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Suppose that you graduate with a good PhD in mathematics, but don't necessarily want to go into academia, with the post-doc years that this entails. Are there any other options for continuing to do "real math" professionally?

For example, how about working at the NSA? I don't know much of what is done there -- is it research mathematics? Are there other similar organizations? Perhaps corporations that contract with the federal government? Companies like RSA?

Other areas of industry? Is there research mathematics done in any sort of financial or tech company?

I've made this a community wiki, since there aren't any right answers...

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    $\begingroup$ I like this question, but as currently phrased it seems to apply mainly to people who are US citizens. (Certainly those of us who are not US citizens will find the question "How about working at the NSA?" admits a very short answer.) I don't think that that necessarily makes the question inappropriate, but it might be nice to make this assumption explicit? $\endgroup$
    – Ian Morris
    Jun 17, 2010 at 16:44
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    $\begingroup$ Hi Ian, You are right. I am a US citizen, and so am interested also in opportunities only available to US citizens; of course other opportunities are also welcome, and it makes sense to distinguish the two types. $\endgroup$
    – Wilson
    Jun 17, 2010 at 16:49
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    $\begingroup$ Ian--Most governments' cryptologic organizations hire mathematicians. See, e.g., the UK's Government Communications HQ, Communications Security Establishment Canada, the Australian DSD, etc. Of course they will have to be nationals though... $\endgroup$
    – Steve Huntsman
    Jun 17, 2010 at 17:13
  • $\begingroup$ I am a UK national, with a reasonable Ph.D., but the evil GCHQ was very unfriendly at the test centre and rejected me. They didn't give any reasons, but it must have been the computerised psychology/security test, since their mathematical tests were pretty easy compared to hard graduate level stuff. So it doesn't matter how good a mathematician you are, you're not guaranteed to get a cryptography/code breaking job at your own country's version of the NSA/GCHQ/CIA/etc. $\endgroup$
    – Zen Harper
    Jun 18, 2010 at 9:51
  • $\begingroup$ @Zen: No job is ever guaranteed. As a rule, it is much easier to get a job if you have personal connections, and GCHQ is no exception to the rule. $\endgroup$
    – Timothy Chow
    Jun 18, 2010 at 14:10

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I have worked in academia, at the research center of a telecommunications company (Tellabs), and at two different FFRDCs (MIT Lincoln Laboratory and IDA). At all of the non-academic jobs, I have done "real math," published papers, attended conferences, given talks, etc. So it is certainly possible to continue doing "real math" outside of academia.

You should be aware, however, that in almost any non-academic job, there is pressure on you to produce results that are "useful" for the company or the government. The amount of such pressure varies, but it always exists, because ultimately that is the main justification for your paycheck. In academia, the corresponding fact is that in almost any academic job, there is pressure on you to teach, since that is usually the justification for a significant portion of your salary. Finding a non-academic job where there is no pressure on you to do anything "useful" is akin to finding an academic job where you have no teaching responsibilities.

Certain high-tech companies and certain FFRDC's recognize that a good way to attract top talent is to give their employees the freedom to pursue their own research interests, whatever that may be. All the non-academic jobs I had were like this. They actively encouraged me to spend some amount of my time doing "real math" regardless of whether the results were of any "use." How much time? Well, if the company was doing well, and if I was doing a good job of producing "useful" results that they liked, then they would give me more freedom. But if the company was doing poorly then they would start to squeeze. During the telecom industry meltdown in the late 1990s, Tellabs eventually eliminated its research center entirely, along with my job; Bell Labs (more famously) suffered a similar fate.

So far I have been drawing a dichotomy between "what the company finds useful" and "real math," and maybe you don't find that satisfactory. After all, if you're sufficiently motivated, you can do "real math" on your own time regardless of what your "day job" is. Maybe what you want is a job where providing what is useful to the company involves doing real math. This is a taller order; for example, at Lincoln Labs I found that there was almost no real math involved in the work they wanted me to do, and I eventually left that job for that reason even though it was a great job in almost every other respect. However, it is still possible to find such jobs, depending on what area of math you are interested in. If you are interested in large cardinals and are hoping for a job where your theorems about large cardinals will be "useful" then you are probably out of luck. However, if your interests lean towards areas with known relevance to computer science or various branches of engineering then your chances are much better. The NSA scores pretty well in this regard since it is no secret that number theory and various other branches of so-called "pure" mathematics are relevant to cryptology.

