It would be nice to make a case for funding pure mathematics projects strictly on their own terms, as good mathematics, but this is increasingly not the world we live in. Legislators and funding agencies want to see "useful" proposals, which puts anyone who does pure mathematics in the position of having to come up with (often tenuous) real-world applications for their work.

My question is: How do review panels perceive this practice? Do they really find largely hypothetical connections with applied sciences convincing? At what point does it become shameless BS that damages an application?

  • This depends very much on the review panel. I think the answer will very much depend on the funding body (and, if the funding body is large enough to have multiple panels, the specific subfield of mathematics). Sep 30, 2018 at 2:32
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    Probably when it starts with “take an imaginary field...”
    – Solar Mike
    Sep 30, 2018 at 6:12

2 Answers 2


I'll give you my perspective as an applied mathematician who has read a fair share of pure math proposals. I do, for example, find it amusing that essentially every number theory proposal stresses the importance of the work to cryptography. One can conjecture that it's rather unlikely that that is actually true for even a significant fraction of these proposals, but it is the common approach in that community to address NSF's requirement to explain the "Broader impact" of the work.

Now, since that is so and everyone does it, it's likely that a selection panel consisting of people well versed in the field is just going to read over that statement and ignore it -- everyone knows that applicants need to pay lip service to the broader impacts, registers that they do, and moves on. In other words, as long as you stay within the norm of your subfield in conjuring up possible connections to applications, then there is no problem for you. Of course, if you promise completely outrageous connections, then even a panel of people who are in the same situation as you will find that you've gone beyond. Where exactly that line is is of course hard to define -- talk to others in the community and let them comment on what you write.

I do want to add one point to this, because I really don't want to come over as a pure-math bashing applied mathematician: The same of course happens in what is generally understood as "applied mathematics". Whether you try to show that the solution of some PDE is in a slightly smaller Sobolov space than was previously known, or whether you show W^{1,\infty} error estimates for finite element discretizations of some obscure equation, the truth is that much of this kind of research is also pure. The only difference is that the equation you are considering might have been motivated by some actual application, but the actual work you do is not motivated by trying to actually solve the problem in the same way as an engineer would want to approach this. In other words, your work may be closer to applications and is often easier to explain to laypeople, but it's not exactly applied in many cases and the "broader" impacts are also quite limited.

  • Thanks, Dr. Bangerth, very helpful. I've heard about grants in number theory (which is not my field) name-dropping cryptography too, but have you encountered any examples which cited specific potential applications of the pure research?
    – user113878
    Oct 4, 2018 at 16:20
  • @user113878: Of course many pure math research areas do have pretty immediate applications. A lot of algebra is used in practice. A lot of number theory is in fact also used in practice. The subdivision of what we call pure/applied math is not driven by what really is applicable, but it's just a rough categorization of broad research areas. Oct 4, 2018 at 18:06
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    What I want to say is that a lot of research in "pure math" areas really is quite applied or at least applicable. My last paragraph makes clear that a lot of research in "applied math" areas really is not applied or applicable. Oct 4, 2018 at 18:07
  • Well, also, "number theory", and "combinatorics", at the low end are catch-alls for diddling around to no larger purpose. No advancement of collective understanding of anything. Likewise, low-end self-labelled "applied math" is all too often just numerical diddling, without any genuine "application" any more than other diddling. Apr 17, 2023 at 21:27

This won't be a direct answer to your query, but I hope will be useful advice. First, if you must do pure math and you must get funding to do it, then you have no option but to apply. If you don't apply, you don't get funded.

Next, being devious isn't going to help you and will likely be recognized, as you fear.

However, on some time scale, all mathematics is applied. The time scale may be long, of course. In my personal case, it took about thirty years for someone to find an application for my "beautiful" but extremely arcane dissertation to find application. I'd predicted that it would never be done. But thirty years isn't very long, so that doesn't help you directly.

You work in some subfield of mathematics, of course. It might be very narrow and it might be a currently popular subfield, or not. But it is likely that others in the same subfield, or a closely related one, have seen applications of their work to some problems. If you can learn what those applications have been, you have the basis for a statement that "Advanced work in X has been known to contribute to Y as evidenced by (citation)". That isn't devious, or phony.

But note that "applied" is actually a continuum, not a discrete thing. Pure is pretty far to one end of the continuum, but it isn't disconnected from the other. But it will take some investigation on your part, and I suspect that you are good at that sort of thing.

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    I don't agree that pure and applied work are on the same continuum. They are quite different approaches toward quite different goals, although there are some questions for which these are complementary. But you are right in the sense that the pure goals need to be contorted into applied goals as if they were on a continuum.
    – user113878
    Oct 1, 2018 at 17:56

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