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Why would a professor doing pure math/ theoretical computer science want to take PhD students?

I understand that in computational and experimental sciences taking students is at least beneficial to them. However, in subjects like pure math, it is not obvious to me that there are any significant advantages to take some PhD students. It is probably expected from the department that you will take some students, and it is nice to have someone who can type things up for you. But other than that, I don't see any reasons.

So, why would they want to do that? What are they expecting from their (prospective) students?

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    Comments are not for extended discussion; this conversation has been moved to chat. – aeismail Feb 14 '18 at 13:51

11 Answers 11

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They get to train the next generation of mathematicians. They get to establish a relationship with young students, guide new researchers when they are up-and-coming and those people become their students. For the rest of time those researchers will be the person trained by Professor Stella and that is intrinsically valuable for several reasons:

  1. Some people like teaching. I love teaching, and if I leave industry and go into university research it will be largely because I want to teach. The ability to work with and mould young minds really appeals to me, and I get value out of knowing that I've helped people learn.
  2. You develop an attachment, and a sense of pride, in your students. Their accomplishments become, not quite yours, but close to feeling like yours. If you’ve ever served as a TA, you’ve probably worked for a long period of time with a struggling student. Maybe you felt a twinge of pride when they finally got it, or got a good grade on an exam. When you’re an advisor, you get a lot more than a twinge.
  3. They are part of your legacy. They will tell stories about being your student for another 20+ years after you pass away, they will toast you at banquets in their honor and thank you for making them the researcher that they are today.

What a professor expects of you is that you try your very best to be a good scientist, that you do interesting work, and that you continue to carry on the torch of mathematics for another generation after they are gone.

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    "Mounding young minds"? Makes me picture a pyramid of skulls in your office... :-) – Nate Eldredge Feb 14 '18 at 4:04
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    This is very good answer and should be rule, unfortunatelly it is not and most ppl are not like this. but good principle thou – SSimon Feb 14 '18 at 4:06
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    @SSimon Alas this is very true. I had the fortunate circumstance of being surrounded by professors who loved teaching and loved their graduate students. For other people they will take on students because they have to, but I think that wouldn’t be an answer to the question at hand. – Stella Biderman Feb 14 '18 at 4:08
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    @StellaBiderman very good answer and nice experience, but if we are honest, most professor are not like that, but the should be! you are right all the way – SSimon Feb 14 '18 at 10:46
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    Such a good answer all around that I hesitate to nitpick, but I'll just mention one reaction: I try to avoid labeling mathematics students as "geniuses", because (in my opinion) that reinforces harmful cultural tropes that math ability is a fixed trait (either you "have it" or you have no hope) rather than, as we know, malleable and trainable. Our culture's equation of math ability with innate "genius" not only drives students away from math but also reinforces harmful gender and ethnicity stereotypes within math. Anyway, again, great answer. – Greg Martin Feb 15 '18 at 17:47
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In addition to the points @StellaBiderman gives, oftentimes PhD students can actually be useful collaborators, not just students.

Yes, the professor will be far more advanced in terms of experience, but a good graduate student is more useful than for just the clerical work you seem to have in mind. Guided well, the student can help explore a research topic and may occasionally have insights or perspective that the professor might otherwise overlook.

This is particularly true for professors who tend to work on many projects at once. They take a managerial role and guide different research projects simultaneously while the students they mentoring focus solely on one of these projects and do much of the legwork.

In order for this sort of relationship to work well, the student will generally be beyond their first year and has a good degree of independence and self-motivation

