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Many professors got their Ph.D degrees from departments that were much more prestigious than the ones they are working in. So when they train Ph.D students in their current department, (I think)they may have the following comparisons, though they will never speak them out:

"When I was an undergraduate student in Princeton University, I could solve such problems much quickly than you graduate students do".

"When I was taking algebraic topology with *** in Princeton University, we covered the whole book within one semester while you guys...."

At the same time, the students may also have thoughts that they can never make achievements as big as their advisers in the future. For instance, they will get positions in departments that are much less prestigious.

So my question is, what motivates professors to train Ph.D students who aren't nearly as smart or excellent as them? Of course, I am comparing students and they advisers in their schooldays. I know professors just can't refuse students who want to work with them without strong reasons, but what makes them "feel happy" to work with students who are not nearly as good as them?

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    Unstated assumption: a significantly weaker student cannot be of help to an advisor. This is false. A lot of student theses are of the nature "understand some ideas the advisor has no time to fully pursue, write them up readably and correctly, and check how far they can be taken". This can be done using much less intelligence or even training than the advisor has (if such a comparison makes sense to begin with), and yet be of much use to the advisor (and others in the subject). – darij grinberg Feb 2 '17 at 22:47
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    I also think a professor very well can refuse a student for being too slow or not smart enough or for various other reasons that might not be the student's fault. And, short of that, can drop many subtle clues to the student that an advising relationship might not work out well. At MIT, we had qualifying exams for essentially this reason: By failing a student or letting them "barely pass", an examiner can tell them "in the current state, you won't make a good student of mine". – darij grinberg Feb 2 '17 at 22:51
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    One remark about slowness: it does not matter as much in research as you seem to think. A slow student might just get less done, but not necessarily at a lower quality or originality. So "we covered the whole book within one semester" might be just an anecdote. (My experience with people who read long books within one semester is: often they don't learn as much from said books.) – darij grinberg Feb 2 '17 at 23:02
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    I wouldn't think much of a professor who thinks they can accurately evaluate their students' "IQ" (or whatever) from the limited interactions they have. I don't think that's the right mindset to use for advising a student. There are plenty of reasons students can fail to perform that have nothing to do with raw ability. An advisor should be trying to address concrete actions by the student, not making assumptions in advance. If you drop a student you do it because of how they perform, not because you're making advance judgments about why they failed to perform. – Trixie Wolf Feb 3 '17 at 0:19
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    "When I was taking algebraic topology with xxx in Princeton University, we covered the whole book within one semester while you guys...." I can assure you that many "average" mathematicians, even with a not-so-decent alma mater think exactly the same. Plus, they also think the same about their own lecture from 5 years ago. Ahh, the good old days… – Dirk Feb 3 '17 at 0:44
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As often happens, I'd disagree with significant parts of the implicit and explicit premises of the question.

For example, "as good as" traditionally makes most "sense" in the solitary-hero model of mathematics, in which there is some mystical gift/talent/genius without which one cannot reach the statospheric regions... blah-blah-blah. Further, "fast problem solving" is easy to see, but not much related to serious, long-term research enterprises, so to think about "solving problems faster" is misguided (I claim).

Yes, it is convenient for an individual (as it was to some degree for me years ago) to do well in contests, because this is an easy "success" to put on one's resume. In contrast, a natural affection for mathematics is hard to document, hard to "show", and so on. Many in my cohort at Princeton years ago had done very well on the Putnam, but turned out to not really like genuine mathematics (without quick gratification) enough to finish their PhD.

And, really, should one care in one's heart about the prestige of one's workplace? Sure, in some practical regards it may have advantages in pay and "status", but there are equally obvious downsides: pressure to generate status, as possibly opposed to genuinely advancing our collective understanding (for example). Tension and stress. Who needs it?

My own preference is for students who are genuinely interested in math, as opposed to viewing it as a career choice or a vehicle for ego. (E.g., some of my students who manifested a sort of mathematical "Oedipal" complex created needless troubles for themselves, and saddened me as well.)

To my mind, the operational difficulty in making interesting, genuine contributions to advancement of our collective understanding is not our "lack of heroic genius", but the commodification of research, and corporatization of universities, insofar as it leads to a fake objectification of "research". For example, success as measured by federal funding dollars. Srsly? The federal government as font of sage wisdom about human knowledge? "Even" NSF is driven by its accountability to congress, and certainly never having enough money to fund all the reasonably worthwhile projects people propose.

In summary, I like talking about interesting (to me) mathematics with people who are willing to not try to sucker-punch me or prove they're "better than me", but, instead, "focus on the math". Likewise, I have no interest in "proving I'm better than" anyone else, because I am less interested in those feral (if typical human...) interactions than in understand mathematics better.

