Much of our computational knowledge of algebraic K-theory is due to trace methods, most notably the cyclotomic trace, which approximates K-theory via cyclic homology. This approximation becomes especially good if we compute cyclic homology over the sphere, the higher categorical analogue of the ring of integers. The resulting invariant, denoted TC, is referred to as topological cyclic homology.

Recent advances in our understanding of TC, due to Nikolaus and Scholze, provide even more computational control. We will discuss this approach to TC and trace maps from algebraic K-theory, as well as some applications to our understanding of algebraic K-theory from the perspective of chromatic homotopy theory.