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    Also, they save time. Time is valuable, and you can only work on so many things. Having "lower level" people lets you delegate tasks and not work on them. I know people who co-authored a paper with a professor they did research with as an undergraduate. Mostly they proved lemmatta that the professor didn't have time for. As an entry-level industry researcher, this is still true in my life. Yesterday I wrote presentation slides someone else is giving because my time is cheapest, per hour. They were full of involved graph theory, and certainly weren't clerical work, but saved my boss time. – Stella Biderman Feb 14 '18 at 6:36
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    @StellaBiderman At least in pure math, for any serious project training students to contribute generally takes much more time than working things out yourself, at least in the short term. So I think this is not a major reason for most mathematicians. Though possibly things are different at the top schools, where many students are relatively independent and stay in academia, and training an army can have long term benefits to your research program. – Kimball Feb 14 '18 at 15:51
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    @Stella: It does depend on the field...to a degree. It also depends on the student. For every student you know who proved things that a professor "didn't have time for," I am confident that I know several more students who needed months or years of the professor's time to be directed to do something that the professor could have done herself in hours, days or weeks. I think this probably true also in combinatorics and logic. – Pete L. Clark Feb 14 '18 at 17:08
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    I am talking about PhD students. "I would fully expect most graduate students to be apt enough to contribute to research. That's what they're supposed to be learning how to do, right?" Right, but at their own pace, not that of their thesis advisor. Just because a student meaningfully contributes to the research project does not mean that the advisor would not have contributed in their absence, because the goal is not efficiency but rather for them to learn to do research... – Pete L. Clark Feb 14 '18 at 17:28
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    In terms of your lots of students: well, I am a bit older than you and also the Graduate Coordinator of my ("above average") department. Is it perhaps possible that you attended an especially elite undergrad institution -- the same one as I did, even! -- and therefore are not actually picturing the generic PhD student in pure mathematics? – Pete L. Clark Feb 14 '18 at 17:30
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Because it's fun? Professors in pure mathematics love mathematics. Working with bright, enthusiastic students who are starting to engage in mathematical research is the sort of thing that someone who loves mathematics would naturally love to do. Your question assumes that this is something hard to explain. I don't see why. There may be exceptions of course, but most advisors enjoy such work.

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    Exactly! Why do math professors at 4-year colleges work with undergraduates on senior projects, reading/research courses, taking students to MAA and AMS conferences to present in the student section, etc.? Why do high school math teachers spend time after school or during their free period working with a strong math student (probably this happens more often in small rural schools), working with students in math competitions and science fairs, etc.? Why do so many people provide such detailed and lengthy answers in Mathematics Stack Exchange? (length limitations prevent giving more examples) – Dave L Renfro Feb 14 '18 at 13:57
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    +1 Too many of the answers here try to emphasize research benefits of having students, but I think the majority of pure mathematicians I know agree that having students slows down one's publications. It's not about making one's own research better. – Kimball Feb 14 '18 at 15:57
  • @DaveLRenfro why would rural teachers spend more time with good math students? – Azor Ahai Feb 14 '18 at 18:36
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    @Azor-Ahai Dave can answer for himself, but I suspect that the most important adjective in "small rural school" is "small", although rural schools also have a tendency to have less discipline problems, which can be demoralizing to a teacher. A teacher at a small, rural school is often less burned out than their urban colleagues and thus more likely to have the resources to help individual gifted students. – John Coleman Feb 14 '18 at 18:57
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    @Azor-Ahai: Sorry, I threw this in based on my own experience (attended a rural school and taught in both types), but mainly during the 1970s and 1980s before the internet. Perhaps the once in every 3 or 4 years strong math student would not be so isolated at a rural school in today's internet era. And I'm not talking about all teachers, but rather the one (typical number is zero or one) mathier-than-usual teacher who, every few years, has a student who could potentially obtain an undergraduate degree in math (if not more). In large urban and suburban schools, such unicorns are not so rare. – Dave L Renfro Feb 14 '18 at 21:35
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I am a professor with a background in both disciplines (Computer Science & Mathematics). I can direct a good candidate in a few sentences to try a new approach, or read somebody else's published work and try to apply it to a new problem.

This is especially true in Computer Science: I have given a student a basic idea that became a published paper; but it took them a year to code the idea, incorporate it into an existing library, find issues and fix them, run experiments to document its performance and capabilities for a paper, etc.

All of that is a learning tool for them to be able to do original research and justify it in papers (and understand the effort that it takes to fully realize an idea that can be described on a whiteboard in ten minutes). The same is true for mathematics, researching for prior methods, finding the exact citations to make are all tedious and painstaking efforts, few of us have photographic memories, we are lucky to remember the name of the person that invented some technique or first proved a theorem.

In Mathematics we also have the working out of a proof. We may be 95% certain a proof can be made, but figuring out its exact form can be a brain buster that takes a great deal of effort. Think of it as similar to an architect given a drawing of an unusual exterior of a skyscraper. Can this be built? I think so, if I can solve X, Y and Z, which I think are soluble, but I can't say I have a solution for sure until I have a solution for sure.

In the meantime, while my student works on his project and consults with me (at least once a week is my practice), I can use my hours working on something else, alone or in collaboration with colleagues, and hopefully both I and my student get a healthy publication from our collaboration.

