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I will be graduating with a Masters degree soon in mathematics. For various reasons I have decided not to pursue a career in academia for now and will instead be working a job in industry that will not require much advanced mathematics on a day to day basis.

Besides MathOverflow, StackExchange, and other online venues, how may one stay engaged with the mathematical community from outside academia? I plan to keep in touch with faculty from my institute, but I would like also to engage with the wider mathematical community.

My goals include having others to collaborate with, and to discuss mathematics in general.

Edit: Thank you all for the wonderful and helpful answers! Seems like there are many ways I didn’t think about.

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    $\begingroup$ if you haven't already, you might try joining the AMS, MAA or SIAM and going to the occasional conference when convenient. $\endgroup$
    – Will Jagy
    Nov 21 at 22:23
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    $\begingroup$ Perhaps, you can convince your employer that mathematics may be useful for them, if that is so. Otherwise, perhaps you can find another employer, for whom mathematics could be more useful. $\endgroup$
    – Iosif Pinelis
    Nov 21 at 22:54
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    $\begingroup$ @WillJagy Oh, I did not know they accepted non academics as members. I will give it a look, thanks! $\endgroup$
    – Nate River
    Nov 21 at 22:55
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    $\begingroup$ As someone who dropped out as an undergrad, MO+researchgate has been plenty of contact with the mathematical community. I currently work as an independent contractor and just passed the physical exam to be a firefighter in St. Louis; the flexibility of these interaction formats makes it possible for me to ‘slip a little math in’ whenever I feel like it, and this has been a wonderful experience thus far. $\endgroup$
    – Alec Rhea
    Nov 22 at 11:23
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    $\begingroup$ @NateRiver Not to be too cynical about it, but most professional organizations are happy to accept as a member anyone who pays their dues. $\endgroup$
    – Timothy Chow
    Nov 22 at 15:10

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If you like computers, you could consider getting into formalized mathematics, which is mathematics done completely formally and verified by computer programs, known as proof assistants. Formalized mathematics has been picking up quite a bit of momentum lately, and large communities have formed around it. I am suggesting it as a possible activity because you can contribute meaningfully without being a researcher in mathematics. There are entire branches of mathematics that are not formalized yet. With some luck, you will be able to tell your grandchildren that you helped computers take mathematics away from humans.

A good starting point is to look around the Lean proof assistant and its mathematical library mathlib. The community is welcoming and appreciates help from everyone. We also have a Proof assistants StackExchang site, and the Lean community has a very active Zulip server (see above link).

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I studied Maths, and did a PhD in Maths (PDEs). I really enjoyed it, and had made good progress, but for other reasons I decided to leave Academia after completion (10y ago), and have been working in Industry since then, in jobs that do not require advanced Maths. Based on my experience, a few comments and partial answers to your question (but also a bit of a counter-point for your expectations):

  • I miss Maths (after all, I spent 10 years working hard on getting good at it). There are jobs in Industry that might require advanced Maths, such as working for Hedgefunds or Defense-Contractors, but opportunities are rare, and the topics might not be for everybody's taste. This is just a comment, noting that you specified you would not go in such a direction.
  • I have tried to continue doing research (or taking part in academic discussion) on the side, and have not been successful. I know of some people who manage to do this besides their day job and might even publish a paper or so, but those people are usually exceptionally good (even among academic researchers), and doing this is usually extremely demanding (even for the brightest), for several reasons:
    • Research takes a lot of time: Researchers tend to get more efficient as they specialize, but it's common for PhD-students and PostDocs to put in 40-80h per week. If you have another job, putting a comparable amount of effort in requires a lot of dedication, and the absence of any other activities (hobbies, family).
    • High level maths (even reading it) is very demanding, and can be even more challenging if the only time you have left to do it is after a day in the office. This is when many people curl up and watch TV. Admittedly, I try to read undergraduate Maths books (mostly on new topics) in the evenings, with fair success, but I don't see myself being able to focus on (and enjoy) the latest papers in PDE-research.

My point is that keeping in touch with Academia on a sufficiently high level beside a day job will usually be very challenging, and it is worth asking yourself if this really is what would make you happy. Personally, I have come to accept that for me it is not possible to have a job and keep up with academia-level Maths. Most of my former PhD-colleagues have come to similar conclusions. The love for Maths is still there, but not the time & energy for it.

