Analytic Zariski structures, predimensions and non-elementary stability

The notion of an analytic Zariski structure was introduced in [1] by the author and N.Peatfield in a form slightly different from the one presented here. Analytic Zariski generalises the previously known notion of a Zariski structure (see [2] for one-dimensional case and [3], [4] for the general definition) mainly by dropping the requirement of Noetherianity and weakening the assumptions on the projections of closed sets. This leads to interesting new fenomena, in particular, the family of closed-in-open subsets forms a hierarchy which starts with analytic sets and continues by induction to more complex ones, called in [5] generalised analytic sets (defined classically on the complex numbers and in the context of rigid analytic geometry). In [1] we assumed that the Zariski structure is compact (or compactifiable), here we drop this assumption (some interesting structures are not compactifiable in the strict sense). The class of analytic Zariski structures is much broader and geometrically more rich than the class of Noetherian Zariski structures. The main examples come from two sources: (i) structures which are constructed in terms of complex analytic functions and relations; (ii) “new stable structures” introduced by Hrushovski’s construction; in many cases these objects exhibit properties similar to those of class (i). Although there are concrete examples for both (i) and (ii), in many cases we can only conjecturally identify a particular structure as an analytic Zariski one. In particular despite some attempts the conjecture that Cexp is analytic Zariski is still open (even assuming this is the same as pseudo-