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Explore the intricate relationship between K-theory and topological cyclic homology (TC) in this 46-minute lecture from the Fields Institute's "Cyclic Cohomology at 40: achievements and future prospects" series. Delve into Dustin Clausen's explanation of recent results demonstrating the tight connection between K(R) and TC(R) for p-adically complete commutative rings with mod p^n coefficients. Gain insights into the definition of topological cyclic homology as a variant of cyclic homology, its application to noncommutative rings, and the natural map from algebraic K-theory to TC. Consider the challenges of extending these findings to non-commutative settings and understand the significance of TC(R) as the closest cyclic homology-like approximation to K(R). Learn about the collaborative research efforts with Bhargav Bhatt, Akhil Mathew, and Matthew Morrow that underpin these discoveries.
