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In pure mathematics (please only answer for this field, as other areas are different), what is the ratio of PhD students finding a project on their own, versus working closely with their supervisor to find a project versus the supervisor giving their student a project?

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    I think this is unanswerable without carrying out a research project to get some evidence. The ratio is greater than 0 and less than 1. I'll guess that it is less than 1/2. There is a third option, which is a jointly developed idea, and a fourth, one proposed by a third party. And, who cares??? – Buffy Apr 30 at 12:27
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    I care. That's why I asked. – Peter Apr 30 at 13:15
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    Why would knowing the ratio matter to you? Are you in an uncomfortable place with your own advisor? That is a different issue. Every interaction is individual. – Buffy Apr 30 at 13:18
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    I want to understand what a PhD student in pure mathematics should expect from their supervisor with regards to helping find a research project. Is it mostly the case that supervisors give their students a project? Is it mostly the case that the student and supervisor work together to find a project? Is it mostly the case that the student finds the project entirely on their own? I don't care about a precise number, I just want to know what is expected and happens most of the time. – Peter Apr 30 at 13:59
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    @Peter If you care, then please ask the question you actually want answered instead of some faux-generalization that is both unanswerable and useless. – Elizabeth Henning Apr 30 at 16:37
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As mentioned by Buffy in the comments, you're unlikely to find an actual ratio to answer your question without gathering some data. However, it seems like your question is actually where your research project would come from as a PhD student in pure math.

The answer is quite simple: it depends on you and your advisor.

  • Have an idea for your project? Tell your advisor about it and get their feedback! A thesis project (or even a side project) needs to be both interesting and manageable within a restricted time frame. One of the main reasons you have an advisor is to learn how to ask interesting but manageable research questions. Your advisor's experience is a valuable resource for learning this skill.

  • Want your advisor to give you a project? Ask them about it! Some advisors have lots of project ideas and are happy to share them with their students. Other advisors will be more hesitant to give you a project -- I know some who think it is important for students to come up with their own questions.

  • Want to come up with your own project but don't know how? Tell your advisor and ask for advice! They will probably tell you to read lots of papers, listen to lots of talks, and ask lots of questions. You can tell them the ideas and questions you have, and then you can ask if any of the questions are interesting. Again, asking research questions is a skill that you need to develop.

A corollary of this discussion: don't just pick an advisor based on their research area or whether or not they're "famous." Every advisor has a different approach to research and mentoring, and you want to make sure that your advisor's style is compatible with your personal working style. If you don't like the idea of your advisor making you find your own question, or if you have your own idea for a research project that you want to pursue, then you should be sure to ask potential advisors about this before deciding to work with them.

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Adding to @Stephen_McKean's answer, and also considering @Elizabeth_Henning's comment that the question is very likely not the real question in the questioner's mind...

My first point would be that it is entirely reasonable for a math grad student to not be able to come up with a research project that is do-able, interesting (to experts!?!), and hasn't already been done (or is viewable as a routine application of well-known methods). Or won't be scooped by more experienced people during the time it takes to get done by a novice. So if this situation seems daunting... well, you're being completely reasonable. Note: very talented people without experience cannot reasonably "compete" with equally talented people with decades of experience. Or at least that surely cannot be what "getting a PhD" is about.

On another hand, sure, if by good luck one has by chance stumbled upon a do-able, interesting, not-easily-scooped project... GREAT! Hopefully one can find an advisor/mentor to give technical advice and so on. And/or, more subtly, to steer from the idea for which you had great enthusiasm to something more professionally interesting and more viable. The latter is a serious trick, and I myself am not able to routinely execute it.

I am skeptical of the (math...) advisors who declare that their students should come up with their own projects. My reaction would be "wow, if you can do that, without advice from experienced people, you are in a different league from me... and you should probably get your advisor's job!"

Yes, now and then, but quite rarely, this can happen. But it cannot be the required or routine part of the business.

EDIT: to clarify, based on comments, I certainly would not at all discourage anyone from coming up with their own projects... But/and professional viability is a different question. In my own experience, I'd come up with some projects that seemed "original" and at least a little bit interesting... but when I got to grad school at Princeton, I discovered that my little projects were literally super-easy exercises for serious people, and that the genuine contemporary progress-in-math was barely intelligible to me.

Yes, one can choose which culture one wants. The high-end biz is rough, not reliably gratifying, and many of the people in it are not really sensible as people. And, all the worse, apparently one has to make decisions about "paths" at such an early point in life that one has insufficient information of and/or appreciation of what the ambient culture will do to oneself...

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  • I know of at least one highly respectable advisor who requires his students to come up with their own thesis problem (virtualmath1.stanford.edu/~vakil/potentialstudents.html). Of course, you are right that coming up with good research questions is especially hard for a new researcher with little experience (speaking from personal experience). But I think learning how to come up with new projects as a grad student enables one to have a more productive postdoc, which hopefully leads to a more successful tenure-track job search, etc. – Stephen McKean Apr 30 at 19:01
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    @StephenMcKean This is neither what Ravi actually does nor quite what he says he does in the page you linked. – Elizabeth Henning Apr 30 at 19:07
  • @ElizabethHenning See the second sentence after "My personal style as an advisor." I'm not saying that Ravi leaves his students to fend for themselves. I'm sure he gives lots of feedback and guidance as his students figure out their thesis problem (for example, by suggesting toy problems earlier in their studies). – Stephen McKean Apr 30 at 19:14
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    As a variant on my remarks, certainly trying to come up with one's own project is a worthwhile project-in-itself. – paul garrett Apr 30 at 19:37

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