Advice on finding good research questions

Question

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.

Answer 1

As you have discovered, such things are very difficult to do alone.

I suggest that you work to develop a circle of collaborators over time. Find a way to have regular conversations within your circle throwing around ideas. In a large department, that can often be with other local faculty, but it probably means going outside the local institution.

Attend conferences as often as you can and talk to people. Then keep talking to them.

Your doctoral institution may have some people who will be happy to talk to you, assuming a large department. Your advisor might be willing to introduce you to their own circle.

If you have funds for it, or can get them, arrange for people in your field to come visit and talk to students. Use the opportunity to talk to them about "stuff" that might develop. Or, get invited to talk elsewhere, paid or not, and start those conversations with people there.

And, keep notes. Lots of notes. Keep them in a form that you can easily review them. I like notecards for such things as they are easy to carry, sort, revise, discard. But keep at least a notebook of ideas that occur to you or that come up in conversations.

Answer 2

It comes down to being curious and asking lots of questions. Some examples:

• Is there a relationship between X and Y? If so, do they share the same properties? If not, then which properties are equivalent and vice-versa.
• Has anybody answered a research question you have in mind?
• Is this property known, true/false? Why is this true/false?
• Why is this problem hard?
• ...

Sooner or later you will find a question that has no answer. You then have to find out whether it is a significant question. If yes, then set forth a plan to answer it. Otherwise, move on. There are some questions that are not within your reach. In that case, you may want to find a low hanging 'fruit'. If you are this stage or research direction, you tend to find many open questions.

Many researchers have a pre-set of questions in mind; equivalently, they have a 'taste' for certain type of research questions.

Some researchers have a 'hammer' in mind, and their goal is to look for 'nails'.

I'm sure there are many other strategies.