For my master’s thesis, I am working on intrinsically describing the $\infty$-category $\mathrm{DM}_{S,\mathrm{res}}^\mathrm{kgl}$ of kgl-linear resolvable motives in terms of only smooth log smooth projective schemes, which, using logarithmic geometry, would make it easier to define functors out of this $\infty$-category.

In particular, we can then more easily construct weight filtrations on any kgl-linear cohomology theory using the methods of Annala–Pstrągowski. With this description in hand, it should be relatively straightforward to e.g. recover a décalaged variant of Mokrane’s pole-order filtration on logarithmic crystalline cohomology. I am also curious about the possible arithmetic implications of the resulting weight filtrations on various cohomology theories.