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Let us suppose that a certain senior Ph. D. student is interested in "securing" a postdoctoral position under a professor whom he does look up to. Would it be right for the student to write the researcher and ask him if he stands a chance of eventually being hired as a postdoctoral fellow in the institution wherein the researcher works? The Ph. D. student is certainly interested in some topics on which the researcher has written in the past; furthermore, there is a certain problem at which the student plans to try his hand once he manages to graduate and this problem is also somewhat near the area of expertise of the researcher in question.

Thanks in advance for any suggestion you may want to leave for me below.

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How close are you to graduating? If you're still a few years away, then perhaps get an introduction from your advisor to that professor and see if there are natural ways to seek feedback or advice from that professor.

If you are thinking about the next year and a half, then your questions should be more concrete. In particular, you might tie your question to funding, as that determines a lot of the possibilities. Even if a professor loves your work, they can't hire you if they don't have the money for it.

  • If you see from the professor's CV that there is an active grant, you could ask about whether there is space on that grant.

  • If there are regularly postdoctoral openings in the person's department, you can ask about whether the professor is looking for another postdoc soon. (You may learn that people in the department tend to take turns hiring postdocs, and the professor is (un)likely to be up this year. Apply anyway, but know your odds.)

  • You could say that you're preparing to apply for an external grant for a postdoc (e.g. NSF in the U.S.) and wondering whether the professor would be open to having you work with them on that. (Often a faculty sponsor is already required to apply.)

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    In addition to feedback and advise, one might also wish to collaborate with the professor. – Tommi Mar 24 '18 at 20:21
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The situation in math is slightly different than other fields and so it is reflected in the hiring process.

As a general rule, it is always good if your potential employers (i.e. the professors in the related math department) will know you beforehand, and definitely if you have means to go and present your work in a seminar go do that. If you happen to meet in a conference, go introduce yourself and shake hands with him.

Approaching a professor directly out of the blue (i.e. just emailing) might not be the best thing, you might get overlooked or something, it might irritate the professor, the usual best idea about emailing someone without prior meeting with him is to arrange a more senior person (i.e. your adviser) to email him and tell him about you and inquire about your status as a future hire. It is also OK to email him with your preprint before putting them up in the Arxiv (again, do everything in coordination with your adviser, as it would be best to write explicitly in the email - my adviser, Prof. MATH, asked me to send this article to you for your consideration) and then end up in p.s - I'm about to graduate, and I would like to consider me as a post doc.

You have mentioned some future work, after the initial contact, you may bring that up (again, it would be best to ask your current adviser about it, maybe there are better experts to contact, maybe things are easy, you don't want to discourage your future employer so it is a delicate situation). Definitely you should write that future direction on your research statement you upload to mathjobs.

Regarding ''securing'' - most post-doc positions in math are (junior) faculty, means that there is some teaching involved, and the hire need to be accepted by the department. There's no extensive voting process like the one done for senior faculty hires, but usually there is some post-doc committee sifting through the applications, short-listing and so on. The professor you've mentioned can ask to consider you to the short list. Moreover, in many cases the spots are allocated between the different ''groups'' namely in there are X places, Y will be analysis, Z will be NT, W will be dynamics, P will be algebra, Q will be applied and so on. So you should check that this professor you mentioned can grease the wheels inside his group in order to get you high up the ranks.

In the end, the best thing you have is your recommendations (and this is what carries the most significance to your hire). Make sure that they are good and when hiring season is in play (say December), make sure your adviser is taking care of you and encouraging his friends to hire you.

Good luck.

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Discussing potential research projects and post-doc positions is good, and one stands a much better chance to get a position with prior contact with the professor. Also, the student can get some insight in how the application process works. It could even be that the university tailor the announcement for a post-doc position to be favorable for the student, if the student is really strong.

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