Galaxy?
We do well to move within our own system. Even getting to another star is problematic.
You invent Warp technology, and we'll talk about it. (We have research labs ...)
Well, I'm thinking long term.
Real, real long term...
We did manage to make a rather neat matter-transmitter-discombobulator once. You zap it, it pops up over there--->, then promptly goes through some weird matter/anti-matter annihilation process.
We have however modified this NASA proposal for a subluminal ZPE engine into a power supply for something a bit more ambitious and along the lines of a Droscher/Hauser/Heim warp engine. Hawking had some curious ideas on the subject, but missed a fundamental point. You don't violate causality with FTL travel. No matter how fast you go beyond light speed using warp, you cannot return to your destination before the time coordinates from which you left.
Consider, set up a warp bubble and FTL to Cirus. This takes you maybe three days ship time. You then return to Earth. No matter how much faster than light you go, you still return to Earth 3+ days after you left. Time for you is relative to your baseline time-space coordinates.
How so many smart folk keep missing this is beyond me. FTL isn't hard, it's Time travel that is impossible.