Model Theory of Valued fields

These notes focus mainly on the model theory of algebraically closed valued fields (loosely referred to as ACVF). This subject begins with work by A. Robinson in the 1950s (see the proof of model completeness of algebraically closed valued fields in [41]). Thus, it predates the major work of Ax-Kochen and Ershov around 1963; and, unlike the latter (and much subsequent work on quantifier elimination for henselian fields), the model theory of algebraically closed valued fields is also well understood in residue characteristic p. The viewpoint given is motivated by ideas from stability and o-minimality. My main purpose is to make more accessible the background to some recent work of Hrushovski and co-authors – see for example [19], [20], [21]. The bulk of the material for these notes is from [14], [15]. There is considerable overlap with some notes of David Lippel and myself, for a similar tutorial series for the workshop ‘An Introduction to Recent Applications of Model Theory’ at the Newton Institute, Cambridge, March 2005. The context algebraically closed valued fields may seem rather specific. However, there is a geometric viewpoint whereby an algebraically closed valued field is a universal domain for an arbitrary valued field (of appropriate characteristics). Model-theoretic results on ACVF will yield information on quantifier-free definable sets in any valued field.