Some ideas in need of clarification in resolution of singularities and the geometry of discriminants

Abstract

This text, which appeared in volume 2313 of the Springer Lecture Notes in Math. dedicated to Catriona Byrne, presents problems in two different areas of algebraic geometry. The first concerns the role of ``infinitely singular'' or ``non Abhyankar'' valuations in the study of local uniformization of valuations with a view to resolution of singularities in positive characteristic. The second one concerns the relationship of the geometry of the discriminant of real miniversal unfoldings in the sense of Thom with the movements of Morse functions on a cobordism which differential geometers use.