What about tetration with arbitrary real or complex bases and exponents, or is that too fringe?
Anyway, if mathematicians indeed have less to do, perhaps they could start working on standardizing tau over pi, to make radian angles less confusing for everyone.
I'd say it's kind of a cool topic and would probably lead to some interesting recreational math but it probably won't lead to new, significant research, which means a new theoretical understanding of some abstract structure or another.
Anyway, if mathematicians indeed have less to do, perhaps they could start working on standardizing tau over pi, to make radian angles less confusing for everyone.