The world problem
What is the shape of our universe? We know that when we look around us, we see 4 dimensions: three are of course length, width, and height; the fourth is time. We also believe that you will see 4 dimensions basically wherever and whenever you go in the universe.
Knowing this, however, isn't enough to know the overall shape of the universe. The surface of the Earth has 2 dimensions, for instance latitude and longitude, but this alone is not enough to know its shape. A two-dimensional surface could be round like a ball, flat like a paper, looped like the surface of a donut, or even worse: it could be twisted in hyperspace so that if you went around the world, you would come back to where you started, but everything would be backwards.
How much harder is it to figure out the overall shape of the 4D universe? Much harder! If one has a complete atlas of the Earth, it is no hard work to determine that it must be roughly spherical. However, even if one had an atlas of the universe, it has been proved that no computer program could be guaranteed to tell you whether it is "simple" like a 4D ball, or has "handles" like a donut.
For those with some knowledge of abstract algebra or computer science, there is a recent paper that gives a beautiful overview of the proof that this problem is undecideable. The proof relies on two cool facts I'd never seen before, which give an idea of how the proof goes. First is that every finite group is the fundamental group of some 4-manifold - one can construct a manifold directly from the group representation. Secondly, any Turing machine can be encoded into a group, and an input mapped to an element which is trivial if and only if the Turing machine halts!