I'm interested in higher height analogues of results in "classical" motivic homotopy theory, i.e. results concerning the motivic filtration on the K-theory of discrete rings; I'm interested in the case where the ring R is allowed to be a non-discrete (e.g. E-infinty) ring spectrum and we replace K-theory with an approximation (e.g. THH, TP, TC).

The even filtration, as pioneered by Bhatt-Morrow-Scholze, and developed in significant greater generality by Hahn-Raksit-Wilson and Pstrągowski, gives a very powerful tool for studying the THH, TP, and TC of ring spectra, which serve as invariants approximating the algebraic K-theory. Bhatt-Lurie and Drinfeld further gave accessible presentations of the associated stacks (i.e. the stacks associated to the graded Hopf-algebroids coming from the even-filtered Čech nerve of a cover) in the case of discrete rings. I'm interested in leveraging the stacky versions of the even filtration to answer questions about K-theory (and friends) of higher ring spectra.

Many questions about algebraic K-theory are only accessible via trace methods, i.e. by passing to TC where things are slightly more computable (see the Dundas-Goodwillie-McCarthy theorem). I'm interested in some of these questions.