What do mathematics researchers do if they aren't good?

Question

I am a high school student so I don't know much about academia, but I would like to clear up some confusion I have over research in mathematics. Please excuse my naivite/ignorance on this topic.

I get the impression that mathematics research at the graduate and post-doc level is hard. It could take months of getting nowhere before you make some progress on a problem, and depending on how good you are and the difficulty level of the problem, you could go an entire year without publishing any papers. I get the impression that proving interesting or important results is even harder and is really only for the best - the real mathematicians. By important I mean results that will be noticeably useful to other researchers in the field.

I am guessing that not every person who gets a PhD and goes into research is good enough to prove interesting or important results, and I'm guessing that the percentage of PhD's who go into research and who will become successful mathematicians is less than 50%. I'm wondering what do these people do? If they can't publish enough papers and they aren't successful in solving any problems, they can't continue like this forever right? I mean at some point the university they are employed by will reject them? Do these people leave academia entirely and go into industry, or switch fields into physics or something like that?

Please let me know if my understanding is correct.

Answer 1

Many people who complete PhD's in mathematics end up leaving academia within a few years after completing the degree. Many others settle into teaching oriented positions at community colleges, four year colleges and regional comprehensive universities where they typically end up publishing little or no research. A small percentage of all PhD's in mathematics end up as tenured faculty in research universities (much less than 20%) and even among these mathematicians at research universities there is huge variability in research productivity (e.g. as measured by papers published per year) and impact (e.g. as measured by citations of these papers.)

Answer 2

Terry Tao (a famous mathematician) has a good answer to this question. The following excerpt goes to the heart of the matter, and you should read the whole post:

even if one dismisses the notion of genius, it is still the case that at any given point in time, some mathematicians are faster, more experienced, more knowledgeable, more efficient, more careful, or more creative than others. This does not imply, though, that only the “best” mathematicians should do mathematics; this is the common error of mistaking absolute advantage for comparative advantage. The number of interesting mathematical research areas and problems to work on is vast – far more than can be covered in detail just by the “best” mathematicians, and sometimes the set of tools or ideas that you have will find something that other good mathematicians have overlooked, especially given that even the greatest mathematicians still have weaknesses in some aspects of mathematical research. As long as you have education, interest, and a reasonable amount of talent, there will be some part of mathematics where you can make a solid and useful contribution.

Answer 3

It's true that not every person who gets a PhD has a successful academic research career, but I want to emphasize that that is different from being a successful mathematician. Many people who get PhDs want to go to work in industry/govt or some more applied field, and many people who get PhDs want to focus primarily on teaching. This does not mean they are automatically not successful, or couldn't be successful academic research mathematicians if they wanted to. (I know many very talented researchers who have gone to industry, or into teaching--not because they couldn't do research, but because they preferred something else--and ended up quite happy. Occasionally people will come back to academic research also.)

Incidentally, there is some survey data on jobs PhD's get, e.g., The Annual Survey of the Mathematical Sciences. For instance, Table E.6 says in 2012 848 new PhDs took academic positions and 456 took govt/business/industry positions. This is out of 1843 PhDs awarded with about 9% unknown employment status and 4-5% unemployed at the time. (Edited: According to Table E.7, 600 of those academic positions are postdocs, not tenure-track, but those on the research track will almost certainly do a postdoc first.) So it may be that a majority of PhDs are successful in a broader sense (I don't know about long-term data or job fulfillment).

PS I know this isn't the kind of answer you were looking for, but you can see Brian's answer for that. I just wanted to clear up a possible misconception.

Added: I just saw this data in the most recent Notices issue, which says that recently a little recently there have been about 850/year tenure-track positions filled in the US in math or stat/biostat. This suggests most people who stay in academia right after their PhD have a good chance a getting a permanent position.

Answer 4

A lot depends on what you consider to be interesting or important results. A lot of research gets used by others and could be reasonably called noticeably useful to other researchers. Often there will be several people working in a subfield of math and they will use each others' work in various respects. Other times there will be a hot field with a lot of things to discover, and many mathematicians will be picking up the "low-hanging fruit" and publishing results that will be used. It is still true that key developments that become major tools for other mathematicians are usually done by leading mathematicians, sometimes in collaboration with students or non-leading mathematicians, but this is fully consistent with the above.

As for the fate of mathematicians who don't become research mathematicians.. there are thousands of colleges in the US that need professors and most of them don't really emphasize research. Many also go into industry, such as the NSA or government labs. Some become actuaries and others go into finance. Others become computer programmers and can end out being quite good at that. And there are various perhaps unexpected directions some choose to go in. For example, I know of not one but two who went to top-notch law schools and became lawyers.

