This question is hypothetical. I have completed my MS in mathematics and started thinking about applying to a PhD in Mathematics in the USA as an international student in pure mathematics. I'm wondering: what if I can not produce original research within 5/6 years? What should a student do if they can't discover anything new or interesting at the end of their PhD?

In pure mathematics, discovering a new thing is extremely difficult and nothing is ever for sure. I personally know a few people who have not published anything during their PhD. Will someone be kicked out if they cannot create anything new?

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    May I ask what gives you the impression that "[i]n pure mathematics, discovering a new thing is extremely difficult"? Regarding "nothing is ever for sure": That's true, but so is it for any other field. If things were already known, it wouldn't be called 'research'. ;-) Sep 18, 2022 at 9:14
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    I know many people who have gone to some top schools for phd and connot publish anything and they work in industry. But all of them wanted to become mathematician and some of them are extremely brilliant ( imo silver medalist). These examples make me nervous.
    – G math
    Sep 18, 2022 at 9:24
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    Hmm, I cannot comment on "top schools", since where I live and work there is no such distinction between "top" and other institutions. Anyway, please note that publishing some papers during a PhD does by no means imply that one will get a position at a university or college later on (simply because there are far more PhD students than faculty positions). From my point of view it doesn't make much sense to start a PhD with the plan to become an academic - you can't know whether you really like mathematical research and are really good at it until you have tried it for a while. [...] Sep 18, 2022 at 10:22
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    FWIW, after controlling for other factors, doing well in math contests seems to have zero or even a weakly negative correlation with going on to have a successful research career. They're not the same talent. Sep 18, 2022 at 16:56
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    @ElizabethHenning - to be sure, maybe there is zero or negative correlation among people who go to grad school, but amongst the population as a whole there is a huge positive correlation. The people who scrape by in a math major without ever really learning to write a proof and still has trouble distinguishing "Not all swans are white" and "All swans are not white" after 2 years of working on the difference - which is the aptitude level of most people - are much less likely to be successful at math research than someone who does moderately well in math contests. Sep 18, 2022 at 20:51

3 Answers 3


In the US, in most doctoral programs, the goal is to produce a dissertation, not publications. This was, I'd guess, more true in the past, but remains largely true. A dissertation is usually a relatively long work on a small, tightly focused, area of math. The dissertation might result in one or more publications after finishing the degree but getting involved in the publication process, along with the waits and revisions, isn't usually part of the process prior to graduation.

If you don't produce a dissertation, you don't finish. But most people finish successfully, though some take longer than others.

But, if you don't produce a dissertation it is likely because you didn't listen to anyone while you were a student. Students have advisors and advisors have a responsibility to get their students through the program. They suggest problems or, at least, small areas of math where exploration is likely to prove fruitful. They also give you feedback on what you do and suggest modifications if needed.

If you ignore your advisor's advice it is harder to advance. You aren't locked in a box, all alone, for a few years expected to emerge with some new mathematical field. Results in dissertations can be quite small or more broad. But they are very focused. Advanced graduate study pushes and guides you to that focus.

However, some problems that students attack prove too hard to find a solution in the time available even with the advisor's help. Some students have to "salvage" a dissertation out of failed attempts. But more students, I suspect, are advised to drop a problem if it seems, early on, to not be fruitful. And note that in math, proving negatives can be as valuable as proving positives.

There is a Three Bears issue in math (too hard, too easy, just right). In my own studies I experienced this directly. The first problem I worked on proved too easy and I could produce theorem after theorem daily. There was no substance. The second problem was too hard. After a few weeks it still seemed like a perfect titanium sphere with no possible cracks for entry. Nada. The final problem was "just right" and with work and developing insight a nice dissertation (later published) emerged.

With your advisor's help, choose a tractable problem. Work hard. Succeed. Or, at least, learn that this isn't what you really want.

Note that working mathematicians seldom work in isolation and collaborate widely, sharing ideas. Colleagues substitute for advisors in a sense. Ideas fly around. Some of them result in nice papers. Some are discarded.

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    +1 Great answer (as usual). It might be worthwhile to add the following corollary of your third paragraph: It's very important to find a good advisor (who, for instance, doesn't let you waste years on an intractable problem without intervening). Sep 18, 2022 at 10:38
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    my biggest lesson during PhD was: Ignore most of the stuff others tell you and separate the useful input/people from the others. I had two post-docs mentoring me on a subject they used for their PhD and it was such a waste of time. Another post-doc, who took me under his hood then, was a major help and he hasn't done anything on this topic. Without him everything would have gone south..
    – Ben
    Sep 19, 2022 at 7:21
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    @ben "separate the useful input/people from the others", this is a mayor skill many PhD students learn in grad school.
    – usr1234567
    Sep 19, 2022 at 11:46
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    I didn't. I was never supervised by multiple post-docs and also the topics within the study were not that specific only a few people know. However, during the PhD I had a working contract so why on earth would you think your supervisors, who got their PhDs on the topic, have no clue? Not obvious at all.
    – Ben
    Sep 19, 2022 at 11:52

There is no reason to sugarcoat the answer: you are correct, if you do not produce original research (in the form of a written dissertation, not necessarily a publication) then you will not graduate. This happens to a fair number of people.