The student is not just a slave, however, they become co-authors of the paper and learn how to do this work, and get their contributions to science out there and accepted by the world, to be applied by academics, governments and the world. This teaches them to make a living and have an impact: In both mathematics and computer science, some of these advances can plausibly save lives and raise the standard of living, either by direct use or as a foundation for new applied work in new disciplines.

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    And being a co-author on the student's paper, which is a good thing in the academic "publish or perish" world :-) – jamesqf Feb 14 '18 at 19:38
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The other answers are excellent. One point that I don't think has been made explicitly is that the question assumes a relatively short time frame. While the expected contribution of a student may not be helpful during their degree, I think over the course of a career, the average contribution of a student will be positive. Others have referred to matters of legacy, but this idea can be expressed more concretely:

1) The student, as they progress in their career, can become a useful collaborator. For example, when Andrew Wiles needed help finishing his proof of Fermat's last theorem, he turned to his former student Richard Taylor.

2) By continuing to work in the area of their advisor, the student might advance the field, introducing new results for the advisor to build on.

3) By succeeding in their field, students enhance the general prestige of their advisor's area of study. The emergence of enumerative combinatorics as a major field in mathematics is driven in large part by students of Giancarlo Rota and his student Richard Stanley. Their work, while profound, could have no where near the same impact without tens of descendants extending it. See also Thurston's essay On proof and progress in mathematics.

On an unrelated note, I once asked asked a senior figure in my field whether having students increased his research output (taking into account the above mentioned points). His answer: "No, but it increases my joy in life."

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Here's a list of reasons which are essentially independent of "artificial" motivations by universities or departments:

  1. Bolstering research efforts into questions/subjects the Professor cares about.
  2. Reciprocating/continuing the chain of ages: The Professor availed him/herself of an advisor to become an independent researcher, and so will want to do the same for another, younger generation.
  3. Fresh, alternative perspective on questions/subjects the Professor is interested in.
  4. Opportunity to literally work together on some problem with someone else
  5. Help in teaching relevant courses.
  6. Help doing programming work - which may occur occasionally even if you're just a mathematician, even a theoretical one.
  • This is the most straightforward and honest answer in my opinion. While the purely selfless motivations sound nice, my experience shows that the majority of successful professors who are capable of managing utilize promising PhD students as cheap labor, in exchange for an education. Regardless of the field, PhD students under the guidance of a good professor can produce valuable contributions and alleviate the burden of the more menial tasks. This arrangement should be mutually beneficial. – Underminer Feb 15 '18 at 17:34
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    @Underminer: Note my answer did not indicate whether the pay is low or not. Usually it is (if there's pay at all)... but you could have the same list for a Professor who pays out decent salaries from grant money. Also, teaching and programming are not menial tasks... they're just labor-intensive :-) – einpoklum Feb 15 '18 at 18:00
  • @einpoklum I'm surprised grant money hasn't been mentioned more. You can only do so much work yourself, so if you want to take on more grants, you need students to help with the work. A professor can add a considerable amount of income for themselves while only adding a relatively smaller amount of work managing the students who will work on the grant. My first hand experience with this is isolated to only a few grants, but in one case 3 professors did virtually nothing on a grant they had income from. Probably varies greatly with subjects that have more grant opportunities. – AaronLS Feb 18 '18 at 0:53
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In addition to the other reasons cited, it may well be that the professor is expected to supervise a number of research students. This might be stated in the professor's contract, and even paid for on a per-student basis.

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    Even if not paid on a per-student basis: Ph.D. students may be a big positive when promotions or raises in salary are determined in the department. – GEdgar Feb 15 '18 at 14:40
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    @GEdgar ...and when you want to move to a different, more interesting or prestigious institution. – henning Feb 15 '18 at 14:43
  • @henning ... Absolutely. Some job searches may even explicitly look for candidates with experience advising Ph.D. students. – GEdgar Feb 15 '18 at 14:53
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Turning a comment into an answer at the suggestion of others. Inspired by the good answer from @Stella Biderman.

There are some selfish reasons for taking students.

Reputation by association

If any of your students gains substantial reputation, you may gain in reputation by association. If this is your intent, then you will likely be very picky about who you allow to be your student.

Merely being smart or being good at what you do is not enough; the student needs to exhibit some kind of quality that is likely to attract reputation. Maybe the student is socially talented and likely to become publically well known, or maybe they are very aggressive and arrogant but good at arguments. The part about aggression and arrogance may sound odd, but I once had a professor tell me that there were a large proportion of people in his field who were smart but pushed their way to the top with their inflated ego just as much, if not more, as their talent.

