Advice for researchers outside academia Perhaps some personal background is relevant to this question. A couple of years ago, I graduated with a master's degree in Applied Mathematics from a good Dutch university. Even though I obtained somewhat decent grades for the courses and the thesis (a 7/10 on average, maybe a B in the American system), I sensed that I probably wasn't good enough to pursue a PhD in a relevant mathematical research direction.
Therefore, I opted for a career in a different area. However, I keep coming back to mathematics over and over again. I am still drawn to some mathematical problems, and I work on them in my spare time. One day, I'd like to be a (co)author of (at least) one publication.
I've now come at a point that I think I have some interesting ideas for a certain problem. In order to flesh them out, I think it would be a good idea to collaborate with someone who has more expertise than me in the relevant domain. I've found a postdoc at my former institution, whom I suspect has the appropriate background to work on this problem with me.
I'm not quite sure, though, how to approach this person. I've never met him/her, and it seems somewhat odd or inappropriate to send him/her an e-mail out of the blue. I've thought of proposing to give a short presentation of my ideas to this person, after which s/he can decide whether the approach seems worthwhile enough to delve into more deeply.
This -- finding and approaching trustworthy fellow mathematicians who have the time and willingness to collaborate -- is one particular problem I've encountered while working as a researcher outside academia. As an “independent researcher”, if you will, I encounter all sorts of other problems, including:

*

*Getting access to some journals, books and research papers;

*Obtaining licenses for expensive proprietary software;

I wonder whether you have any advice to independent researchers on these matters, and whether you can perhaps give some general guidelines or nuggets of wisdom to those working outside academia.
 A: One point (mentioned in parenthesis by Alexandre Eremenko, but worth highlighting) is that since this postdoc is at your former institution, there's a chance there are professors there who remember you. (I myself is a professor who is terrible with names of people I meet in real life, but there are those who seem to remember everyone.) If so, that could be another way "in".
One way to play it could simply be to add a professor as a Cc: in your email to the postdoc, as a way of saying "BTW Prof, this is what I'm up to these days" without requesting a response on that part. Furthermore that does give the postdoc a chance to ask the professor about you before responding, should (s)he feel that advisable.
Another way to go ahead, especially if the postdoc you found ends up not being interested in your ideas (this happens; what research one wants to engage in is as much a matter of personal taste as one's choice of music), is to explore the possibility of you giving a talk at your old institution; maybe there is someone else there who would be interested. How feasible this is depends on the character of the institution; in a large department with research seminars every day it might be hard for a hobbyist to squeeze in, but in a smaller department (or one which for some reason doesn't have that many seminars going on) it can be easier to find an opening. Being an alumnus (even if at a low level) is a plus for you — many professors find it somewhat novel to hear of a former student who isn't himself a math professor doing math anyway. Talks by alumni addressing current students is also its own genre in academia, which can be given better resources than research activities.
One matter that can be tricky is knowing where to put the emphasis in your talk, when you do not know your audience. (Some things they all know and do not need to be reminded of, other things could be elementary to you but novelties to them.) If you get as far as being invited to give a talk then you may however be able to get the host to comment on an outline for your planned talk, if you ask nicely. They usually want the talk to be interesting as well.
A: Contact out of the blue; the postdoc will be proud to have been noticed. But remember that postdocs are in the tightest bottleneck of their lives, having to produce research of high quality quickly in order to get the next job right now. If the postdoc does not see how to get a paper out of your idea immediately, the postdoc should reject the opportunity for collaboration without wasting your time or hers. Do not be offended; starting a research collaboration with a new collaborator can take a lot of time, just to learn how to work together, even if both are very experienced in their fields. Many very top notch researchers have been rejected when they have asked to collaborate with even far less reknowned researchers who were too busy.
You will need to have a collaborator within the academic world, to make certain that your results are not already known, and to get access to academic journals. Obtaining that collaborator is the only problem you have to face before you can really join the international mathematical conversation.
A: I think it's a reasonable question. The quick answer is that there's nothing wrong with emailing someone you think might want to work with you.  The first email should be brief, polite, and give a very short explanation of who you are.  ("I graduated with a master's from X university in 201X and continue to have an interest in subject X.") Then it should give an again, very brief description of the problem and what you hope to achieve.
To your other question, I myself left academia eight years ago because of a two-body problem, and the biggest surprise for me was that MathSciNet was no longer available to me.  Not that an individual subscription would cost an exorbitant amount --- I was directly told by the AMS that if I am not affiliated to an institution, there is no way to subscribe, period. The best recommendation I can give you here is to look for SciHub, if I am allowed to say the name directly.
A: I agree with the other two answers.  I'm an independent researcher myself.  Like you I only have a Master's degree.  I've been lucky to coordinate with another coauthor in academia.  I might not have been able to continue publishing if it weren't for that connection.
So, yes, contact this person out of the blue.  The worst thing that can happen is that they'll tell you no.
A: The question of how to contact a person you were never introduced to, is not a question about mathematical research, so it is out of place here. (Perhaps you may contact some of your professors of fellow students with whom you studied, and who do research in mathematics, all depends on individual circumstances).
I will address the second part: how to get access to books and journals. Of course, this is absolutely necessary for study and research.

*

*There is arXiv. www.arxiv.org. It is a free resource, and more and more people
upload their results to this place. I believe that a majority of mathematicians already do this, and want to use this opportunity to call all the rest to join this majority. This is the only remedy against ongoing "privatization of mathematical knowledge" by big corporations.


*You should try to get access to some good university library. This depends on a country of course. In principle, university libraries are supposed to be accessible to the public. You enter, you browse, and you can copy whatever you want, perhaps paying some charge for copying.
Unfortunately this free access is curtailed more and more with years, and especially
bad situation is now, because of the epidemic. Paper subscriptions are eliminated,
and for electronic access you need a university ID. Unfortunately, this is a worldwide trend now. The only thing I can advise in this connection: contact someone in your university (where you studied, or some other university) and ask whether you, as a former student, or as an independent researcher, can have
an access to their electronic library resources. Some libraries may have
computers on their premises which allow you access to their resources without an ID.


*There are many books and journals on Internet with free access. And finally there are "pirate sites". I do not recommend any particular one, because some of them may be illegal, in some countries. But finding them is not a big problem.
They have very many books and journals for download.

