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I'm in pure math and I'm trying to come up with interesting research projects to work on. How can I find interesting problems?

Also, I'm not even a grad student or postdoc or assistant professor. I completed graduate school. I was going to start a postdoc, but I was prevented from going, due to a pandemic that has killed over 6 million people. I eventually gave up the offer. Now I am staying at home and working a part-time job, but I am completely isolated. I want to publish good papers in good journals but I don't have any interesting problems to work on and I am not talking to anyone. I don't have a publication record and I don't know what to do. At this rate, I will never have any publications! I cannot go on like this. I need to do something different.

Should I just ask people if they have any problems that I can work on? If the answer is no, then what should I do?

To become a serious mathematician I need papers. To produce papers I need results. To get results I need to solve interesting math problems. To solve interesting math problems I need interesting math problems to solve. I am stuck at this step.

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    You should revise your thesis according to the referee reports, and resubmit, possibly to a different journal... Sep 26, 2022 at 16:58
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    You're not alone. Finding a (good) problem to work on is a challenge for everyone, even for established mathematicians. Many smart professors told me that they had a weak semester\year during their first postdoc, so you are not alone, its a shock for everyone
    – Yanko
    Sep 26, 2022 at 23:50
  • Related: academia.stackexchange.com/q/71810/19607
    – Kimball
    Sep 27, 2022 at 0:14
  • If you do not have a natural continuation for your phd work then you cannot hope to advance alone. Did you have an advisor? Discuss with him/her about this. Usually, PhDs can get jobs teaching in universities, in some parts of the world. If you manage to do this then you could have opportunities to discuss with other professors there. It is difficult to give advice. The situation is not as simple as you describe in your last paragraph... Problems to work on come on their own if you are on the right path. Sep 29, 2022 at 8:17

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I'm not in math, but my understanding is that math is at least like other fields in that new work builds on old work, so the best place to find new problems is to look at solutions: read work that is being published in your field. Papers may either explicitly identify places where they've fallen short or implicitly there may be openings in the areas they do not reach.

Unfortunately, knowing which of these problems are likely to be solvable or are of the highest interest is really a matter of experience, and there's no shortcut to experience.

You say you are isolated and not talking to anyone; it does seem to me to be nearly impossible to continue in research without academic connections that you can talk to people through. However, even without an academic appointment, universities may host talks and other events that are open to the public. You'll need to make more effort to find these sorts of things if you aren't part of a department that communicates them to you.

Should I just ask people if they have any problems that I can work on?

This option does not seem appropriate to me for a person in your situation, though if you have close personal contacts it may be reasonable to discuss with them. Cold emailing or similar is probably a waste of your time and everyone else's.

If you haven't successfully published as a PhD student, it seems to me that you may be holding your expectations of yourself too high if you expect to publish "good papers in good journals" all alone. Academic publishing isn't easy to learn, and it's supposed to be a central part of the apprenticeship process that is a PhD degree.

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    Especially if there are universities near you that run seminars... these are usually open to the public, and can be a way to meet people, see what current activity is about, and maybe ask for a little advice from more senior people in a seminar that touches your specific research interests. Sep 26, 2022 at 17:01
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    Also, some seminars not even near to you may be "hybrid", meaning you can attend by Zoom, if you just get the address from the organizers. Go to depts websites to see what seminars they offer, etc. Sep 26, 2022 at 17:14
  • @paulgarrett To become a serious mathematician I need papers. To produce papers I need results. To get results I need to solve interesting math problems. To solve interesting math problems I need interesting math problems to solve. I am stuck at this step. I will see if there are seminars. I don't know how to phrase things. It may be weird to e-mail an organizer and say, "I'm not affiliated with your university whatsoever; can I join the seminar?"
    – cgb5436
    Sep 26, 2022 at 17:32
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    You don't have to emphasize that you're not affiliated with the university! (Especially for hybrid/remote seminars.) Honest intellectual endeavor is the real point, not "affiliation"... Sep 26, 2022 at 17:33
  • @paulgarrett I went to a Zoom seminar last week. I still don't get how to get from this to an open problem that will lead to a publication. That's because I am an outsider and I think I need to become an insider. Most of the people who attended the talk were already insiders (meaning, they knew each other, wrote papers on the subject, etc.).
    – cgb5436
    Oct 4, 2022 at 17:34

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