  In this dissertation, we discuss mainly the corresponding geometric and
representation theoretic aspects of relative $p$-adic Hodge theory and $p$-adic
motives. To be more precise, we study the corresponding analytic geometry of
the corresponding spaces over and attached to period rings in the relative
$p$-adic Hodge theory, including derived topological de Rham complexes and
derived topological logarithmic de Rham complexes after Bhatt, Gabber, Guo and
Illusie which is in some sense equivalent to the derived prismatic cohomology
of Bhatt-Scholze as shown in the work of Li-Liu,
$\mathcal{O}\mathbb{B}_\mathrm{dR}$-sheaves after Scholze,
$\varphi$-$\widetilde{C}_X$-sheaves and relative-$B$-pairs after Kedlaya-Liu,
multidimensional rings after Carter-Kedlaya-Z\'abr\'adi and Pal-Z\'abr\'adi and
many other possible general universal motivic rings or sheaves. Many contexts
are expected to be sheafified, such as over Scholze's pro-\'etale sites of the
considered analytic spaces by using perfectoids or the quasisyntomic sites by
using quasiregular semiperfectoids as in the work of Bhatt-Morrow-Scholze and
Bhatt-Scholze.
