I'm going to be applying to grad schools next year (I have an undergrad degree in math and phyisics), and I have narrowed down my areas of interest to two fields: functional analysis and it's involvement in QFT; and computational/theoretical neuroscience. I find pure math more enjoyable, but I'm concerned about job prospects. The specific area of math I'm interested in doesn't seem to be that popular. Unless I'm blind, there appears to be very few mathematicians working in c* algebras and operator theory. There are many in Europe (in both math and physics) but not in America. Is there a reason for this? Is it career suicide to go into this field? I would think with the yang mills mass gap problem still unanswered, the field would be more populated. The "safe" option is to scrap math and go into computational neuroscience. I've always been fascinated with the brain and see wonderful things happening once the brain is understood completely. Computational neuroscience has at least some mathematics involved in it, especially statistics. MIT has a statistical neuroscience group that works in this area. There is also people like William Bialek and Michael Berry at Princeton who use a lot of math in their theoretical models of the brain. One positive aspect of neuroscience is that getting into a grad program at a top school would be much much easier than in math. I can actually consider applying to MIT, Harvard, Caltech etc... which would be kind of refreshing. Any advice? Are there any other fields out there that are "in demand" but use a lot of mathematics? Are there any areas of pure math that fit this description? Any areas of theoretical physics that are not overly saturated (quantum information?)?