We briefly discuss some recent analytical results in 4D simplicial quantum
gravity: (i) The characterization of the singular structures sampling a nei
ghborhood of the critical point; (ii) The transition between weak and stron
g coupling as driven by the dynamics of a polymeric interface; (iii) The po
ssible connection between triangulations counting and the geometry of the m
oduli space of singular Euclidean structures on 4D manifolds.