Dustin Clausen: Some special features of K(n)-local stable homotopy theory

The "chromatic" perspective provides a filtration of stable homotopy theory, whose graded pieces correspond to what's known as K(n)-local stable homotopy theory.  One of the remarkable features of this story is that these K(n)-local categories, although they explain all sorts of delicate torsion information in stable homotopy theory, exhibit many simplifying formal phenomena which make them more analogous to a characteristic zero analytic theory.  These include exponential and logarithm maps, canonical "divided power" operations, and a weak analog of semi-simplicity of finite group representations.

I will explain an analogy with elementary calculus (a different one from Goodwillie calculus) which helps to frame these phenomena and generate new questions about them.  This is a result of discussions with Gijs Heuts, Akhil Mathew, and Vesna Stojanoska.  The talk will mostly be a survey talk aimed at discussing the work of others, notably Bousfield, Hopkins, Kuhn, Ravenel, and Rezk.  It will also be focused on concrete phenomena which arise, and their connections with other areas of mathematics such as p-adic analysis and formal groups.  In particular we will see in examples that K(n)-local homotopy theory generates number-theoretic constructions which are as interesting and primitive as possible given the constraints of the situation.