The backstory on my possibly-extreme context for valuation of mathematics is that while I greatly enjoyed reading a variety of mathematics books in the public library as a kid, I had no idea at all that a person could make a living by doing such stuff, and was entirely prepared to just "think about math" as a hobby after 5:00 pm at some engineering job. The idea of teaching high school math was out of the question (crowd control), and community college was barely plausible (and pay not so good). The idea that one could teach "as little as 15 hours a week" and think about math the rest of the time and get paid pretty well was amazing and unfathomable to me at the time.

Plus, I claim that "math is not so hard", if portrayed less strictly-orthodox-ly than is common when "requirements" are used as a club to beat people with. That is, the ground question is interest and aesthetic, not "genius" (which is kinda a fake thing anyway).

(It is true that some people think their students should go off by themselves and come back only when they've done something amazing, but this seems to me a pathetic waste of the (supposed) accumulated knowledge/wisdom of more experienced people... This ultra-Spartan model connects to the bad hero-model, in my opinion.)

Wait, what was the question? :)

EDIT: ... and I should add that the main benefit of going to a fancy place for grad school was to get the idea that mathematics was not the dreary, grim, tedious, ugly thing that it may be portrayed by people who feel that they are inadequate, or don't really like it after all, but are "in too deep". That was a great relief!!!

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    Great answer, IMO. I just want to emphasise that (1) you don't have to be a genius to do research, and (2) there is simply so much work to be done in almost all researchable areas that even "average" students can very much make significant contributions. – 101010111100 Feb 3 '17 at 0:11
  • @101010111100, yes, indeed. Good to put quotes around "average"! :) – paul garrett Feb 3 '17 at 0:17
  • All of your answers are really wonderful to read, as a student. Thank you so much. – Nitin Feb 15 '17 at 19:29
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Professors (in my experience) only push PhD students to right direction. It is not like they are some research buddies that work a lot alongside.

Professor is often the manager that takes some executive tasks in administration and manage the courses and their dominion on PhD and postdocs pushing the research to where the grants are made. They may have some research ideas that they execute on their underlings. In tenure track they increasingly move from your postdoc to a full professor that has only a coordinating role.

The hierarchy in academic organization is really flat, and thus there may be an illusion that the professors are actually doing research, but they seldom do have hands on the real work.

And how this links to the question is that the PhD is simply a normal underling for a professor. In this way, universities are similar to business. The professor can still get something done with the less talented person to him, that is worth value for the professor. The professors seldom do any research, so they delegate the work to their underlings. The less talented ones get tasks or research problems that they can cope with.

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Prestigious education, especially at the undergraduate level, is often an indication of a student who is (from a stable fortunate background and) hard-working and bright. But I never assume that a lack of Big Names on a CV means the opposite. At the graduate level in the United States in particular, name-brand prestige is mostly hype. The best choices of graduate programs usually aren't the ones with the most impressive names, but the ones with a given field's esteemed faculty members who are top-notch supervisors.

An example: my subfield (sociolinguistics) is quite sporadically represented among the Ivy League and other super-prestigious American colleges. As part of a search committee, if I were looking at an applicant with a Ph.D. from Ohio State or North Carolina State, that would likely be a solid attestation of potential because those schools have excellent sociolinguistics labs and some prominent people in the subfield would likely have written those recommendation letters. If I were looking at an applicant with a Ph.D. from Princeton and letters from people I've never heard of, I would frown and think, "Okay...if you wanted to do sociolinguistics, why the heck did you go there?!" I mean, you never know (maybe it could be that the student is firmly tied to the local area and had no choice), but these days that would be a pretty strange route to go.

I never think about relative 'smartness' of my students; and even if I wanted to, using the name of the institution that granted the Ph.D. would be a pretty poor proxy for a measure of that. What makes me feel happy are seeing my grad students excited about the concepts and coming up with their own research ideas and joining the scholarly community and completing big papers and learning to give presentations and getting jobs of their own (in academia or otherwise, depending on the circumstances). Couldn't care less about whether the place I work has prestige in general, especially as that's mainly about undergraduate education.

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So my question is, what motivates professors to train Ph.D students who aren't nearly as smart or excellent as them?

In most circumstances, research is not a competition. If I have a student who is working on a scientifically valuable project, then their work does not become less valuable or worthwhile if I have even greater accomplishments myself. Conversely, it doesn't become more valuable if I don't.

When I supervise grad students, I hope they will make the world a better place through research and teaching. How likely they are to do good work matters, as does how much time and effort I need to put into mentoring, but it doesn't matter whether they seem more or less promising than I did at the same age. I might make comparisons with other students to decide how to allocate my time (for example, we do this all the time in graduate admissions). However, if a reasonable investment in mentoring will enable someone to do good work, then I have no reason to care how they compare with me.

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