Local and global methods in homotopy theory

Massachusetts Institute Of Technology, Cambridge MA

Investigators

Abstract

ABSTRACT The PI will investigate how systematic phenomena in the homotopy groups of spheres is reflected in arithmetic phenomena occurring in modular and automorphic forms. The stable homotopy groups exhibit chromatic layers of periodic behavior. The PI will continue his work studying how the moduli space of elliptic curves, and the theory of modular forms, detects the second chromatic layer. The PI, with his collaborators, will investigate generalizations that relate the higher chromatic layers to Shimura varieties, and their automorphic forms. The unstable homotopy groups of spheres will be investigated within a similar algebro-geometric framework using the Goodwillie tower. The PI will study the relationships between ramification phenomena in number theory and unstable phenomena in homotopy theory that arise. The PI's work studies higher dimensional geometry using the theory of numbers. The spaces of elliptic curves and their generalizations that the PI is studying are central objects in number theory, playing a pivotal role both in Wiles' proof of Fermat's last theorem and in Harris and Taylor's proof of the local Langlands correspondence. Geometry in higher dimensions occurs in economics and physics. The study of number theory gives rise to the encryption systems that allow for the secure transmission of data, such as those used by secure web pages. The links between the seemingly disparate fields of geometry and number theory that the PI proposes to investigate will encourage dialog that breeds new science, produce problems suitable for engaging students in research activities, and stir public interest in the mathematical sciences.