Now to the constructive part of the answer:

  • It is not uncommon for people to go back to Academia, and pursue a PhD etc.
  • There are Maths-Olympiads in most countries. Although you seem too old to participate ;) many of the trainers/coaches are volunteers. The problems are often very hard (even for Maths-graduates), but in more digestible portion sizes, so getting into this might satisfy your desire to do Maths.
  • There are communities of people who enjoy doing Maths and solving puzzles, such as https://www.mathsjam.com/ . I met them in the UK, and it's a fun opportunity to discuss Maths with other people.
  • Find a professor at your (?) university, who would be willing to be your mentor for a project. He might be able to assess what topic would be suitable for your circumstances.
  • Generally, it's hard to follow Academic Maths from the outside. You are missing out on workshops and conferences, and on the personal connections/discussions. The prior is hard to do if you have a job, and don't want to spend all your vacations on them, but I think the key to keeping in touch is to have connections inside Academia who are interested in discussing their research with you. That being said: Most researchers are extremely busy with teaching, getting tenure, writing grants, ..., so that they will most likely only take you on in the role of some kind of student, or collaborator for research (but you would have to convince them that you could contribute).
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I can partially answer this, although this is based on my own experiences which are a bit different than yours. I completed a PhD and then did a few postdocs before transitioning to teaching high school. I stay in touch in part by going to conferences on occasion, going to talks at local universities when they interest me, and regularly looking over the nightly arXiv updates to see what is interesting. I also am continuing to do research which helps also since that means I'm still publishing, and I get requests to referee papers which keeps me more aware of what is going on elsewhere. In my case, I also have a slight additional advantage because the high school I am teaching at has some very advanced students that are mathematically as good as a lot of undergrads, so I can also write papers with some of them. (We have a "Seminar" for mostly seniors and a few juniors where the first semester we cover a topic area like number theory or graph theory, and then the second semester we try to do a research project in that area.) But aside from the details of my own experience, it seems like the main thing one needs to do is to just put in the effort and to make sure to stay involved. How one does so is likely less important than that one does make the effort.

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    $\begingroup$ +1. My situation is very similar; I think the only thing I would add is that a teaching job (as opposed to a job in industry/tech) has the advantage, for this purpose, that work "feels" very different from math research and so the latter can be easily construed as "play." At least for me, I suspect that a job which feels more like mathematical research would make it much harder to continue doing actual mathematical research. $\endgroup$
    – Noah Schweber
    Nov 21 at 23:04
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    $\begingroup$ @NoahSchweber What kind of jobs feel like mathematical research to you? Uh, other than mathematical research. $\endgroup$
    – Nate River
    Nov 21 at 23:35
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    $\begingroup$ @NateRiver So I haven't actually had a job in tech or industry, so I'm guessing here, but in my limited experience programming feels quite a bit like math (especially when looking at machine learning). In general, anything with a heavy problem-solving focus "uses the same sort of energy" - for me at least - as math does, and having a job involving such would make math feel less relaxing. Of course this is all highly subjective and variable; the real import of my comment is that anyone in this position should think seriously about how different activities will impact their experience of math. $\endgroup$
    – Noah Schweber
    Nov 22 at 1:09
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    $\begingroup$ @NoahSchweber As someone who worked in IT for industry, for nearly 30 years, before moving to a research job still in industry, I would say programming is very different from maths. Unless, yes, you do math-intensive things such as machine learning (and do not use machine learning blindly without understanding the maths involved, like many people do). Most programming in the industry is business software, and that uses mainly addition and sometimes multiplication... $\endgroup$
    – Jean-Armand Moroni
    Nov 23 at 9:56
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    $\begingroup$ @Jean-ArmandMoroni “Sometimes multiplication” cracked me up. $\endgroup$
    – Nate River
    Nov 23 at 12:22
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I don't know if it can be done. First, the only way to have any kind of connection with the mathematics "community" (the word is rather over-used) is to do useful work in mathematics. Second, to do useful work in mathematics outside of a day job requires an unusual amount of either talent or dedication.

