TITLE: Advice for researchers outside academia
QUESTION [34 upvotes]: 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.

REPLY [3 votes]: 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.