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I'm a pure math PhD specialized in hyperbolic topology, doing a research postdoc right now which ends in August of 2018. So I'll be starting to apply for my next job in the next couple of months. I've been thinking about looking into applied math jobs in addition to the usual math research and teaching jobs.

I realize that there are jobs one could get in industry just for holding a PhD in math, but I am interested in jobs where my particular areas of knowledge/interest have potential for important scientific applications. Barring that, I would most likely accept the best teaching or research position offer I got (work which I also enjoy).

Below are some areas of math that I have studied. I've started with some specific topics I've studied closely, then put some more general areas that you could say I work in. My question is: what are (if any) some known applications of the following disciplines from pure mathematics, to the sciences?

  1. Hyperbolic 3-manifolds
  2. Hilbert-Blumenthal surfaces or Hilbert modular varieties
  3. Dynamical systems using Hamilton's quaternions
  4. Quaternion algebras (not just Hamilton's quaternions, but generalized quaternion algebras as defined by Dickson).
  5. Low-dimensional topology
  6. Arithmetic manifolds
  7. Algebraic number theory, especially quadratic fields, lattices, rings of integers
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    $\begingroup$ NSA among others $\endgroup$ – PsySp Sep 27 '17 at 10:52
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    $\begingroup$ The question in the title and the question in the body do not match: Many companies I know hire people in pure math (in any area), but no company I know applies recent results of any topic on your list. $\endgroup$ – Dirk Sep 27 '17 at 13:25
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    $\begingroup$ I'm voting to close this question as because it is either a shopping question or about the contents of an academic discipline (and should probably be asked separately for each subject on Mathematics or MathOverflow). $\endgroup$ – Wrzlprmft Sep 27 '17 at 14:06
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    $\begingroup$ This question is definitely off-topic for math.SE. This isn't a question about math, it's a question about companies which apply math. It would almost certainly be closed under the "Seeking personal advice" reason. $\endgroup$ – Najib Idrissi Sep 28 '17 at 8:30
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    $\begingroup$ @anomaly The reason has mostly to do with occasional existential crisis about the effect of what I do on the world. I love math but there are days when advancing theory that has not found applications in over 100 years feels kind off like just a sophisticated form of intellectual masturbation. $\endgroup$ – j0equ1nn Sep 28 '17 at 19:53
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In most fields of pure mathematics, there aren't a lot of industry options that rely directly on your expertise or involve continuing with your research program. Instead there are jobs which use the skills you've learned as a researcher: problem solving, abstraction, ability to learn complicated ideas quickly, etc.

There are occasional exceptions: NSA or other cryptography if you work in certain areas in number theory, quantum computing labs for certain areas of topology and algebra, etc.

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    $\begingroup$ Good. I think it is almost impossible to be able to work as a pure mathematicien in the industry. But on the other hand, lots of industries prefer people with certain exposure to pure abstract thinking. $\endgroup$ – PsySp Sep 27 '17 at 15:49
  • $\begingroup$ Thanks, I am aware of that and this is the generic response to mathematicians thinking about applied jobs. I know it's rare to find a job directly relevant to one's mathematical expertise but I was hoping (maybe naively) to find out something about what types of applications exist of the stuff in my list. Since folks are voting to close the question, I feel I should have asked this on math.stackexchange. $\endgroup$ – j0equ1nn Sep 27 '17 at 23:50
  • $\begingroup$ @j0equ1nn I am not sure anyone in the academia can answer what are the industry applications of these pure math topics. I am not even sure that math.stackexchange people would be able to answer that. $\endgroup$ – PsySp Sep 28 '17 at 9:15
  • $\begingroup$ @PsySp Is this doubt based on the topics I listed or would you say this about any list of pure math topics? I think it's worth a try in the math community! I've heard, for instance, that quaternions are used in navigation and 3D graphics, e.g. robots that explore deep sea and certain video games and CGI animation. I bet someone more knowledgeable than me could say more, at the very least about quaternions! $\endgroup$ – j0equ1nn Sep 28 '17 at 16:59
  • $\begingroup$ When you guys say "industry" do you think of the sciences as a subcategory of that? Like, do these comments imply that you think there is basically no applications of pure math to the sciences? $\endgroup$ – j0equ1nn Oct 7 '17 at 2:12
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You're not going to find anyone in industry willing to pay you to work on low-dimensional topology (and I say this as someone whose PhD is also in low-dimensional topology). You can certainly find companies who would be interested in the fact that you have a PhD and thus have demonstrated the ability to conduct research, work independently, handle difficult and abstract concepts, etc. You might even find a few companies that think in the fields you mention are nifty, though they won't have anything you can actually work on in them. Quaternions, for example, are used in computer graphics, but the level of what you describe would be overkill for what a company would be willing to pay you for. Frankly, looking for important scientific applications is not something that industry often does, at least in pure math.

To answer your specific question: The NSA does some nontrivial work in cryptography, which ties into the algebraic number theory point in your list. (That's a very broad subject, though, and the NSA is not exactly forthcoming about its research.) There are some places in industry that might be able to tie one of the points on your list to machine learning (say, face recognition), which is something they're very interested in funding. More broadly, you might find a government research lab that has something involving, say, low-dimesional topology, but that's probably one particular project rather than a full career. I've heard that biotech companies are interested in topology for various reasons, but that might be difficult to get into without a background in biology. If you want continue doing actual math and math research, your best bet by far is to continue on the academic research track.

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  • $\begingroup$ Biotech is interested in knot theory exclusively, is my understanding: RNA bundling. They need a lot of PDE people tho to work in inverse problems regarding tomography/imaging. $\endgroup$ – John Samples Sep 28 '17 at 17:52
  • $\begingroup$ Thank you! While a little discouraging, this is the kind of answer I was hoping someone could share. I remember seeing a talk on applications of knot theory to RNA but yeah, it seemed like the math part was pretty easy and the real prerequisite might be more in biology. Thanks again for sharing! $\endgroup$ – j0equ1nn Sep 28 '17 at 19:48
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    $\begingroup$ @j0equ1nn: No problem. As someone who's worked in a similar field of pure mathematics, I wish I could give you a more encouraging answer. The truth, though, is that there's really no opportunity to do serious pure math outside of academia. $\endgroup$ – anomaly Sep 28 '17 at 19:54
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While this does not address the availability of jobs in a specific industry, cosmic topology is an area of applied math to which many of these pure math topics apply.

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