There are various "representation theorems" for lattices such as Birkhoff's Representation Theorem that states that *every finite distributive lattice is isomorphic to a quasi-sublattice of the lattice of subsets of some set*. There is an analogous Stone Representation Theorem for Boolean algebras.

Here is a fact that inspires this question: given a vector space $V$, the Grassmanian, $\mathrm{Gr}(V)$, is the lattice of subspaces of $V$ where joins are given by subspace sum, and meets by intersection. Fact: $\mathrm{Gr}(V)$ is a modular lattice (this holds more generally for the lattice of subgroups of an abelian group, for example).

Here is my first question: *given a modular lattice, $L$, does there exist a vector space $V$ such that $L$ is isomorpic to a quasi-sublattice of $\mathrm{Gr}(V)$*? This post may be a partial answer to this question, but perhaps there is more progress towards an answer in less generality.

My second question concerns cellular sheaves:
let $\mathcal{F}: X \rightarrow \mathbf{Mlc}$ be a functor from the face relation poset of a cell complex (i.e. $\sigma \trianglelefteq \tau$ if $\sigma$ is a face of $\tau$ with transitive closure...) to the category, $\mathbf{Mlc}$, of modular lattices and (Galois) connections.
*Does there exist a cellular sheaf $F: X \rightarrow \mathbf{Vec}$ such that $\mathcal{F} \cong \mathrm{Gr} \circ F$?* In a way, this question is the natural generalization to the first question to a particular class of simple diagrams of modular lattices and connections.