Zariski structures and simple theories

Abstract

In this thesis, I consider generalisations of geometric stability theory to minimal Lascar Strong Types definable in simple theories. Positively, we show that the conditions of linearity and 1-basedness are equivalent for such types. Negatively, we construct an example which is locally modular but not affine using a generalistion of the generic predicate. We obtain reducibility results leading to a proof that in any w-categorical, 1-based non-trivial simple theory a vector space over a finite field is interpretable and I prove natural generalisations of some of the above results for regular types. I then consider some of these ideas in the context of the conjectured non-finite axiomatisability of any w-categorical simple theory. In the non-linear Zariski structure context, I consider Zilber's axiomatization in stable examples, and then in the case of the simple theory given by an algebraically closed field with a generic predicate. Comparing Zariski structure methods with corresponding techniques in algebraic geometry, I show the notions of etale morphism and unramified Zariski cover essentially coincide for smooth algebraic varieties, show the equivalence of branching number and multiplicity in the case of smooth projective curves and give a proof of defining tangency for curves using multiplicities. Finally, I give a partial results in the model theory of fields which supports extending the Zariski structure method to simple theories.