From Quillen's to Waldhausen's K-theory

In the first part of this talk, I will propose a quick trip in the algebraic K-theory of structured ring spectra, or "brave new rings". In this setting, Quillen's algebraic K-theory of rings, which captures number-theoretical information, and Waldhausen's A-theory of spaces, which captures geometric information, become instances of the same construction. In the second part, I will explain some steps in the attempt to interpolate from one theory to the other. This is in part joint work with John Rognes (Oslo).