Dévissage, hearts, traces, and units in K-theory

I will discuss the developments of the last 10 years toward the existence of a fiber sequence K(BP) -> K(BP) -> K(E(n)) proposed by Ausoni and Rognes in 2002. Here, E(n) is the Johnson-Wilson theory and BP is the truncated Brown-Peterson spectrum. This sequence generalizes the famous fiber sequence K(F_p) -> K(Z_p) -> K(Q_p) due to Quillen. After touching briefly on the key role of dévissage in Quillen's theorem, I will talk about the n=1 case of the sequence established by Blumberg and Mandell, and of the generalization of their methods to Barwick's Theorem of the Heart. Finally, I'll discuss the role of trace methods in K-theory computations and explain how to prove that no such fiber sequence exists for n>1 using a new calculational tool: the computation of the trace on the units. The new material in this talk is joint work with Barthel and Gepner.