Applications of Topological Cyclic Homology in Algebraic K-Theory

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.

Introduction
Prehistory
Dennis
Ktheory
Additive Ktheory
Rational homology
topological cyclocamology
Rational theorem
Bergstennelson
Prime field
State of the art
Impact on number theory
Smooth case
Ktheory of ring spectra
Localization
Closed excision
SuslinGabber
The norm
Two important developments
Single atomic spectra