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How are (pure) mathematicians paid to do research? Let's say my interests lie in the foundations of mathematics and I want to do research in that area. How am I going to get paid for that in the traditional context of academia?

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    Become a professor and perhaps find some grant money if needed. – Jon Custer Jul 8 at 22:03
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    Not exactly "research on the side". At some universities, research is the main job and teaching is a necessary byproduct in some sense. – Buffy Jul 9 at 0:00
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    Actually it's even worse than that. Your "research on the side" may consist mostly of writing grant proposals to fund others in your lab who do the actual research. In fairness though, this fits under both research and teaching categories as you are supposed to be training those lab members via this process. – A Simple Algorithm Jul 9 at 8:00
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    @ASimpleAlgorithm "fund others in your lab who do the actual research" does not apply to mathematics. – Andrés E. Caicedo Jul 9 at 15:51
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    @ASimpleAlgorithm While that may be applicable to some STEM fields, that's not how math research works - there aren't labs at all in the sense that you're imagining them, and solo projects are extremely common if not the norm. So that's just not accurate. – Noah Schweber Jul 9 at 17:39
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In the US, 99% of long-term positions that involve being paid to do research in pure math are tenure track faculty positions, colloquially known as professorships (in the US, they follow the progression Assistant Professor -> Associate Professor -> Professor). A professor is paid to teach, do research, and to a lesser extent, to do a variety of other vaguely related things that are discussed in many places on this website and elsewhere.

Professors teach and do research. It is not true that (quoting from one of the comments) “mathematicians are not paid to do research, but instead do research on the side and teach to make a living”. It is also not true that (quoting from another comment) “at some universities, research is the main job and teaching is a necessary byproduct”. Perhaps some professors have the mindset that their job is “mainly” about one or the other thing, but that’s simply a matter of personal perception rather than an objective truth. The objective truth is that professors teach and do research, and are paid to do those two things. Nothing is “on the side” or is “the main job”.

It is also not the case that (as seems to be implied by another answer) all math professors have, or need, grant funding to do their work. Grant funding is good to have, and getting it is both a catalyst for and a side-effect of career success. It can also give your salary a modest boost. But most of the funding doesn’t go directly into your pocket, and there are plenty of math professors who have steady employment and do quite well in their research without having it.

Finally, there is a very small number of mathematicians who have permanent, full time positions doing only research. Examples of places where such positions exist are the Institute of Advanced Study, and Microsoft Research. Those positions are very prestigious and rare, so hoping to land one of them is not a viable career plan.

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    This is a US-centric answer. In the UK and elsewhere "professorships" are something different. Most academics are paid separately for research and teaching, and don't necessarily have to do much (if any) of the latter. – Stop Harming Monica Jul 9 at 13:54
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    @user3209815 this is discussed in various other places on academia.se. I answered the question at the level of generality that OP seemed interested in, but if you or others want to chime in with more specific details, you/they are welcome to do so. – Dan Romik Jul 9 at 16:55
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    @OrangeDog good point. In the same vein, it also occurred to me that in some countries (France being the main one I know about) there are quite a lot more full time research positions for mathematicians. – Dan Romik Jul 9 at 16:57
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    As you said, pure mathematicians (like me) are paid for research, teaching, and service. But how that works is more complicated. If I unilaterally and arbitrarily decided to do no research in the coming academic year, I could expect a lousy (perhaps negative) raise a year from now and perhaps a lecture from the department chair. If I unilaterally and arbitrarily decided to do no teaching in the coming academic year, I could expect to be put on leave without salary and/or to (have my tenure revoked and) be fired for not doing my job. (see next comment) – Andreas Blass Jul 9 at 17:12
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    I emphasize that the preceding comment is about "unilaterally and arbitrarily". If I get sick and can't teach, or if I arrange with the administration to have no teaching duties for a semester, that would be entirely different. (Both of those things have happened to/for me and my department and college administrations were very good about it.) – Andreas Blass Jul 9 at 17:14
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The current top answer is simplistic and US-centric.