In summary, jobs where you do "real math" do exist. When considering such a job, though, you should first ask yourself, will I enjoy producing what this company considers to be "useful" results? If the answer is no, then you will probably not be happy at the job even if they give you some freedom to do "real math." However, if the answer is yes, and the company gives you some amount of freedom to do "real math," then it will probably be an excellent fit for you.

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    $\begingroup$ Incidentally I think this is related to the source of some of the questionable research you see in industry.. oftentimes people want to pursue their own research interests and not necessarily those of the company, so they try to spin their own research as useful for national defense or whatnot. Due to government largesse in things like the war on terror, it sometimes works. $\endgroup$
    – Michael Greenblatt
    Jun 18, 2010 at 1:48
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    $\begingroup$ Michael, "spinning your research" happens in academia too. When applying for an NSF grant, people try to make their research seem more attractive by drawing connections with more fashionable areas. This is a fact about human nature and I don't think it has much to do with industry/government per se. $\endgroup$
    – Timothy Chow
    Jun 18, 2010 at 2:30
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    $\begingroup$ Sure.. but I think it's easier to succeed at this sort of thing in industry, at least in the defense contracting industry, due to the huge amount of money available and the concomitant lower standards (not to mention that so many contracts were only open to US citizens/permanent residents which restricts the applicant pool). I may be somewhat biased due to my own experiences here but there really was a lot that I saw that was just jaw-dropping. $\endgroup$
    – Michael Greenblatt
    Jun 18, 2010 at 2:44
  • $\begingroup$ I think the analogy with teaching works best if you mean those colleges that value teaching higher than research. $\endgroup$
    – Bjørn Kjos-Hanssen
    Oct 26, 2010 at 22:55
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Defense/aerospace, government, and finance (perhaps less so now) are all industries that engage mathematicians. I have personally worked in defense and defense-related industry for my entire (still short) career (think tank, defense university, R&D), and found a great deal of intellectual freedom and interesting work in every job I've had.

The Agency mathematicians with PhDs I've met over the last decade or so are clearly very competent. While I won't speculate on what they do on a daily basis, the people I have met uniformly seemed to be engaged in activity that they found interesting (though at least two have since left to other government posts). They go through an introductory program to teach them relevant techniques and rotate through various offices for a few years to learn the lay of the land. NSA also has affiliated FFRDCs that support internal mathematical research. I have met people affiliated with these as well and they all seem to enjoy the work and to find it fulfilling.

As an aside or two, the field of symbolic dynamics was basically inaugurated by Hedlund while working at one of these FFRDCs, and James Simons (of Chern-Simons and hedge fund fame) is also a well known alumnus. There are many other prominent mathematicians who have worked at or continue to consult for these centers, and if you search CVs you can probably find quite a few examples.

Finally, regarding the Agency, I would suggest looking at this.

Regarding other defense/aerospace work, you're likely to find things that involve a lot of DSP or probability (or both).

In finance, there will be a heavy emphasis on probability, particularly stochastic differential equations and martingales. I haven't had any direct experience with or inclination to try to land a spot in this field, but I suspect that (given the pay) it is quite selective. Certainly you'll need to program (probably in C).

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    $\begingroup$ FYI, at least one well-known cryptographer (Victor Miller) has contributed quite a bit to MO. $\endgroup$
    – Steve Huntsman
    Jun 17, 2010 at 17:45
  • $\begingroup$ Steve, thank you very much for the list of NSA related arxiv articles. I am not a US citizen and am really amazed at the subjects of the articles. They are very, very far from my former (clearly uninformed) intuition of NSA tolerance for abstract/pure mathematics. Live and learn! $\endgroup$
    – Georges Elencwajg
    Jun 18, 2010 at 9:22
  • $\begingroup$ One thing I should add would pertain to a job that I've recently started. There is a lot of interesting and current mathematics that can be fruitfully applied to data mining: e.g., persistent homology and nonlinear dimensionality reduction, for starters. I have already found things that will be genuinely useful to do that have not been explored in the literature. $\endgroup$
    – Steve Huntsman
    Feb 11, 2012 at 13:13
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Having worked briefly in the defense contracting industry, I concur with the other commenters. One thing worth pointing out though is that unlike with things like NSF grants, the overall standard to get government money for defense contractors is pretty low, and you get a lot of "bogosity" in many contracts awarded, including those that are funding PhD's in math-related areas who are supposedly doing research-type activities. As a result there is a really big difference in the type of work that goes on at say the NSA or the IDA compared to many other places. I saw projects that got funded for years that had very obvious flaws in their models that most of the posters here would easily point out. Even if companies eventually figure it out they are sort of stuck pursuing their old methods because that's what will bring in government money, not to mention what conceding they're wrong would do to their reputations.

