There's a nice rule from mathematics that states that, when making some types of decisions, one can sample the first (100/e)% of the things available to get a feel for what is out there, then take the next option that is better than everything that came before, resulting in one being more likely to reach the best option possible; this is a simplified description. For a Numberphile video on the topic, see here.

I wonder how this applies to looking for postdoctoral fellowships or to an academic career in general.

My guess is that, since postdocs often require one to have completed a PhD within five years of the application and (100/e)% of five years is approximately 22 months, if one does not get a postdoc offer within that time, perhaps it is time to take action.

About Me:

I aspire to be a mathematician. I'm halfway through a PhD in group theory, in the UK. Job prospects are dire. Teaching below university level does not interest me and yet the majority of jobs out there are teaching roles. I want a job that requires interesting mathematics. The other major job type out there is some form of programming.

I'm planning ahead. If I don't get a postdoc within 22 months of the completion of my PhD, I'm tempted to retrain with a Master's in computer science, and I would still look for postdocs all the while. I will have gotten some form of job, I hope, but I like challenging myself, so perhaps another degree, part-time and online, would be the best option for a lifelong learner like me; I already have a place in mind.

An advantage of this approach: I would stay competitive, should I still want to be an academic. Another: I would have structured learning in marketable skills. I know publishing is important for an academic career but I am a pure mathematician and pure mathematicians, on average, publish less often than (other) scientists.

The Question:

What do you think of this plan? Does it hold water?

Clarification: This is if I have no offers in 22 months. The sample would then suggest that the next option is best.

  • 11
    Since one starts looking for a postdoc before you are awarded your PhD, 22 months afterwards is a clear sign it isn’t happening. Particularly since the actual timing of the PhD is often flexible. You are worrying about the wrong things.
    – Jon Custer
    Commented Jun 8 at 15:58
  • 23
    It's very unclear to me why finding a postdoc is related to the type of decisions you mention. Those are where you see things in order and must choose one before getting to the end of the list and cannot go back to a previous option. Commented Jun 8 at 16:06
  • 10
    I'm a bit confused by what kind of answer you expect. Ime, the decision on when to give up on academic jobs is less about some mathematically optimal search time and more about visa/personal finances/waiting on grant outcomes. I'm not in pure math, but from friends who've transitioned out of it to academia, finding a satisfying non-academic job also takes some time and effort, which should be considered in your timeline. Commented Jun 8 at 16:21
  • 37
    "I wonder how this applies to looking for postdoctoral fellowships or to an academic career in general." It doesn't. Literally the defining feature of the mathematical situation you describe is that you unilaterally decide which option you choose. The defining feature of applying for postdocs, in contrast, is that other people unilaterally decide whether or not they want to accept you. There is literally zero analogy between the mathematical situation and waiting for N/e months to see if you get a job offer. I would strongly recommend completely forgetting about this line of argument. Commented Jun 8 at 16:59
  • 6
    "pure mathematicians, on average, publish less often than (other) scientists." Sure, pure mathematicians publish less, but that's in comparison with other scientists. But your competition is limited to pure mathematicians, so you want to publish as many as or more articles than the average pure mathematician (precisely, the ones in your subfield of pure maths). Do you think you can handle this while studying for another Master? (And of course, the quality of articles matters, but producing good work also takes time)
    – Taladris
    Commented Jun 9 at 3:51

5 Answers 5


The 1/e rule that you are referencing is for a highly idealized model (the so-called secretary problem) involving many assumptions that bear almost no resemblance to your particular real-world scenario. For example, in the secretary problem, the decision-maker wants to optimize a specific parameter: the probability of hiring the absolute best secretary. With this definition of success, hiring only the second-best counts as just as much of a failure as hiring the very worst one. This has always seemed to me as a badly flawed measure of success, even if mathematically the result is elegant and the 1/e rule is a fun thing to talk about at cocktail parties. In real life, anyone hiring a secretary would be pretty happy hiring the second-best one, and similarly I assume you would also be quite happy to end up with the second-best postdoc among all the postdocs you can imagine applying for. This issue alone is a reason to strongly distrust the 1/e rule (And I've described only one among many problems with the analogy you are drawing between the theoretical model and the real-world scenario.)

For this reason, I'd say using the 22 months as your threshold for rejecting an academic career is an example of the fallacy of false precision. You arrived at this threshold through an analogy that doesn't really hold much water, so it's unlikely to be optimal or even close to optimal for the actual problem you are trying to solve. It would be much more sensible to have your choice of threshold be guided by more practical concerns, such as the amount of money you have saved that you can use to support yourself during your postdoc job search.

If you do want to use advanced mathematics to optimize your decision-making, look into the theory of optimal stopping. There is a large literature on this subject, with many results that go well beyond the highly idealized and unrealistic secretary problem. You may find that someone has proved results about models closer to your academic job search scenario, which may inform your decision-making.

Good luck in any case! I hope you fulfill your dream of getting employed as a mathematician, in academia or elsewhere.