First, it glosses over the obvious. Tons of PhD students and postdocs are paid to do research and nothing else. It is difficult (and not really desirable, or usually possible) to be a postdoc forever, but this is certainly doable for many years, even more than a decade if you count PhD+postdoc.

Second, and this is the part which is US-centric in the current top answer, many researchers are employed as full-time tenured researchers. In France for example, more than 10% of all tenured mathematicians affiliated to the CNRS – i.e. almost all French mathematicians in public institutions – are full-time researchers (see this document by the Insmi, 400 researchers out of 3600 full-time researchers). Similar positions exist in many European countries (I know it's the case at least in Belgium, Switzerland, Spain... with various titles). Yes, these positions are enviable and difficult to get, but they are not some unattainable holy grails that only exist at IAS, IHES or the likes.

Third, many private companies employ math researchers. Not many employ researchers on the foundations of mathematics of course, and you would need to do research on what the company wants, but you'd still be a full-time math researchers. These positions exist anywhere, in small startups, global corporations, and anything in between. These aren't positions that exist only in "elite" institutions. Things like data analysis, machine learning, etc are hot right now, and you need serious math for them.


Finally, let me address a misconception in your comment.

So, usually mathematicians are not paid to do research, but instead do research on the side and teach to make a living?

I believe you are confused by the word "professor". While etymologically it means "teacher", nowadays, a professor at a university has a double job: teacher and researcher (and a third job called "administration"). The same goes for a lecturer, a reader, a maître de conférences, a førsteamanuensis, or whatever your job title is. Many academics are relatively blasé and consider that teaching is actually a chore that is imposed on top of the main job, research – a discussion for another time.

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    I doubt many companies employ pure math folk as asked by the OP, the circumstances you are describing all seem to be applied. – Bryan Krause Jul 9 at 17:19
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    Tons of PhD students and postdocs are paid to do research and nothing else. — That may be true in other fields and in other countries, but not in math in the US. An overwhelming majority of American mathematics PhD students and postdocs teach. – JeffE Jul 9 at 18:10
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    @JeffE But the question nowhere specifies that it’s about the practice in the US specifically, and so information about the situation in other countries is equally relevant. – Mike Scott Jul 10 at 6:56
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    "Tons of PhD students and postdocs are paid to do research and nothing else" this is clearly incorrect: the majority of PhD and postDocs programmes all around the world contain teaching, tutoring, reviewing papers, doing some other useless administrative task and so on. – gented Jul 10 at 16:57
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This funding comes from two main sources:

  1. Employment, e.g. by getting a professorship somewhere. This means a steady salary. Note professorships aren't just teaching duties - professors are also expected to output research, mentor PhD students who output research, and so on (see this recent question for what happens to "unproductive" professors).
  2. Grant funding. You write proposals to whoever is funding mathematics research (e.g. the NSA if you're in the US). You tell them what you intend to do, how you intend to do it, how much money you'll need, etc. If they approve of your proposal then they send you money to do the research.
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    With #1 by far as the more common option. And if you have such a position you can also (should also) seek grant funding, maybe to support students. – Buffy Jul 8 at 23:58
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    Probably worth noting that #2 often requires #1 in the US, since most of the grant agencies must abide by certain federal laws which require you to abide by certain federal laws, and demonstrate compliance with in some fashion. As such most of them won't just hand over funds to an independent researcher, instead requiring there be some sort of office that insures your compliance with the laws and terms who will then handle disbursement of funds. Indeed, I'm not aware of any source of grant funding in the US that doesn't require this. MacArthur Grant maybe? But you don't apply for that... – zibadawa timmy Jul 9 at 0:33
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    @Allure Well there's the "General Restrictions" immediately above that: "Awards will be made to non-profit institutions located in the United States only and will be based on a formal proposal submitted by an organization on behalf of the principal investigator(s)." – zibadawa timmy Jul 9 at 9:51
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    @ASimpleAlgorithm Have you done this? I think it is really quite more complicated than you are making it sound to start a corporation competent to receive federal research grant funding, excusing cases of corruption. – Bryan Krause Jul 9 at 16:04
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    @BryanKrause Yes I have. And others I know have as well. It's just paperwork. Though it may take a couple months as you now need to register with a range of organizations (for DUNS number etc.). In the past you didn't even need to found the company until after you won an award (depending on the funding agency). I am not saying things are uncomplicated, only that the system does not have built-in restrictions to block one-person labs run from a garage. Many companies have started this way from federal funding. – A Simple Algorithm Jul 9 at 23:15
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Lot of great answers, already. Some different thoughts (not as good, but hopefully additive):

