Motivated by some problems of theories of valued fields and algebraically closed valued fields (ACVF), the book presents a model-theory approach combining stability-theory ideas and the o-minimal context because “type spaces work best in stable theories and definable sets and maps in o-minimal theories”. The abstract section, developed in Part I, is applied in Part II (in particular to ACVF). In Part I, an extension of stability theory is introduced. Theories that have a stable part are considered and the crucial notion of a stable dominated type is defined. Such a type can be controlled by a very small part, lying in the stable part, analogically to how a power series is controlled with respect to the problem of invertibility, e.g. by its constant coefficients. Moreover, it is shown that there exist o-minimal families of stably dominated types with the property that any type can be seen as a limit of such a family. Furthermore, a notion of metastable theory is defined. In Part II, the theory of ACVF is studied; valued fields are viewed as substructures of models of ACVF. Notions of independence are introduced and applied. For instance, the metastability of ACFV is proved and the stable dominated types are characterised as those invariant types that are orthogonal to the value group. Notice that the key notions and ideas of stability theory are presented in chapter 2 and chapter 7 is an outline of classical results in the model theory of valued fields. The book is comprehensive and stimulating.

Reviewer:

jmlc