(Joint work with Jochen Koenigsmann.) There has been a lot of recent progress in the area of definable henselian valuations. Here, a valuation is called definable if its valuation ring is a first-order definable subset of the field in the language of rings. Applications of results concerning definable henselian valuations typically include showing decidability of the theory of a field or facts about its absolute Galois group.

We study the question of which henselian fields admit definable henselian valuations with and without parameters. In equicharacteristic 0, we give a complete characterization of henselian fields admitting parameter-definable (non-trivial) henselian valuations. We also give a partial characterization result for the parameter-free case.

We prove a number of results around finding strongly normal extensions of a differential field K, sometimes with prescribed properties, when the constants of K are not necessarily algebraically closed. The general yoga of interpretations and definable groupoids is used (in place of the Tannakian formalism in the linear case).

This is joint work with M. Kamensky.