Tame valued fields, partial quantifier elimination, and NIP transfer

Work of Kuhlmann and coauthors has established AKE principles for tame and separably tame valued fields, extending for example the work of Delon on the narrower class of algebraically (or separable-algebraically) maximal Kaplansky valued fields. These principles, and their underlying methods, have had striking applications, for example to existential theories of henselian valued fields, the transfer of NIP from residue field to valued field, and the recent work of Jahnke and Kartas on theories of perfectoid fields. The "Generalized Stability Theorem" is even an ingredient in Temkin's inseparable local uniformization. In this talk I want to explain some extensions of the known AKE principles, and related partial results on relative quantifier elimination, all in various special cases. This includes work joint with Boissonneau, and work of Soto Moreno.