The model theory of valued fields has been extensively studied after, in 1965, the work of Ax-Kochen and Ershov had found remarkable applications in number theory.
To a valued field the value group (an ordered abelian group) and the residue field are naturally associated. The theorem of Ax-Kochen and Ershov can be stated as follows: the theory of henselian valued fields of residue characteristic 0 is complete relative to the value group and the residue field.

Model theorists have later asked what model theoretical properties of henselian valued fields can, as completeness, be understood at the level of the value group and the residue field. I will focus in particular on the problem of quantifier elimination.

It turns out that the statement “the theory of henselian valued fields of residue characteristic 0 admits quantifier elimination relative to the value group and the residue field” is in general false. In this setting, the value group and the residue field do not always carry enough information about the original valued field. As a consequence, other structures associated to valued fields have been considered. In 2011 Flenner obtains quantifier elimination for henselian valued fields of residue characteristic 0 relative to the RV-structures which he introduced with this name,
while developing some less recent ideas of other authors such as Basarab and F.-V. Kuhlmann.

During the talk, while discussing with more details the problem of quantifier elimination for henselian valued fields, I will also argue that RV-structures are, in essence, the same structures that M. Krasner studied in 1957, not in relation to the model theory of valued fields, and that led him to the definition of his hyperfields.

Everyone is welcome and encourage to attend.