Abstract: We introduce low dimensional K-theory first, and briefly explain Quillen's plus construction. Then we recall the Hochschild homology of an algebra, and its relation with K-theory. If time permits, we give a briefly introduction on stable homotopy theory, and $v_n$-periodicity.

Abstract: The Lichtenbaum--Quillen property comes with an arithmetic background, Waldhausen reformulated this property as a telescopic homotopy problem. The chromatic redshift introduced by Rognes generalized this idea to higher height ring spectra. And algebraic K-theory for ring spectra can be well understood by trace method, we will briefly recall the trace method. By the work of J. Hahn, D. Wilson, et al., the redshift problem can be reduced to Segal conjecture and (weak) canonical vanishing problem. We recall the techniques to prove Segal conjecture, and we present examples that Segal conjecture holds. We use the cyclic decomposition to demonstrate some examples that Segal conjecture fails, and thus Lichtenbaum--Quillen property fails also.