I am an amateur mathematician. I have trained myself on math-related websites and other resources. However, the lack of technical knowledge, methods, and sociological culture usually shared by professional mathematicians has prevented me from going further and publishing rigorous and interesting results.
Many amateurs, often outside areas where universities are easy to reach, face the same hurdles. I know a lot of them are cranks but not all of them are: some have extremely good computational skills, an extremely original vision of maths and sometimes a lot of time on their hands.
What can be done to bridge this gap between amateurs and professionals? How can we give amateurs the codes and techniques needed to enrich the corpus of mathematical results with new, fresh and interesting contributions?
For example, could retired professions create a free resource that links universities and labs to amateurs? In this way, the retired professionals could act as "pre-referees," teaching the amateurs how to write a paper, and collaborating on the research itself? There could even be a magazine in which publications resulting from this are published.