The Dundas-Goodwillie-McCarthy theorem, one of the most celebrated results in trace methods, provides a systematic comparison of the algebraic K-theory and topological cyclic homology of ring spectra. A key input to the proof of this result is a robust understanding of the topological Hochschild homology of a trivial square zero extension of ring spectra. In this talk, I will give a rough overview of the proof of the Dundas-Goodwillie-McCarthy theorem, as well as a description of the cyclotomic structure of the topological Hochschild homology of a trivial square zero extension in terms of the relative Tate diagonal, a new notion due to Lawson. Time permitting, I will explain how one can use this description to calculate the Goodwillie-Taylor tower of topological cyclic homology and (integral) topological restriction homology.