Pure Mathematics PhD without any publication

Question

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?

Answer 1

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.

Answer 2

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”.

Answer 3

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.