  1. You could do it as an avocation. Go and earn a bunch of money elsewhere and then do it on the side. Like Fermat.

  2. (US answer) Look at government service, especially the NSA, but also national labs or FFRDCs (quasi government as they have contractors running them). It's not a "great answer" as you don't have total freedom and really they like applied guys more. All that said, there are pure guys going in there and you might find a niche (all you need is enough pay for one). And the pay/benefits/hours/security are great. At least take a look.

  • The NSA doesn’t really care what your specialty was for your PhD. They spend three years retraining you and moving you between projects to make you more well rounded. At least that’s what they told me when I interviewed with them years ago. – Joel Jul 11 at 14:15
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I will argue that research is the primary job that most salaried professors get paid to do. From Wikipedia: Professor:

Professors often conduct original research and commonly teach undergraduate, professional and postgraduate courses in their fields of expertise.

Note that "research" is listed first. From an excellent page by Mark Tomforde, University of Houston, "Job Responsibilities of Professors":

In the UH math department, the responsibilities of a typical tenured or tenure-track faculty member are usually allocated as 40% Research, 40% Teaching, and 20% Service.

Note that research is again listed first. (The approximate allocation matches what I've heard expressed many times, in many places.) Steven Krantz in How to Teach Mathematics (Ch. 6) quotes the Chair of the University of Chicago Mathematics Department, welcoming a new faculty member in the 1960s:

Remember: Our job is proving theorems.

At my institution, faculty promotion is officially based on the standard triad (research, teaching, and service); but I've been told by those involved that in practice, it really just boils down to number of research publications (justified by the fact that research publications are easier to identify and count than quality teaching or service).

One might argue philosophically that the "emphasis" of faculty work derives from the funding source. Traditionally most funding in the U.S. came from state governments (arguing in favor of a research focus); over time state support has shrunk, and student tuition increased, such that today it approaches a 50/50 ratio (arguing in favor of parity emphasis with teaching). See Figure 8 here.

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    Can you please define what it means for research to be the “primary” job? It seems like a meaningless statement to me. Cherry-picking quotes from some guy in the 1960s has zero supportive value for your argument (I could easily make up a quote by myself saying “our job is teaching” - actually I don’t need to, you can simply quote from my answer above). And the statistically meaningless fact that when two words are mentioned together one of them has to come first isn’t very convincing either. – Dan Romik Jul 9 at 5:26
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    @DanRomik: The quote you're nitpicking on is the keystone of a whole chapter in a major AMS publication. You can't seriously think that a self-made quote would count as a citation for a counter-argument. – Daniel R. Collins Jul 9 at 5:36
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    There's a lot more numbers needed to make this argument convincing. Only a fraction of universities grant doctorates and have such low teaching loads. And a large fraction of colleges are private and tuition-driven. I assume you are also excluding two-year schools. – A Simple Algorithm Jul 9 at 7:22
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    @Daniel I haven’t read this AMS publication. Perhaps the chapter you were referring to says some convincing things, but, respectfully, the quote by itself, representing the opinion of a single, arbitrarily picked individual (in the 1960s!), has zero value to support your argument. By the way, to be clear, I’m not even trying to make any “philosophical” arguments about the relative importance of research and teaching. I simply asked you to define what you mean about research being “primary”. Until you do, I don’t think we can meaningfully discuss whether your claim has merit. – Dan Romik Jul 9 at 12:48
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    @ASimpleAlgorithm: No. E.g., my institution is a two-year school (re: promotion standards case). – Daniel R. Collins Jul 9 at 13:00

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