Most of the advanced math in economics in current research is either in econometrics or game theory. Not saying econ is math-less, far from it, but most of the time we can't simply solve our problems by applying advanced math like physics can.
Equilibrium concepts in game theory are a tough thing. The holy grail is still getting a unifying concept of a "stable equilibrium" (see Kohlber & Mertens '86) which is pretty much a guaranteed Nobel (but I'm not sure it even exists, so many geniuses worked on the problem without success).
Equilibrium concepts in game theory are a tough thing. The holy grail is still getting a unifying concept of a "stable equilibrium" (see Kohlber & Mertens '86) which is pretty much a guaranteed Nobel (but I'm not sure it even exists, so many geniuses worked on the problem without success).