In these notes, I present a new, strong notion of stratifications which describe singularities of sets in Henselian valued fields. In the first part, I give the definition, some examples, and I state the main result about their existence; in the second part, I explain how such a stratification in a valued field induces a classical Whitney stratification in R.

We extend the characterization of extremal valued fields given in [1] to the missing case of valued fields of mixed characteristic with perfect residue field. This leads to a complete characterization of the tame valued fields that are extremal. The key to the proof is a model theoretic result about tame valued fields in mixed characteristic. Further, we prove that in an extremal valued field o...