Answer 5

Having completed PhD, with or without a post doc, you are a trained mathematician.

Disregarding having found any new results, one should hopefully have the ability to understand existing results.

This means one can apply mathematics (as opposed to research applied mathematics) in a number of fields, be it banking, IT, defense or many other fields.

There is an additional key skill required: an ability to translate a real world problem into a mathematical format. This is in itself usually the most challenging part of being a working applied mathematician.

Answer 6

It should perhaps be noted that in mathematics there are a lot of problems to consider. If you're a high school student it perhaps does not feel this way to you, but it's really so (I know I didn't feel that way until about the time I was finishing my undergrad). As mathematics develops, new objects get defined - and new questions become possible.

In addition to interesting problems, there is also an endless supply of... other problems. Of course, there is no good definition of interesting, and it varies a lot depending on who you talk to.

In any case, there are much more problems than the really good mathematicians can hope to solve, so there is enough work for the others as well. There are even a lot of problems that the "experts" basically know how to solve, but they never really bothered (it would seem that those whose position is fairly secure care more about quality than quantity). There are many universities where people can get a fairly permanent (research or teaching) position without publishing ground-breaking research.

Answer 7

I think this boils down to the question of how to become/be/stay successful in science.

First of all success can be defined in many ways. If we just disregard the field of mathematics here for a second and look at the whole natural science branch, being successful is always a difficult trade off between good science and quick and dirty publications.

Without a good portion of time investment, there will never be any good science down the line (You may get lucky and profit from the time your PI has spend on her/his field, formulating a genius thesis though, but that's just underlining the importance of time investment).

As a matter of fact, a good portion of basic ground breaking discoveries have already been made. As time/research goes by things get more and more complicated and intertwined. That is true for Math, Physics, Biology and Chemistry, as for any other field.

That does not mean that there are no new things that can be discovered, but the pool of new insights gets deeper and deeper as research goes on (solving one problem just opens a new space of many more new problems, harder to solve than the initial). If you want to publish new amazing research you have to stand on a lot of giant's shoulders, and being that far away from the ground makes the air dangerously thin.

New ideas are needed for new success stories. Nowadays, these ideas are coming from inter-field communications (cross branch collaborations), where e.g. math talks to physics, taking insights from biology, which has borrowed from chemistry and so on. That again needs time.

IMHO, bottom line, it's very naive to assume a publishing rate stays the same, with the same astonishing impact, over time. There has to be a slow down. So taking that as a measurement of success, as appealing as it may be, is flawed.

Luckily, everyone has to deal with this, and as a mathematician, assuming you love what you do, you have a very analytical brain, which is, to say the least, a good starting point for being "successful" in anything you do.

Answer 8

Very astute question from someone who at the time of asking it was only a high school student!

Mathematical research is indeed difficult, and even overcoming this rigor, there is a good percentage of chance that your work depends on. Being at the right place at the right time, making the right mental connections (possibly even before someone else has beaten you to the theorem), etc.

I have the perspective of someone who started out as an engineer in training, fell in love with pure mathematics, completed a Ph.D. at a very respectable public school, did not get a postdoctoral position, and ended up in a rather unenviable adjunct instructor position at a community college in Illinois. Six years and many minimum wage jobs later, I decided to go back to engineering, as it is a much more rewarding career, and to undergo post-math Ph.D. graduate-level training. In these six years, I worked on mathematics in places unusual under extreme hardship, even at times living in a vehicle, on (a) theorem(s). Ten years after I was introduced to the problem, I found a partial solution and feel very proud!.. albeit with less than $2000 in my bank account.

This is what happens if your research isn't there in time for an academic job and when you have a pathological obsession with proving theorems. Whether you would like something like that, I don't know... but there is that "career path".

Answer 9

I get the impression that mathematics research at the graduate and post-doc level is hard. It could take months of getting nowhere before you make some progress on a problem, and depending on how good you are and the difficulty level of the problem, you could go an entire year without publishing any papers. I get the impression that proving interesting or important results is even harder and is really only for the best - the real mathematicians. By important I mean results that will be noticeably useful to other researchers in the field.

To a great extent, one could say that this is true in any of the STEM fields.

It is not true of some of the social sciences. I have read some truly awful theses in the field of education. And those people got doctorates based on that drivel!

By the way, your description seems to me to fit pure math more precisely than applied math.

It is natural to feel some uncertainty -- will I be good enough? Will I cut the mustard?

Fortunately, the path from high school towards the PhD is one that can be adjusted each semester. It's not necessary to pick the exact path and then stick to it no matter what!

Answer 10

in the past, a lot of maths phds who found that academia wasn't for them went into banking and became quantitative analysts or "quants". They then got paid several multiples of an academic salary.