Abstract: Work of Bhatt?Morrow?Scholze highlights a ?q-deformation? of algebraic de Rham cohomology, and moreover a striking relation of this invariant to stable homotopy theory via topological Hochschild homology, i.e. Hochschild homology relative to the sphere spectrum. In this talk, I will discuss an ongoing investigation into a related phenomenon involving Hochschild homology relative to the complex K-theory spectrum, drawing a link with chromatic homotopy theory.