Interestingly, the question of the geometry of the universe (not the same as its shape, but similar) has recently come up against since some measurement showed a value which would be inconsistent with a flat (Euclidean) universal geometry. This is the thing he was talking about with the sum of angles in a triangle.

Beyond that, though, the question of whether the universe is truly infinite or finite, yet unbounded is generally open. The idea being that if the universe started from a singularity, and is still expanding from that, this implies that it must have a finite total volume at any point in time. This is sometimes described in terms of the the surface of a balloon being inflated: It is finite, yet unbounded, and has a finite area at each point in time.

The multiverse thing never really seemed interesting or relevant, to me. It is more of a way of explaining how seeming randomness works, within the universe, while everything else appears to be deterministic. Given that these are never considered accessible, it doesn't really matter in even a theoretical sense if they exist. Plus, the laziness of modern movie writing has made me instinctively roll my eyes whenever this comes up.

All of that said, it would be interesting to get a better sense of the fundamental structure of the universe, as it would get us much closer to understanding how all interactions are mediated, which would be both satisfying but also useful. Of course, this is just more of my dream of someday waking up to a reality where someone has managed to observe a graviton. Failing that, put forth a satisfying theory of everything. I am not holding my breath, though, as it is unlikely I will live to see either thing happen.