On the Telescopic Picard Group

Chromatic homotopy theory aims to study cohomology theories through a hierarchy of simpler layers, organized by a notion called height. In this talk I will introduce the basic ideas behind this viewpoint and explain two approaches to analyzing these monochromatic layers: the classical K(n)-local category, which is closely related to one-dimensional formal group laws, and the T(n)-local or telescopic category, which is more directly tied to periodic phenomena in the stable homotopy groups of spheres. I will then describe a framework for understanding periodicity inside the chromatic layers, and explain how this allows one to lift Picard elements from the K(n)-local setting to the telescopic setting. Finally, I will present an application to chromatic Galois theory, leading to the construction of a first example of a non-abelian Galois extension in the T(n)-local world.