Explore a 57-minute lecture on the applications of topological cyclic homology in algebraic K-theory, delivered by Bjorn Dundas from the University of Bergen at the Fields Institute. Delve into topics such as cyclic cohomology, Dennis K-theory, additive K-theory, rational homology, and topological cyclocamology. Examine the rational theorem, Bergstennelson, prime fields, and the state of the art in this field. Investigate the impact on number theory, smooth cases, K-theory of ring spectra, localization, and closed excision. Learn about Suslin-Gabber, the norm, and recent developments in single atomic spectra. This talk is part of the "Cyclic Cohomology at 40: achievements and future prospects" conference, offering a comprehensive overview of the subject's history, current state, and future directions.