Abstract: Certain fibered hyperbolic 3-manifolds admit a layered veering triangulation, introduced by Agol in 2011. These triangulations have nice combinatorial and dynamic properties, and one is naturally led to the question: can the combinatorial invariants of these triangulations tell us something about the geometry of the 3-manifold? We will present some intriguing experimental results that address this question, and outline a proof that random layered veering triangulations (i.e., coming from a simple random walk on the mapping class group) are non-geometric with probability approaching 1 as the length of the walk goes to infinity. This is joint work with Dave Futer and Sam Taylor.