Work of Hahn and Wilson demonstrated that many qualitative questions about algebraic K-theory and the arithmetic of ring spectra can be answered using a suitable filtration of topological cyclic homology, which closely approximates algebraic K-theory. Hahn, Raksit, and Wilson later observed that this filtration fits into a general paradigm, which also recovers filtrations previously studied by Bhatt—Morrow—Scholze, Antieau, Morin, and Bhatt—Lurie. I will report on some applications of this technology to ring spectra such as real topological K-theory, Morava K-theory, truncated Brown—Peterson spectra, and topological modular forms. In particular, these explicit computations give examples where the Mahowald—Rezk telescope conjecture holds and the Ausoni—Rognes—Lichtenbaum—Quillen conjecture holds. Each of these projects are joint with some subset of my co-authors Ausoni, Bruner, Davies, Hahn, Rognes, Wilson, and Yang.
