The FAI problem needs people doing math.
I think there are various biological problems where the math is underdeveloped. Physicists are as a community quite good at math but in biology there aren't that as people working on the underlying math.
Problems such as calculating cancer survival rates could benefit a lot from better statistical tools. Depending on what you mean with pure math statistics might not be included, but statistics matters.
A high school student with an interest in math asks whether he's obligated on utilitarian grounds to become a doctor.
The commenters pretty much say that he isn't, but now I'm wondering-- if you go into reasonably pure math, what areas or specific problems would be most likely to contribute the most towards saving lives?