Just as background, I am going into my senior year of undergrad and I have a pretty light load this coming semester but I want to fill this time with something that will really make my grad school resume sparkle. As you might guess from my username, I am fascinated by wavelets (and harmonic analysis in general) and I have decided to try to at least attempt some research over the next year to close out my UG studies. I spoke with a professor about this and he said he'd be more than happy to help me as much as he can and get me credit for it but I'd need to come up with the topic and expect to do the vast majority of it on my own (I actually prefer working more independently).
Anyways, I have been trying to find a problem that interests me enough that I can see myself working on 5-6 hours a day but I haven't had any luck. While I feel confident in my knowledge of wavelet theory, I don't really know which way is up when it comes to the actual research end of things. I know the major players like Daubechies, Mallat, Morlet, etc. but I am having a hard time figuring out what constitutes "cutting edge".
Obviously, I expect that any research is going to be more on the applied end of things but I'd prefer it to be as "pure" (i.e. not computational) as possible just because I am limited in how much computing power I have access to (I've had my computer crash many many times from trying to do wavelet projects in MATLAB) and I am skeptical that the department will let me anywhere near the big guns. My knowledge base is sufficient that I can understand most of Mallat's "A Wavelet Tour of Signal Processing: The Sparse Way", though I find it to be quite a slog just because of the notation and lay out. I am particularly interested in some potential interplay between stochastic processes (sufficiently nice ones at least) and wavelet analysis. I do not have a great background in stochastic integration quite yet but I learn fast enough that I feel confident it wouldn't pose too much of barrier.
I would be indebted to anyone who could point me in the right direction.