First off, I have no interest in being a mathematician. Math was always and continues to be quite difficult for me.
So, as an outsider to advanced math, it blows my mind that there are people who’s entire job title is mathematician. How does that work? What does a mathematician do?
Pure mathematicians often answer questions that really only other pure mathematicians care about, but occasionally their results or techniques have relevance in other fields, so universities will pay them to work on this stuff and publish papers. Usually part of the job is applying for grants to fund your research and teaching students.