I would like to work on a research project in my particular field of mathematics. While I'm familiar with the area in general, I'm having a bit of trouble choosing an inquiry narrow enough to be productive. There are lists of open questions, but open questions are generally open for a reason, and I don't have the luxury of spending a decade on a single narrow problem. (This is also the sort of thing one normally picks up in grad school, but my advisor handed me a list of problems he found in a paper somewhere and insisted that I choose among them. Also, I don't have access to any sort of mentor or other useful figure to steer me in a useful direction.)
How do I go from "I want to learn about X" or "It seems like there's a useful connection between X and Y" to some specific, publishable result?
I would like to avoid--- to the extent that one can with something as unpredictable as research--- going down an alley that turns out to be trivial or intractable.