As part of my undergraduate degree, I had to produce a thesis. On the advice of my advisor (naturally), I wrote about the MLC Conjecture in Complex Dynamics. It's a brilliant topic that draws on so many branches of mathematics, and the conjecture is one of mathematics' most difficult and consequential as-yet-unsolved problems. [Note that my name was different at the time].
Background gif (Animated Starfield Tile) created by ArtBIT, distributed with the Creative Commons Attribution-ShareAlike 3.0 License