Logarithmic topological cyclic homology

Logarithmic ring spectra are a simultaneous generalization of logarithmic rings from algebraic geometry and structured ring spectra from homotopy theory. Working with logarithmic ring spectra allows the construction of interesting intermediate localizations of rings or structured ring spectra where one can make an element invertible without making its powers invertible. In this talk, I will outline how topological Hochschild homology and topological cyclic homology can be defined for logarithmic ring spectra. These extended versions of THH and TC take part in interesting localization sequences that do not exist if one only looks at structured ring spectra. A prominent example participating in various localization sequences will be the fraction field of topological K-theory, which can be realized as a logarithmic ring spectrum with logarithmic structure generated by a given prime and the Bott element. This is report on joint work in progress with John Rognes and Christian Schlichtkrull.