Abstract: Computation of the stable homotopy groups of spheres is a

long-standing open problem in algebraic topology. I will describe how

chromatic homotopy theory uses localization of categories, analogous to

localization of rings and modules, to split this problem into easier pieces,

called chromatic levels. Each chromatic level is a symmetric monoidal

category, and we can study their Picard groups. I will talk about classical results about these groups,

and about current work on understanding the Picard group at the second chromatic level.