I have completed my masters in mathematics. I want to pursue PhD in mathematics. I want to frame a research question and write papers in my field.

But some questions are bothering me:

  • How does one know that a research problem is interesting and after solving it, one will be able to publish in a good journal?

  • I know the famous journals in my field. But after reading them, how will I decide myself which problem will be beneficial to solve?

  • Also I have seen there are lots of unsolved problems given at the end of a paper in a journal. If I don't understand or find any interesting problem should I try to solve the problems given at the end of a paper?

  • Does it mean the problems stated at the end of a paper are interesting that's why they are stated as problems?

  • Related: How to choose a good research problem? – Nate Eldredge Sep 9 '18 at 14:16
  • Just few thoughts. Try to look at retrospect from already solved problems, which generated more enthusiasm by the community (e.g., larger number of citations). If you domain is applied math, then probably think of the problems that might have larger potential applicability (e.g., number theory in cryptography). – student Sep 9 '18 at 14:30
  • Just so you know, being interesting is neither necessary nor sufficient for publishing in a good journal. They are correlated, yes, but if your goal is one or the other, you don't have to optimize for both (especially starting out). – Jeffrey Bosboom Sep 10 '18 at 3:42

These questions are best answered by a more experienced person than yourself, in particular your advisor. It is why you have an advisor in the first place. But let me be a bit more specific to your (many) questions.

First, you don't know what you will learn when starting research. That is why it is research and not just writing stuff down. A problem may turn out to be difficult or easy. It may turn out to have an interesting solution or just a run-of-the-mill grind-out-the-result solution. But you don't know until you do it. I think that is true in any field, but is so in mathematics, certainly.

For the second bullet, all problems are valuable to solve. Even if the solution is just to conclude the answer is trivial. You learn from solving them, and that is one of the main benefits of doing math in the first place.

Usually, if a problem is listed as unsolved in a recent paper it is probably worth pursuing. But such questions generate interest, so look, also, for more recent results. That doesn't mean that it isn't also valuable to solve the problems yourself, as suggested in the previous paragraph.

If a paper has been published in a reputable journal, it has been reviewed by other mathematicians. Presumably they agree that the problems stated at the end have at least some interest. But, solving them also increases your skill.

In general all mathematical questions, unless trivial, are interesting to at least some mathematicians. It can be a very small number, of course as some of the silos in math research are very narrow and deep. But if they are interesting to you, especially as a student, they are interesting by definition.

But get a doctoral advisor and explore some of these things with him or her.

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