Maths around the worldWatch
Also depends on how contemporary it should be. You'll have a lot of trouble investigating at depth the vast majority of modern mathematics unless you're comfortable with the language used by mathematicians. Also even if you were to engage with all the jargon, you'd find it hard to write much about things done recently because of the amount of pre-requisite knowledge required to talk about it at depth. But if you were to say, talk about the gaps between primes, you could mention the twin primes conjecture, some work done by older mathematicians and perhaps some outline of modern methods. Maybe this won't be contemporary enough, but I don't know how you'd do any better really.
Some ideas: Maybe you could write on one of the Clay Institute's Millenium problems. These problems get so much attention that there is a lot of accessible material that you can read to get a feel for what these problems are about, especially the Riemann hypothesis which gets the most heads turning.