Moral of the story: there is interesting math of an applied bent out there in the defense industry, but make sure you know what you're getting into.

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    $\begingroup$ This is true. But the yardstick for defense contractors measures different things. There aren't a lot of contracts for applying étale cohomology to ballistic missile defense or what have you. Meanwhile, the LockMarts and SAICs of the world have comparatively few real mathematicians and a LOT more engineers and computer scientists (and some physicists). And since defense contracting is more "pull" than the NSF "push", somebody's got to do the work. This is why frequently you'll see silly stuff like ad hoc Bayesian methods with crummy priors and no stability analysis (this is common) in defense. $\endgroup$
    – Steve Huntsman
    Jun 17, 2010 at 17:56
  • $\begingroup$ Also I just remembered an interesting example that belies a possible reading of this answer. Hochschild was working at Aberdeen Proving Ground when "On the cohomology groups of an associative algebra" was published in the Annals (jstor.org/pss/1969145). Though it's an old example (and Aberdeen is not a Beltway bandit but part of the Army proper), I think it's safe to say that this work was not applied, and in particular was neither motivated by nor directed towards the war effort. $\endgroup$
    – Steve Huntsman
    Jun 17, 2010 at 20:19
  • $\begingroup$ I did see a few funny examples of this phenomenon. My favorite was someone who got funding from the Missile Defense Agency to apply his graduate work in category theory. He made the connection from category theory in math, to category theory in computer science, and thence to computerized guidance of missiles. But mostly what I'm talking about is just very unrealistic/incompetent mathematical and statistical modeling as you described above. $\endgroup$
    – Michael Greenblatt
    Jun 17, 2010 at 20:42
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In the United States, it is definitely possible to do mathematical research at some industry labs, and publish/collaborate just as an academic would. (This is much easier if your research straddles math and computer science, but still possible in some other cases.) My employer, IBM Research, has labs in the New York and San Francisco areas which hire mathematicians to do research with a strong computer science slant. Also, Microsoft Research in Redmond, WA, Mountain View, CA, and Cambridge, MA (and the other Cambridge!) hires people with similar background. Google and Yahoo also have labs but I am not sure they do as much theoretical work.

In the San Francisco area, there are also government-affiliated labs like Sandia, Lawrence Berkeley, and Lawrence Livermore. However I am not sure how much of their work they can publish, and what kind of security clearance they need to work on interesting projects. (They do publish at least some papers, though.)

Every group has different foci, so your chances of working in industry really depend on how much your interests and work intersect with some group. It will require some effort on your part to find appropriate places for you.

Note the above groups are just those that I am personally familiar with. Also, the above labs hire postdocs as well (in fact I am currently one).

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    $\begingroup$ At a government-affiliated lab, you will need at least a low-level security clearance in order to do classified work, and the most interesting projects will usually be classified. Before you can publish anything, the lab will need to okay it first, but if there is nothing classified in it then there is unlikely to be a problem. The vast majority of what we would consider interesting, publishable, pure mathematics will very likely be given the green light by the lab. $\endgroup$
    – Timothy Chow
    Jun 18, 2010 at 1:47
  • $\begingroup$ Google has a division called the "labs", but it doesn't really do research in the sense that Bell Labs or IBM Labs does. I can't remember ever reading a peer-reviewed paper listing that affiliation (which isn't to say they don't exist -- I'm sure they do). $\endgroup$
    – Adam
    Jun 18, 2010 at 8:18
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    $\begingroup$ Google hires a lot of PhDs, but is incredibly secretive about what they do. Word of mouth is they do a lot of systems research and algorithmics -- but they regard this stuff as trade secrets, and so their people don't publish. OTOH, MSR is really wide-open: I'm doing a postdoc at Cambridge 1.0, and it's basically like a postdoc at a university, only with better pay and travel budgets. MSR is strongly CS-focused, though they are happy to hire theoretical CS-oriented people. $\endgroup$
    – Neel Krishnaswami
    Jun 18, 2010 at 10:11
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If you do not want to spend many years in postdocs, do not mind to move in another country, and speak french fluently you can try to get hired in a french university. Some people get permanent positions right after the thesis, and most of the hired people defended their phD only one or two years before. The competition is harsh, but the postdocs are not mandatory.

