One fantasy for physicist and mathematicians is to work out the math to show that something is possible, mathematically, regardless of the practicality of it. Yes it can be done, at the cost of the GNP of Earth for the next 12000 years...and then the hard part starts.

Warp drives: yeah, Star Trek. This physicist thought he might have the numbers worked out. To use slashdot's summary:

...while relativity prevents faster-than-light travel relative to the fabric of spacetime, it places no restriction on the speed at which regions of spacetime may move relative to each other. So a small bubble of spacetime containing a spacecraft could travel faster than the speed of light, at least in principle.

But when quantum effects are considered, it falls apart. Dang.

And in a different but quasi-related slashdot article, a video simulation of what falling through a black hole would look like.