Which mathematics are done? I was wondering what fields of mathematics are no longer actively researched. I would imagine algebra (not the abstract kind) and arithmetic would be two (and ironically they are the most useful now.) Or aren't any done yet. Are all fields being expanded?

A field is "done" when all of its questions are "done."
A question is "done" when everyone is "satisfied by the answers found."
While I think that many fields have famous interesting questions which may be considered "done" because they have satisfactory answers, I think it's almost never the case that all of the questions are "done" in a field. Usually, while you are doing research on a question, lots of questions arise along the way. And then when you finally answer your question, it immediately suggests new questions.
The importance of questions depends on the interest people have in them, and so part of a field being "alive" is whether the topic is "in style" or not. That is going to fluctuate as time passes!
