I'm a pure math phd student and the supervisor gave me two papers so as I find an idea to research but the subjects are unrelated and I can't concentrate on one of them and can't decide which one is likely to be a fruitful subject, What should I do?
First, it depends on the country you're in, the type of funding you're applying for, and probably to a smaller extent on your field of research.
In case you already have an agreement with a supervisor, then you should probably look to him for advice on choosing a topic, but make the choice yourself. Given that he sent you papers on these two topics, you can probably assume (absent any evidence to the contrary) that both would make a good and fruitful subject. Thus, choose with your heart: what problem is most exciting to you?
This is a tricky question. In principle, of course, it's best if one comes up with one's own question/project. However, it is all too easy to accidentally, from inexperience, choose a too-hard, or boring-to-experts, project, or, more subtly, a project in which neither you nor your advisor has any special insights to give you an edge over anyone else.
Especially if you are hoping to make a run at the "research post-doc" game, you'll want to have done Ph.D. work that is of interest to experts in that field. A novice has the obvious difficulties ascertaining whether a given topic might be of interest to experts, but your advisor should be able to clarify this. I have seen examples of people working enthusiastically on a PhD topic, and finish the degree, only to have no one care at all, and no job offers result.
Also, I don't think it's the case that a "smart-enough person" can pick random problems (that are of interest to experts) and expect to make significant progress merely by wanting to. While it's true that "fresh eyes" can be helpful, literal inexperience is rarely useful. I tend to think that one's advisor should give "insider advice" about un-obvious possibilities for progress.
Thus, in reality, I think that the true choices available are something like first choosing an advisor based on their expertise. Then ask them for guidance. Presumably they will not dictate a project, but merely suggest, giving you choices... but it is very important to continue the iteration of "getting advice", rather than just privately making a choice and assuming it is wise. Lacking the information and experience your advisor should have, it is essentially impossible to make good choices on your own. Tentative choices, to be discussed with your advisor, yes.
I understand your situation because I recently started my PhD.
You do the following:
- Try to come up with ideas from those two papers
- Apart from that try to come up with own ideas of yourself
- Take all the ideas with you in a meeting with your supervisor and tell him frankly what you want to work on.
I think in general the role of the PhD should increase trough the course of the project. In the beginning, the supervisor will have a significant role, both in selecting the subject and performing daily duties. This is needed imo to get the unexperienced PhD going. Later, the role of the PhD increases, and the supervisor takes a more passive role.
In the Netherlands, it is customary to write four journal papers as a PhD (4 years). While the first might be an idea that your supervisor came up with, your last should probably be your own idea.