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Does anyone know if it is possible for a Ph.D. graduate of mathematics to successfully do research in his/her spare time?

The work environment and conditions in academia seem somewhat deplorable at the moment---no offence intended, but that simply very much appears to be the case from an outsider's perspective---and it just strikes me that mathematics would surely be the subject most suited to Ph.D. graduates who don't want to go there, but still want to research, as "only pen and paper" are required.

If one were really, truly devoted...how much of a disadvantage would they have compared to someone who chose to stick it out?

Are seminars entirely barred?

Is it impossible to publish?

Collaborate?

How much (more) time would someone who stayed in academia have in a day to spend researching?

Also, what about people who have not got PhDs? I would think it's more or less impossible for them to contribute very much of value if they only have a Bachelor's or a Master's degree, but I really don't know.

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    What do you mean by "research"? There are lots of non-academic jobs where your job will be to do research.
    – Kimball
    Jul 2, 2015 at 11:18
  • Should be moved to Maths SE forum. But still a good question (+1).
    – shivams
    Jul 2, 2015 at 12:46
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    @shivams I disagree: I think it is a fine fit for this site.
    – jakebeal
    Jul 2, 2015 at 13:11
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    @shivams I'm from Math.SE. If migrated, this question would be likely to be voted down and closed pretty quick. It is certainly a much better fit for this site.
    – apnorton
    Jul 2, 2015 at 18:55
  • "I would think it's more or less impossible for them to contribute very much of value" -- Ramanujan didn't have a degree until 1916, which is after many highly valuable contributions including collaboration with Hardy. But obviously, just because Ramanujan did something doesn't mean it's not more or less impossible.
    – Steve Jessop
    Jul 3, 2015 at 23:58

4 Answers 4

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Publishing

In mathematics, most people will judge your research on its merits, not based on your credentials or affiliations. So this is not an issue, if you are able to generate significant results and write them up in a clear way.

Seminars

I have seen many members of the public regularly attend seminars. I've never heard of them being kicked out. It's not an issue.

Funding

The primary drawback is that you won't be paid for your research if you are just doing it in your spare time. Unless you are already independently wealthy, that means you will have relatively little time for research since you need a day job.

It is of course possible to get grants to fund your research without being in academia; there are a number of private entities in the US funded entirely by grants, and many of them include mathematicians. But you said spare time only.

Collaboration

Another drawback is that you won't be in close contact with other researchers by default. It wouldn't be impossible to establish that contact, but again you must consider whether you will have the time. And people won't take you seriously unless you have serious results.

Not getting a PhD

In mathematics, even more than in other fields, it is almost impossible to make a contribution without spending several years learning what is already known. In principle you could do that without getting a PhD, but why would you?

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    I totally disagree with your first point. The reviewers of your paper will google your name and they will turn up their noses if you do not have an institutional affiliation.
    – Flounderer
    Jul 2, 2015 at 23:08
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    @Flounderer Some, maybe, but not as a general rule. I've published papers when I've lacked an institutional affiliation before. I even asked my advisor, who has served as an editor on several high-ranked journals, if I was doomed from the start doing this, and was assured that I was not. And that in fact journals both receive and publish papers from authors with no academic affiliation fairly regularly. A reviewer may have a higher level of scrutiny of your work, but if they reject it simply due to no academic affiliation the editor will ignore them and get a new reviewer.
    – zibadawa timmy
    Jul 3, 2015 at 1:26
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    @Flounderer - I have also had a paper accepted with no affiliation.
    – P.Windridge
    Jul 3, 2015 at 12:28
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    @DavidKetcheson - I agree with your last point. I.e. if you've done the work necessary to publish (in decent journals), then you've probably done the hard part of getting a PhD. So you may as well be enrolled on a part time degree and get some credit (and yes, affiliation and access to seminars and a library)
    – P.Windridge
    Jul 3, 2015 at 12:32
  • Another obstacle to overcome as a non-affiliated researcher would be lack of access to powerful computers and software, something that can really help speed along research.
    – 727
    Aug 14, 2015 at 0:52
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It should be possible, but it will be hard.

  • Whether seminars are barred to you will depend on your locale. Even if non-students are officially not allowed to attend seminars, you may find a sympathetic professor who may turn a blind eye.

  • Of course you can publish as an independent researcher. You may come in for some extra scrutiny until you have established a bit of a reputation.

  • Similarly, nobody will care specifically about whether you have a Ph.D. when you submit a paper, as long as you have a command of the specific specialty you are trying to publish in.

  • Of course you can collaborate, once you have convinced someone that you have worthwhile ideas. Getting a Ph.D. may be helpful to distinguish you from random cranks in establishing contacts, as well as start you out on building a network.

  • As to the time you will have to do research, this will vary. If you are at an R1 institution, you will have a lot of time (but of course getting in is hard). If you are at a teaching college or in a research job in industry, you will have more time. If you have a completely unrelated day job, you will need to rely on your spare time. And when you come home after a typical job, you will not be as fresh and as enthusiastic for research as if you started out in the morning, at your desk in a university. (If you ever decide to start a family, expect your spare time for mathematical research to drop to zero immediately.)

