Abstract

The theory of topological cyclic homology, on the one hand, is a vital tool for the computations of algebraic K-theory, on the other hand, is itself interesting in respect of the relation to the various p-adic cohomology theories, or essentially to the prismatic cohomology theory. The purpose of this talk is to briefly provide a general introduction of topological cyclic homology with the aforemetioned perspectives, including mainly the basic definitions, the cyclotomic trace relating algebraic K-theory and topological cyclic homology, and the (motivic) filtrations relating topological (negative, or periodic) cyclic homology and prismatic cohomology. If time permits, a computation of topological cyclic homology for local fields through the descent-style technique might also be touched upon.