Q&A: How to Find Open Problems for Math Research?

by Justin Skycak on

Cross-posted from here.

Question

Though not an undergraduate student, I just wanted to know where can I find hard new types of problems regarding the problems in graduate level mathematics.

Where can I find all kinds of research and open problems? I know that Wikipedia gives a list of problems but I still doubt that there are many problems on some authentic math-research websites.

Answer

I am happy to answer, though I need to preface my answer with a couple disclaimers:

  • It's really unlikely that you'll be able to rigorously solve any of these problems without a thorough background in both undergraduate and graduate-level math. Even graduate students generally need to spend some time specializing before they're in a position where they can make a contribution towards the "edge" of humanity's collective mathematical knowledge.
  • I'm not a professional research mathematician. I just had a phase during undergrad when I was really interested in learning more about what kinds of problems are "open" in math today -- in particular, the "typical" kinds of open problems, not just genius-level Millennium Prize problems and the like. (I was curious to see what the "edge" of math looks like nowadays, since today's math goes so deep that it can be hard to know just how deep.)

Anyway, here’s my answer.

The site openproblemgarden.org gets mentioned a lot for these kinds of questions, but it’s heavily slanted towards graph theory and isn’t exactly comprehensive and up-to-date.

I think you’ll have better luck searching Google Scholar with the query "unsolved open problems in [insert specific field of math here]" and putting some date constraint to limit the search to fairly recent results.

For example, you might search “unsolved open problems in combinatorics” and limit the results since 2020. (Of course, this is just a starting point, and you’ll want to play around with the date range and the specificity of the query to get better results.)

Note that if you identify an open problem that you’re interested in learning more about, you’ll also need to do follow-up searches to learn more about the work that has been done on that problem, especially any work that has been done since the “open problem” article that you’re reading was published (who knows, the problem may have been solved since then, or some other kind of progress may have been made).

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