How to identify future directions in a saturated field without getting discouraged
If you browse through top 10 schools (or researchers') websites, you might get a "feeling" on where (or what) their research is heading towards! For example, you can check what their newest post docs are working on, PhD students' thesis titles etc. You can also do the same thing by checking NSF newly funded proposals. This may not give you a definite answer, but it might help you establish a "directional sense"

PS.
I'm in engineering not math.