Edit in 2017: in the past years, among other things the importation of the postdoc system to France has changed things. Now it is rarer (though possible) to be hired just after PhD, and the competition has become even harsher. Salaries have not increased, so I guess the attractiveness for US applicants might be quite low.

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    $\begingroup$ @Benoît: I cannot help but notice that the faculty at French universities are almost 100% French citizens and/or obtained their degrees from French universities. (I don't blame anyone for this.) I would think that as an American, say, applying for such a job, the competition would be especially harsh. Also, from what I've heard, salaries for a Maitre de Conference position are somewhere between 1/2 and 2/3 of that of an American assistant professor position. (Nevertheless part of me finds such an option attractive. If only my French were better it might be a serious consideration...) $\endgroup$
    – Pete L. Clark
    Jun 17, 2010 at 16:58
  • $\begingroup$ @Pete L. Clark: you're right that it seems especially difficult for a young mathematician from abroad to get hired in France without any preliminary contact (and I know at least one case of a very good candidate that is unlikely to be hired this year given her rankings); however, a fair number of successful candidates are, for example, from Italy (I think of candidates that had their PhD there). Concerning the salary, you are perfectly right but it is still sufficient to live quite comfortably, except if you need to find an apartment in Paris. $\endgroup$
    – Benoît Kloeckner
    Jun 18, 2010 at 7:17
  • $\begingroup$ @Pete L. Clark: there is also a possibility to be hired in France without speaking French so well (although you should be willing to learn): CNRS offers each year a few positions (one or two !) partly dedicated to attract mathematicians with some experience (typically, four years in academia) from abroad. It is called CR1 (Chargé de recherche de première classe), and like any other position in CNRS there is no teaching load, although you can easily give a course (usually at graduate level) if you want to. The starting salary is something like 2300 euros a month. $\endgroup$
    – Benoît Kloeckner
    Jun 18, 2010 at 7:26
  • $\begingroup$ @Benoît: thanks for your response. I sense you're right that I was overstating things a bit (and certainly you know far better than I): in the last decade the liquidity of students and jobs within the EU has increased considerably. But for an American (or Russian, Japanese, etc.) to get a job in France: that seems like a tall order... $\endgroup$
    – Pete L. Clark
    Jun 18, 2010 at 11:48
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    $\begingroup$ I once tried to figure out the relative value of US vs French salaries and found it exceedingly difficult, mostly due to uncertainties in the US social benefits' costs. Even ignoring the health care costs, the universities have different and hard to calculate pensions (say, UMN very generously paid me extra 13% to my 401K, while MIT paid only 5% but had a separate pension plan). Some universities pay tuition for children, some don't. Often, the salaries tend to lag behind inflation with some full professors getting less than new assistant professors, which makes comparisons even harder... $\endgroup$
    – Igor Pak
    Jun 18, 2010 at 18:04
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I would browse some journals in your particular research area to see if any of the authors work for a non-academic institution.

In my experience most of the research work done in industry/gov't is considered properitary or non-disclosable. So there is not much opportunity to publish results.

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  • $\begingroup$ I agree that it seems there is little opportunity to publish outside of Academia -- this is why it is hard to learn what is done there. But I suppose that publishing is not a necessary condition for satisfying work, so long as there is a good internal community of mathematicians with whom to work. $\endgroup$
    – Wilson
    Jun 17, 2010 at 16:50
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    $\begingroup$ I've always found the ability to do interesting work without the bother of publishing to be a plus. $\endgroup$
    – Steve Huntsman
    Jun 17, 2010 at 17:23
  • $\begingroup$ Is there any such industrial places in Europe where you can do 'real Mathematics'? I'm now in my first year of academic postdoc, and I'm actively trying to switch to some areas like that, but before that, I want to possibly do another postdoc in a much more applied area relating to what I'm doing now. But I want to limit my search in Europe only. If you know any, please do let me know, thanks a lot in advance! $\endgroup$
    – Learning math
    Nov 26, 2013 at 22:00

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