Local and global methods in homotopy theory

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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.