I like your style. Right now I'm distracting myself from solving a PDE arising in quantum mechanics using techniques from compressed sensing.
Of the 3, compressed sensing is the easiest to pick up as it's more of a new signal processing trick than a fully developed research area. Checking out Candes' ICM talk (http://dsp.rice.edu/cs) as well as Igor's obsessive blog (http://nuit-blanche.blogspot.com/) does not require that much of a time commitment and can be a lot of fun. Finding research problems is another story.