Algebraic K-theory of strict ring spectra

We view strict ring spectra as generalized rings. The study of their algebraic K-theory is motivated by its applications to the automorphism groups of compact manifolds. Partial calculations of algebraic K-theory for the sphere spectrum are available at regular primes, but we seek more conceptual answers in terms of localization and descent properties. Calculations for ring spectra related to topological K-theory suggest the existence of a motivic cohomology theory for strictly commutative ring spectra, and we present evidence for arithmetic duality in this theory. To tie motivic cohomology to Galois cohomology we wish to spectrally realize ramified extensions, which is only possible after mild forms of localization. One such mild localization is provided by the theory of logarithmic ring spectra, and we outline recent developments in this area.Comment: Contribution to the proceedings of the ICM 2014 in Seou