I have been a tenured faculty member at a math department in the US for a few years. Recently two colleagues of mine asked (apparently independently) why I do not have PhD students. I honestly explained my reasons to each of them. Also I added that despite that, if it is necessary I CAN take students. Neither of them commented my answer.

Should I take these two questions at face value? Or did the colleagues try to let me know in an indirect way that I am expected to have students? Is there a way to make sure whether my department is unhappy that I have no PhD students?

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    The most effective way to find out what your colleagues meant by the question is to ask them about it. – GrotesqueSI Sep 9 '19 at 18:21
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    I am curious as to the reasons you gave your colleagues... not so much out of my own curiosity, but to estimate your actual situation as well as their (at least semi-predictable) reactions... which may be as helpful to you as an answer to your literal question. That is, as I think you guess, "why don't you...?" is often a polite form of "you really should...", in the first place. And there are many "reasons" that can be given that will antagonize and alienate your colleagues. Can you add these details? – paul garrett Sep 9 '19 at 21:35
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    @eykanal you have deleted the best answer (by @GrotesqueSI) to the question "Is there a way to make sure whether my department is unhappy that I have no PhD students?" – Anonymous Physicist Sep 10 '19 at 0:02
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    You're tenured, so I assume you've been at the university for a while. Do you have a more senior colleague who you can go to to see if the question is likely to have had undertones you've missed? Perhaps over a beverage of choice (be that alcoholic, caffeinated, or other) as an excuse to have a casual conversation? – Van Sep 10 '19 at 3:50
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    Title is vague. Rewrite to summarize your specific issue. – Basil Bourque Sep 10 '19 at 3:56

It is impossible to read their minds, of course, but I think that they are at least expressing surprise that you don't have doctoral students. There may be a (very) mild rebuke in it, especially if it is thought to be a necessary element of the work.

But, there may be an implied warning, that, although they can't sanction you for not having students, there might be long term negative consequences for your career if you get by without accepting students. You don't say whether you have been asked to be an advisor and you don't say whether you have turned people down. If either is true then the warning, as well as the rebuke, might be a bit stronger.

But whether there are consequences or not depends on local custom at your place of employment as well as your other contributions to the profession.

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    Yes, in PhD-granting departments (in math, in the U.S.) the expectation is that faculty supervise at least a few PhD's. At the same time, yes, doing other things instead may be viewed as doing one's share. – paul garrett Sep 9 '19 at 20:22

By and large, mathematicians have a tendency to express themselves logically and literally, much more so than the general population. If your colleagues asked you why you don’t have PhD students, it is reasonable to assume by default that their intent was to find out why you don’t have PhD students. That explanation is not the only conceivable one, but is at least as likely as any other.

As for any underlying subtext, hidden motives, subtle hints, passive aggressive behavior, a twisted sense of humor, or any number of other reasons why people (even mathematicians) sometimes say X when they mean Y: you’ll just have to ask them.

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