| Résume | Model theory has a lot of applications to the study of valued fields, beginning with the celebrated Ax-Kochen-Ershov Theorem, proved in the 1960s. It relates the first order theory of a Henselian, equicharacteristic 0 valued field to the theories of its value group and residue field. Since then, many generalisations has been made with a similar philosophy.
In this talk, we will introduce the AKE principle, go over the key ideas of the classical proof, and finally, look at some new developments in this area. In 2023, Anscombe, Dittmann and Jahnke proved a version of the AKE theorem for mixed characteristic, finitely ramified fields. We will talk about the the challenges of this generalisation, and how this is solved by enriching the residue field with an additional structure. |
