Geometric stability theory in non-elementary classes

Non-elementary model theory studies classes of structures which cannot  be axiomatized in first order l
ogic but to which some model-theoretic  conventions might be applicable. Such classes can be for
example  classes definable in some formal language extending first order logic.

Geometric stability theory is known to be one of the most resourceful  sources of tools for the applications
of model theory. Generalizing  these tools to non-elementary frameworks does not only widen the scope 
of applications but also gives new ways to understand the existing  concepts and conventions.

We will concentrate to the framework of finitary abstract elementary  classes and generalize to this
framework a 1980's result by Hrushovski  about interpreting groups and fields acting on the geometry
induced on  the realizations of a regular type. This is joint work with Tapani  Hyttinen.