Perhaps if you are analytically inclined then analysis on fractals is a good place to look. A good portion of the work done is with second finite difference equations leading towards a limit definition of a ``second derivative'' and uses some linear algebra such as inverting small matrices. Strichartz's Differential Equations on Fractals is a good place to start, especially the first few chapters where spectral decimation is discussed. As an aside signifigant number of papers in this area have come out of REU programs.

In general focusing on your courses is important but that should still leave you with some time to think about other topics as well. This kind of curiosity will help you see what's out there and give you a sensible way to choose a specialty when the time comes.

Best of luck.