Sometimes, people state a theory that they haven’t proven – most famously, Fermat’s Last Theorem. They might work on this to solve it.
There are “problems” in which we can predict what the answer can be, but we haven’t conclusively proven it yet.
They might also take problems or demonstrations from more [practical examples](https://youtu.be/rXfKWIZQIo4?si=x-1fb13y8xBE5_x1), or applied fields like physics, and work out the maths for them.
There’ll also be a lot of teaching, marking, writing, supervision, grant applications etc. going on.
I really like the channel [Numberphile](https://youtu.be/d8TRcZklX_Q?si=GwQwk_C3wIBbYwMx) if you want to find out more about both serious and more whimsical stuff – a lot of it just seems “ooh cool!”, but may have practical applications in the future.
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