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

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