看似不同的领域之间的联系，将鼓励催生新科学的对话，产生适合学生参与研究活动的问题，并激起公众对数学科学的兴趣。

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.