  • 6
    Another reason the model is flawed: typically there's one postdoc hiring round per year, so it's not like the OP has 60 continuous months of samples—they have 5 discrete samples. They're also not independent, etc. Commented Jun 9 at 19:37
  • 3
    In the UK (where the OP indicates he currently is), the postdoc timings are much more ad-hoc than PhD admissions, not neccessarily clustered as you suggest
    – penelope
    Commented Jun 10 at 19:07
  • 4
    Even if the model was spot-on, the conclusion still isn't right; the conclusion would be to reject all postdocs offered in the first 22 months and accept the next offer you get that's better than any of those. There's just so many mistakes in the reasoning chain! Commented Jun 10 at 19:50
  • The stipulation is that I get no offers in that time frame, @DanielWagner.
    – Shaun
    Commented Jun 11 at 21:04
  • @Shaun A mathematician should have no trouble understanding the generalization if that universe obtains: you reject all 0 postdocs offered in that time, and accept the next offer you get in the remaining three years (which is therefore guaranteed to be better than any you've seen so far). Commented Jun 12 at 19:14

The 1/e rule only applies in situations where you have no information about the population before sampling. Here that is not the case. By the time you are writing up your PhD, your advisor and other folks around should have a good idea how competitive you are for a postdoc in math. If you get a couple of opinions that you're not likely to get a postdoc, you really should look at other options immediately.

Furthermore, unless you continue to write good papers, after a year or two without a position, you're not going to be hired for a postdoc. The limit for 5 years is mostly aimed at preventing people from taking a 3rd postdoc, not to give a new PhD more time to look for a position. Realistically, you have 2 years, so 1/e is a few months.

  • 17
    The rule also applies to a setting where you want to maximize the probability of getting the best "thing" among the one available. For positions, if you get the second-best option it's surely worth more than zero. Commented Jun 8 at 19:14

I think it's way to early to formulate a plan that calls for a binary decision 22 months from a future date now unknown.

You seem to be enjoying your doctoral research while learning some programming skills potentially useful outside academics. That should suffice for now. Keep your mind and your options open as you finish your degree and start your postdoc search.


Generally speaking, you don't need retraining for programming. Your programming experience is already more than enough to get and do an entry level programming job. Remember that the vast majority of programming jobs are about moving the names displayed on this site 5 pixels to the left, or adding another option to "Flag", not anything that requires serious knowledge of data structures or algorithms or compilers or cryptography. Indeed, anything that really requires knowledge of computer science is hard, and anything hard is expensive and prone to failure, which means industry tends to stay away from it.

Now, if you want to do interesting computer science, then the Masters is useful, but your interests would lead you into areas of computer science where jobs are almost as scarce as in pure mathematics.

  • 1
    So the vast majority of programming jobs, as you describe them, are pretty much lost to ChatGPT. Maybe CS knowledge would provide some job protection, at least temporarily? Commented Jun 9 at 11:40
  • 2
    This answer confuses me, though I agree that an additional masters isn't helpful unless this person wants to go into the more data science/machine learning type route. There are tons more jobs in cryptography-related work than research type jobs in math, at least if you're are a citizen of a country with a robust military hacking division like the US and UK. Maybe you don't consider that "interesting computer science", but perhaps that should at least be made clear?
    – user176372
    Commented Jun 9 at 19:53
  • 1
    Counting traditional tenure-track jobs at PhD granting departments as research-type jobs, the number of cryptography related jobs in the national security establishment is quite small compared to all pure mathematics jobs, or even to jobs in algebra/number theory/combinatorics Commented Jun 9 at 20:14
  • 2
    @AlexanderWoo That sounds plausible, though the NSA has 30k employees or so total (obviously distributed between different jobs, and on a cursory search). More precisely the supply/demand ratio must surely be much better for laborers if you fulfill the citizenship requirements. There are also contractors who hire in that space. I've just never heard of anyone having a particular problem landing that kind of job out of a PhD if they were willing to take one.
    – user176372
    Commented Jun 9 at 20:18
  • 3
    The OP has a Master's in mathematics and is halfway through a PhD. They are not looking for an entry-level programming job. There are a gazillion jobs in so-called "applied mathematics" where companies are looking for someone who can both write good programming code and solve mathematics problem. If the OP can learn to write good code then they will be sought after by recruiters and they'll have no problem finding an interesting job in that field, because the vast majority of applicants either suck at math or suck at writing good code.
    – Stef
    Commented Jun 10 at 9:16

Others gave you general advice. Adding the following:

  • Traditionally the telecommunications industry has some need for people who know their way around abstract algebra. Yes, you need to pick a number of other things (and do a bit more programming). I do need to warn you that my impression is "dated" at best (at 2008 I realized myself that my head is too deeply inside a teacher's hat that industry is not for me, but you are younger than I was at the time).
  • You are in grad school, so you are ideally placed to see what the current trends are. See what other grad students at your university are doing to learn how the land lies!! Many companies acknowledge that a math PhD is prime evidence of your ability to use your brain. When I was in grad school, here's what I saw: a complex analyst hired by AT&T, a model theorist hired by GM, a differential geometer moving into the defence industry after a quick programming stint in telecomm. Transitions I have seen later (leaving out my many connections to telecomm, because that is potentially distorted by my personal history): a number theorist + a coding theorist launching careers in actuarial science, another number theorist (who went to UK for a grad degree) becoming a quant, a cryptographer working for military (no surprise, really). I don't know where the traffic is nowadays. My point really is that your fellow grad students will do quite a bit of legwork for you (as will you for the benefit of grad students junior to yourself).
  • 2
    A further reason I'm downplaying telcomm a bit is that nowadays IoT is one of the big drivers, and I simply have no idea what kind of math they need. Of course, the physical layer is still based on older ideas, possibly needing to be adapted to a different realm. Commented Jun 9 at 6:35

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