I'm a relatively newly-minted PhD (in math, algebraic geometry specifically, if that matters) approaching the end of their first year in a TT position at a smaller teaching-focused state school.
Like many people that I know, my thesis question was suggested to me by my advisor. As a result I didn't get much experience or guidance on how to go about developing a research program and finding good questions to tackle. Further, since I (for better or worse) skipped the postdoc phase of an academic's life, I feel as though I missed some sorely needed scholarly-development.
My current position requires a very modest amount of research for tenure, and so I find myself in a predicament where I need to produce publishable research, but I am having an enormous amount of trouble finding questions that I can actually tackle in the small amount of research time available. I know that the standard advice is to start with follow up questions to one's thesis, but my thesis resolves a handful of remaining cases of a question in my subfield, so there is very little to nothing left to extend or follow up on. Recently I felt as if I had a good question, but some recent digging on my part has me doubting that the project will turn out anything at all. While I accept that this is part of the research process, I really can't afford to spend 6+ months working on something to have it not pan out, since I already have so little time to devote to research to start with.
My question and hope is that some of you would be willing to share your process of finding a question that eventually turns into a publication.