On the complexity of the bifurcation sets of critical points of smooth functions

A complicated critical point of a real function can be morsified in many topologically distinct ways. I will promote a combinatorial algorithm enumerating such Morsifications. The upper estimates of the complexity of this algorithm (in particular, a proof of its finiteness) follow from the estimates of the local degrees of certain bifurcation sets related with our singularity, such as the caustic, the Maxwell set, and the Stokes' sets. I will demonstrate some such estimates; some of them are nearly sharp, and the other ones probably can be improved a lot.