Everyone here is talking about applied fields in industry, but there’s also plenty of research mathematicians who don’t work in any of these fields, who simply further the study of mathematics in universities or research centers.
At the core, mathematicians prove conjectures and develop theories. I know that sounds vague, but mathematics is such a massive subject that you really can’t narrow it down much further.
For reference, here’s a (far from complete) [list of unsolved problems in mathematics](https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics). Here’s the [Millennium Prize problems](https://en.wikipedia.org/wiki/Millennium_Prize_Problems), which will net you a million dollars if you manage to resolve one of them. Mathematicians will explore techniques and directions of research in problems like these, making incremental progress, proving weaker versions, noticing connections with other parts of math, and occasionally having breakthroughs that result in a proof.
This may seem unmotivated and pointlessly academic, but actually as fields like physics and computer science grow, they continue to take advantage of more and more recent mathematical theories and fields. Not that the mathematicians working on new theories necessarily care about its applications, but today’s mathematics could easily be essential for tomorrow’s physics.
Other than this, research mathematicians will likely teach at least some courses, give some talks, and a few will author textbooks.
Latest Answers