Reputation by celebrity-like treatment

Alternatively: "Feeling of importance by feeling needed"

If you already have reputation, students competing for the opportunity to work with you increases your reputation further and boosts ego.

Popularity and celebrity status is one of those things that snowballs, boot-strapping itself once you have it. In this case, people desiring you for your reputation increases the very reputation that causes them to desire you.

I had included an example of a specific and well known researcher that you have all heard of, but I deleted it before submitting to avoid the appearance of being negative. People who fit into this category definitely exist though.

It is not necessarily just an increase in reputation that may be desired though, but rather could just be the celebrity feeling or even just a feeling of importance at all. If this is the case, you might not even be reputable, but you might still feel more important if people are coming to you as a superior. In this case, it is possible that the relationship is good for the teacher if they need a boost in self esteem due to a lack of feeling self-worth elsewhere.

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There are many good answers here. I would like to offer a rather different perspective: Teaching can make you a better researcher. I would include in this not only supervising PhD students, but supervising undergraduates (e.g., senior theses) as well as teaching lecture courses, including introductory undergraduate courses.

There are exceptions, of course -- for some teaching is truly a counter-productive distraction. But I think that a significant majority of researchers are helped (or would be helped) by teaching -- even those who dislike teaching, and claim that it interferes with their research.

There's the old saying that you don't really understand something unless you can explain it to someone else. Teaching gives you the opportunity to review what you thought you knew and fill in the gaps you didn't know you had (or pretended not to have) -- it's like an ongoing oral exam.

Fields tend to develop certain blind spots -- things that are taken for granted and everybody knows are "true." It's bracing to have a student come along and say "I still don't follow what you're saying -- could you go over that again? How do we know it's true?" (Out of the mouth of babes ...) There's just a vibrancy that comes with teaching and having students around, even if it seems like a pain in the neck in the present.

Richard Feynman (theoretical physicist) addresses this, at least indirectly, in his "Surely You're Joking, Mr. Feynman!" book when he talks about struggling with his research as a young professor at Cornell and the advice he got from (from recollection) Hans Bethe.

A couple of comments: Here I define "research" as pushing the boundaries of knowledge forward with insight and creativity. If you define "doing research" as, say, getting grants, then time spent teaching instead of writing proposals probably does lower your productivity. (Thanks for listening. K. Frank) Also a heavy teaching load, say four-plus courses per term or supervising eight or ten students, will likely be a research negative. But a light load, say one or two courses per term and two PhD students plus an undergraduate senior thesis, leads to better research, and I think many professors recognize this (even as they grumble ...).

  • I strongly agree. And we can note that there is one long-standing tradition of complaining about teaching, in certain mathematics milieus, and even peer_pressure in those milieus to complain. Hence, many of these complaints are nearly purely rhetorical. – paul garrett Feb 20 '18 at 19:29
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Well, from my experience as a student, and from what the current dean of my alma mater and a top researcher said (not a literal quotation): You need students to do the work while you are busy not working. And he did not mean that the profs are not working. He just meant that they are busy doing stuff like sitting in PhD/master juries, writing grant proposals, teaching, giving talks here and there, going to necessary social events here and there, sitting in scientific councils, academic senates, preparing whatever they need to prepare etc.

This is, IMHO, the most pragmatic reason behind it. Of course, the reasons the other answers give are also important, and I wouldn't judge what is the primary drive for having PhD students as it's a rich mixture of many reasons.

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Other answers dance around this, but do not state it explicitly. Professors are the managers of academia. Their own individual scientific / mathematical contribution is expected to be limited. Instead, they are expected to lead group of researchers in the form of Master's students, PhD students, post-docs, and, if they are more senior, assistant / associate professors.

The only professors who really get the freedom to think deep thoughts all day are professors emeriti.

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    This seems highly field specific, and not applicable to the fields the OP mentions. Maybe in a field where professors run laboratories, but not in mathematics. – Stella Biderman Feb 16 '18 at 4:16
  • @StellaBiderman Having a laboratory or not is not a factor. In several European countries, for example, the hierarchical structure is very rigid regardless of field. A research group may have only one full professor, under which there are associate / assistant professors and under them post-docs and PhD students and under them masters students. – Eric Feb 20 '18 at 20:37

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