Appears in collection : 2023 IHES Summer School – Recent Advances in Algebraic K-theory

I will discuss motivic filtrations on trace invariants of commutative ring spectra, defined in joint work with Jeremy Hahn and Dylan Wilson, and I will report on further work in progress concerning the associated graded objects of these filtrations, which constitute an extension of the theory of prismatic cohomology to the setting of commutative ring spectra.