Somehow, it feels like working on a recreational problem that no research mathematician could possibly care about makes me more a member of the "mathematics community" than trying to force myself into the straitjacket of what happens to be a popular research pursuit nowadays. So much of what drives academia is tied up with money, reputation, and the need to make a living. If you want to stay pure, put mathematics and the enjoyment of mathematics first, and staying in touch with the "community" a distant second.

I'm sorry if this is not what you want to hear or not useful to you. I do recognize your yearning, but I have a feeling it may be misdirected.

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    $\begingroup$ This is the first time I recall someone other than myself commenting that a particular word is overused. But I'm not sure this one is. A community is a group of people who communicate among each other. $\endgroup$
    – Michael Hardy
    Nov 22 at 17:29
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    $\begingroup$ @MichaelHardy Well, maybe I was being too negative. I guess my sense that mathematicians can't really be said to form a community really says more about me than about them. But since this is exactly the kind of question that invites a number of different perspectives, including cynical ones, I'll leave my answer as-is for now. $\endgroup$
    – R.P.
    Nov 22 at 20:15
  • $\begingroup$ Cynical or not,, the advise to work on recreational problems is very good $\endgroup$
    – Vincent
    Nov 29 at 9:45
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Let me answer this from a personal perspective. I have not been in academia since 2004. In two ways I stayed connected to professional maths (in a broad sense):

  1. I wrote a paper in the slow rhythm of like one publication every 3-4 years.

  2. Starting 2010, I participated here on MathOverflow, mostly asking a lot of questions, some rubbish, some appreciated by the community. So far this has been a fun ride!

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Several other answers have focused on mathematical research, but I notice that you did not specifically say anything about research in your question. If you have an interest in mathematics education, there are ways to stay involved without being a professional teacher. For example, if there is a math circle in your area, they will probably welcome qualified volunteers to help. Or your local university may have an undergraduate math club or summer REU that you could volunteer at. Or you could help host a Julia Robinson Math Festival.

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    $\begingroup$ Thank you for the suggestions! I maybe should have specified - I am interested in research but more importantly I would like to have a community to discuss math with in general. $\endgroup$
    – Nate River
    Nov 22 at 15:37
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    $\begingroup$ Research is the most difficult row to hoe, and even if you have a collaborator, successful research usually requires long hours of solitary work. If community is your priority, then I'd recommend getting out there and having as many face-to-face interactions as possible. Find out all the people nearby who have similar interests. Attend large national meetings such as the JMM or MathFest. Face-to-face interaction is always better than virtual interaction as far as community is concerned. $\endgroup$
    – Timothy Chow
    Nov 22 at 15:54
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I think the answers to this question: Advice for researchers outside academia offer some useful insights.

In addition to the useful answers already provided here, I suggest:

Factors to consider:

  • you will need to balance the demands of your job with your desire to work on mathematics. (This is crucial if you're going to be a reliable collaborator.)
  • per the answer I linked to above, getting access to research materials may not be easy.
  • remember that most people in academic life work under a considerable amount of pressure, so you need to exercise care in the demands you make on their time.
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I will take a bit of a pessimistic view.

You can't, at least not really.

Experts' time is valuable. They have several demands on their time already with teaching, research, advising, service, admin, etc. I have several colleagues who I might wait 3 months to get a Zoom call with that lasts a couple hours. During that time it is extremely productive and helpful but it takes months to get in. I know people more senior than me who might be able to get in with those experts a bit sooner but still takes months.

Math is a bit different than other interests because in order to do math (not just learn about it) you must be committed full time. People can recreationally play basketball but you can't recreationally solve open problems - at least not really. The existence of the Wikipedia page on contributions made by amateur mathematicians proves my point - it is so rare that there can be a Wikipedia page on it.

In short. You can learn about math. You can maybe talk to some experts occasionally. But it is highly unlikely you will be able to do math and many experts won't be interested in talking to you.