HOWEVER: the way in which it happens, and the reasons why it happens, are typically not the ones you’re imagining. What usually happens is that people discover that they don’t have the passion for research that they thought they did before they really understood what research is. They come to realize that research is not a good match for their interests and abilities. Then they leave the program, typically after around 2-3 years and with a Master’s degree instead of a PhD, having learned a lot of interesting mathematics, and having learned useful things about themselves.

I would say that it is pretty rare that someone stays for 5-6 years and then fails to graduate. If they lasted that long, they must have advanced to candidacy, found an advisor, and must already have been making decent progress with a research project. The system is set up to monitor people’s progress and not allow them to reach a point where they fail catastrophically right at the end after investing a large sunk cost. Which is not to say this never happens — all sorts of things happen, including various life events, physical and mental health issues etc — but rather that this is a worst case scenario that almost always has much more complex causes than “I didn’t discover anything new”.

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    The situation can be different for international students who need to stay in the US for one reason or another - despite it being clear to them that they won't finish, they stay in order to maintain their immigration status until they run out of funding, making enough progress to not be kicked out but not enough to finish without a miracle (which does not occur) at the end. Sep 18, 2022 at 16:00
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    Discovering you don't have the passion for research is usually correlated with not getting results. These are not independent and have some complex causal relationship with each other. Sep 18, 2022 at 16:02
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    @AlexanderWoo yes, good points, everything you say is true (and consistent with what I wrote). It would be hard to enumerate all the complicated issues that cause people to leave a grad program, either early or at a later stage. But the main point is that the simplistic scenario OP is describing is not a typical one.
    – Dan Romik
    Sep 18, 2022 at 16:18
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    Not in maths, but I know at least of one case where someone was given a problem with an incorrect assumption by an advisor, then spend two years (phd is 3 years here) researching it before getting to the level where they could realize the problem with the assumption, and basically (from a publication pov) wasted 2 years. He did end up publishing some 'salvaged' article out of this, but given the experience he quit soon after, as he felt he would be working 80 hours per week for the last year to finish his PhD. Compared to literally his brother who (cont.) Sep 19, 2022 at 6:25
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    'lucked into' a great publication after less than a year and little effort (half that year he wasn't working on his PhD at all, but just enjoying life)... it doesn't feel unfair to say there is a lottery aspect to PhDs. Point is: No doubt there are people who would say he just 'didn't have the passion', but I don't think that's fair... Now I don't know how common these things are, but I have seen enough PhD 'students' who worked <10 hours per week and succeeded whilst others worked more than full time and quit. Sep 19, 2022 at 6:27

It depends on what your goal is.

Yes, a PhD requires writing a dissertation that contains original research, but the bar for this is actually very low. I have seen many dissertations which are less substantial than my undergraduate honors thesis. (Admittedly, my undergraduate thesis was substantial enough to be publishable in a decent journal.) I have seen other situations where all of the non-trivial ideas in the dissertation came from the advisor. You should always be able to find an advisor who will let you graduate with very little (but not zero) in the way of original work, and the original work could be of a very routine nature. The resulting dissertation might not be publishable, or might be publishable only in a very low ranked write-only journal, or it might be obvious in some way that it is really the advisor's work (because the advisor doesn't mention your contribution in their letters and you can't talk about your work intelligently in interviews).

If your home country has a shortage of PhDs and any PhD is sufficient to get you hired, then this is all okay (and you have genuinely participated in research, which is helpful to your future students). If you're looking to continue a research career in the US or Europe, such a dissertation will leave you with almost no opportunities. No one will hire you for a postdoc. (Well less than half of pure mathematics PhDs in the US get an offer of a postdoc.) Increasingly, you also need some research success to be hired into a permanent position at a teaching-oriented university.

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    but the bar for this is actually very low: that probably depends on the university. For example, I have never seen a dissertation of the “it might be obvious in some way that it is really the advisor’s work” category, and I wouldn’t advise OP to factor something like that as an option, even if it’s an option of last resort, when making a decision to start a PhD.
    – Dan Romik
    Sep 18, 2022 at 23:11

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