Many of you are passionate about mathematics and study it as amateurs. Problems are one possible starting point, often leading to open questions or interesting generalizations.
Others have a favorite subject, sometimes carried over from their student days, which they explore in depth—first through popular mathematics books, then through graduate textbooks and research papers.
By amateurs, we mean non-professionals: people who do not belong to a research organization and are not paid for their research. They are therefore free to explore the subjects they enjoy, at whatever pace suits them, without the pressure of reporting requirements, deadlines, or publication targets. The downside of having no official status is isolation, more limited access to publications and information, and the near impossibility of communicating their results or having them validated.
We should also mention the "semi-amateurs" who come from the research world, are often retired, and decide to use their free time to pursue research of their own. They already have the contacts, expertise, and access to publications.
Finding a topic --------------------