Ax-Kochen/Ershov principles in positive characteristic

A major open problem in the model theory of valued fields is to gain an understanding of the first-order theory of the power series field F((t)), where F denotes a finite field. For sufficiently "nice" henselian valued fields, the Ax-Kochen/Ershov philosophy allows to reduce questions of elementary equivalence and elementary embeddings to the analogous questions about the value group and residue field (or related structures). In my talk, I will present a new such principle which applies in particular to a large class of algebraic extensions of F((t)), albeit not to F((t)) itself. The talk is based on joint work with Konstantinos Kartas and Jonas van der Schaaf.