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    $\begingroup$ Thanks for the contrasting view! For those who disagree, please do not downvote but instead write a comment explaining your disagreement. $\endgroup$
    – Nate River
    Nov 23 at 12:24
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    $\begingroup$ This is not really that much of a contrasting view. By "do math", user479223 means "make a significant original research contribution," and this is indeed very difficult to do as an avocation, especially if you don't already have a lot of experience with mathematical research. But if you're satisfied with other types of mathematical activity, and if you're not concerned about producing original results that are publishable in a research journal, then there are many ways to stay engaged. $\endgroup$
    – Timothy Chow
    Nov 23 at 17:11
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    $\begingroup$ @TimothyChow I also interpret "community" as "community that regularly make significant original research contributions." I think it is extremely hard to be engaged with that community without being in it. $\endgroup$
    – user479223
    Nov 23 at 20:46
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    $\begingroup$ @user479223 Well, I suppose it depends on your definition of "engaged with." If you mean being in frequent communication with, and getting rapid responses from, then even professional researchers aren't always "engaged with" other researchers. Plenty of researchers work mostly in solitude, and even those who collaborate will often be engaged only with the people that they are currently collaborating with on a specific project. That's just the nature of the work. $\endgroup$
    – Timothy Chow
    Nov 24 at 0:44
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    $\begingroup$ (continued) I think it has less to do with whether you are "in" or "out" of "the community," if by that you mean that people will pay attention to you if and only if you are employed by a university. Maybe they'll grant you some initial credibility if you have an academic affiliation, but unless there's some specific research project where it's clear that both parties are contributing substantially, staying engaged with a specific researcher on an ongoing basis is unlikely regardless of your social status. (The exception is that advisors have an obligation to stay engaged with their students.) $\endgroup$
    – Timothy Chow
    Nov 24 at 0:46
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I will add my personal experience, because I think it can add a couple of relevant bits without pretending it to be a full answer: I left academia after a PhD in mathematical physics a bit more than 10 years ago, in part because I realized what I really enjoyed was learning new stuff and not researching (my situation was somewhat different from a typical maths student's in that coming from theoretical physics and having done my thesis in a physics department, my knowledge of undergraduate math outside the areas I needed for my work was limited, to make an example I had no idea of what a projective module is).

So, the bit of advice i would add is: you also have an advantage wrt a professional mathematician, as you can learn any math you find interesting. Had I pursued a career in academia, I would still not know what a projective module is.

If this bit of advice is interesting to you, here a couple of random learnings / things I would say to myself 10 (rather 13...) years ago (in no specific order):

  • Do a plan, buy the books and be disciplined about the time you are able to allot;

  • (This should be obvious): do a lot of exercises;

  • Be disciplined does not mean becoming obsessed: it's your hobby now, not your profession. Your math session should be something you are looking towards during the week;

  • Sticking to the plan will be difficult, almost impossible. I found out that learning new things becomes more difficult as you age, on the other hand you will be compensating with mathematical maturity and experience. This is not a problem because, well, it's your hobby now, not your profession;

  • Feel free to change your plan: make a detour, take a pause to learn something else you met an found interesting, ... it's your hobby now, not you profession;

  • Learning purely from books (especially math books) is extremely challenging: there are now lots of very good lectures online - go to youtube, search for your interest area and often you will find something excellent (here one needs to mention Prof. Borcherds' channel, a real gem). Also a lot of math departments put material online (not necessarily on youtube).

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  • Pick up a hobby which will help you meet and become friends with mathematicians. This could be chess or go, or it could be volunteering in math popularisation events or writing articles for math popularisation journals. I play the game of go and I've met many mathematicians during go tournaments, and now I'm friends with them. Although there is an immense knowledge gap because they study advanced topics professionally every day and I don't, we still have interesting conversations about mathematics on a regular basis.

  • Subscribe to mailing-lists for mathematicians. There exist many mailing lists for researchers, organised by topic. As a result of my subscriptions I receive invitation links to interesting webinars where mathematicians present their research. Depending on the topic and target audience, these webinars range from super interesting to absolutely unintelligible, but you can usually guess which is which just from their title. Warning: you also need to set up filters in your mailbox to avoid wasting time on all the irrelevant emails you might receive, like advertisement for tenure tracks and post-docs.

  • If you like teaching and your schedule allows it, teach! Many universities will be happy to employ you to teach even just one class per week even if you're not a professional mathematician. You don't need to be a professional researcher to be able to teach what's an increasing or a decreasing function to sophomores. But it is crucial that you love math and enjoy teaching.

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You can pay to take course work you are interested in just for your own sake and learning, as un-matriculated which means you are not enrolled in a degree program usually.

Also you can request to audit courses. These things allow you to kind of still learn as a student as you please.

Some schools may have math clubs and extracurricular things. Like group theory club or various things.

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