Academia still has advantages, for instance:

  • Your library will likely have subscriptions to the major journals, so you don't need to bombard authors with requests for their papers.

  • You will likely have at least some budget to attend conferences. If you go the independent researcher route, you will need to pay for these out of your pocket.

  • If you are not at a university, you may have to pay higher registration fees (which depends on the conference and may be negotiable).

  • If you hold a day job and want to attend a conference, you will need to use up vacation days for that.

  • Although you can collaborate as an independent researcher, it will still be easier at a university. There will be seminars right next door, guests, access to students you can mentor and develop into collaborators etc.

I work in a math-heavy research job in industry and have been able to do some more-or-less serious research, mostly on the side. (And I obviously still have affinity for Academia.) It can be done, but it's hard.

If what you want to do is research, do your Ph.D. first (I did). This will give you an idea of what's ahead of you and give you a bit of a network. If afterwards you don't find a job in academia, you can still get a day job and try the research-on-the-side route. (However, your priorities in life may have shifted by that time.)

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    "nobody will care specifically about whether you have a Ph.D. when you submit a paper" This is true in a literal sense. However, without a PhD, it's hard to pick up the necessary skills to write competent-looking mathematical papers, or to select suitable topics. You'd need to find some other way of acquiring those skills.
    – David Richerby
    Jul 2, 2015 at 13:29
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    If you are at a teaching college or in a research job in industry, you will have more time. Did you actually mean less time (compared to R1)?
    – just-learning
    Jul 4, 2015 at 10:40
  • @just-learning: "more" time compared to the time at an unrelated day job, but of course less time than "a lot" of time at R1. Sorry, I see how that could have been confusing.
    – Stephan Kolassa
    Jul 4, 2015 at 14:03
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A supplement to other answers: no one seems to have explicitly mentioned what might be biggest "disadvantage": you will probably have a lot less motivation to do "academic" research being outside of academia. Research is hard, and external motivation (e.g., constantly being around people interested in say pure math research) can make a big difference in your productivity. Here, where you work and what kind of community you are in and culture you are a part of can play a big role.

My personal impression is that quite a few people in industry positions with a more academic math-heavy culture (like NSA, Microsoft Research, Google, IBM, once upon a time AT&T/Bell Labs, National Labs, etc) do a reasonable amount of academic publishing in their spare time. But at other kinds of companies, even where you do research, like banks or hedge funds, academic publishing seems to be less common.

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    In addition to this, it is hard to find the time for research if you have a regular 9 to 5. Research takes chunks of time, not just snippets here and there. It's doable but not easy.
    – Cameron Williams
    Jul 4, 2015 at 4:09
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As an academic mathematician, I can assure you that I do about half of my research in my "spare time" (that is, outside of paid hours) and I have done so for over 30 years. I expect most academics do similarly. If you are truly devoted, then just do it. I recommend getting up at 4am or 5am daily so you have a couple of hours before the family awakes.

Regarding access to libraries, seminars, etc: contact a university (probably best near where you live) and ask about an "adjunct" appointment or affiliation that gives you access to facilities.

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    If you're an academic mathematician, then surely you're salaried, in which case the concept of "paid hours" doesn't apply to you.
    – zibadawa timmy
    Jul 3, 2015 at 1:29
  • @zibadawatimmy contracts usually say you have to work a certain number of hours per week. If you work more than that, you are working unpaid hours.
    – Davidmh
    Jul 3, 2015 at 8:01
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    @Davidmh I would strongly disagree. You are paid to do your job to a certain minimum specification, but if you go over that's not something that's external to what you're being paid for: it is exactly what you're paid for. My parents are salaried professionals, and sometimes they work long nights or weeks or even months. They don't get paid extra, and it's not "unpaid hours": they're doing the job they are specifically salaried to be paid, and sometimes that requires extra time and workload. That the work can't be easily characterized in work-hours is precisely why you get salaried.
    – zibadawa timmy
    Jul 3, 2015 at 8:51
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    @zibadawatimmy my contract does specify a workload of 39 h a week, but no specific work output (it is very difficult to define, anyway). It doesn't mean I stick to that schedule, but when I work more than that, I am not getting paid for it, hence unpaid hours. Some postdocs here even have contracts for a certain number of hours a year, and they can distribute them as they please.
    – Davidmh
    Jul 3, 2015 at 13:45
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    @Davidmh That's a strange specification. That sounds like an explicit attempt to classify you as a part-time worker to avoid requirements to give you various benefits and the like. Looking at your profile confirms what I suspected from that: you're a grad student. My grad student contracts also involved talk of work hours, in the form of saying I was to work less than 20 hours a week on my TA duties (rarely a problem). It's mostly a stipulation the university puts in to cover their own butts, and to avoid full-time employee classifications.
    – zibadawa timmy
    Jul 4, 2015 at 1:12

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