Currently, a warp drive is forbidden by relativity. So I would say, all research trying to find where relativity breaks. So I reckon the gravity wave detectors like LIGO and LISA are more relavant than CERN.
That may seem like "equivalent to forbidden", but some of the interesting surprises of quantum physics were originally theorized by Einstein and company as a way to disprove quantum physics because they were clearly too weird to be true. Maybe not as weird as "Oh, you thought an event's cause had to precede its effect? LOL.", but bizarre and non-intuitive nonetheless.
Of course, that's not a small assumption...
The spacecraft doesn't actually move through space it pulls the space where it needs to go towards the spacecraft and pushes away the space behind it.
I'm no rocket surgeon so don't quote me.