For my master's thesis, I am working on extending recent work of Annala–Pstrągowski regarding canonical weight filtrations on resolvable motives. In particular, I hope to establish a more robust and functorial construction of the "motivic" weight filtration.

More specifically, my goal is to define an $\infty$-category $\mathrm{DM}^{\mathrm{log}, \mathrm{kgl}}$ of logarithmic kgl-motives over the spectrum of a Dedekind domain, using only smooth log smooth projective schemes as input. A suitable sub-$\infty$-category $\mathrm{DM}_{\mathrm{res}}^{\mathrm{log},\mathrm{kgl}}$ of logarithmic resolvable kgl-motives should then be canonically equivalent to Annala–Pstrągowski's $\infty$-category of resolvable kgl-motives. Then, using logarithmic motivic homotopy theory, we should be able to leverage this equivalence to produce functorial weight filtrations.

As an application, it should be relatively straightforward to e.g. recover a décalaged variant of Mokrane’s pole-order filtration on logarithmic crystalline cohomology. I am happy to discuss this project in more detail.