An Isaac Newton Institute Workshop

An Introduction to Recent Applications of Model Theory

Analytic Zariski Geometries and the Hrushovski Collapse

Authors: Assaf Hasson (Oxford University), Boris Zilber (Oxford University)

Abstract

In recent years much progress has been made along the 'Zilber Programme' intended to explain Hrushovski-type constructions in terms of analytic geometry. These results, however, have only dealt with Hrushovski-type structures of infinite rank. we will show that considering appropriate expansions of the infinite rank ab initio structrue, Hrushovski's strongly minimal sets can be obtained as infinitesimal neighborhoods (in terms of specializations) of carefully enough chosen points. We will also show that these results can be extended to other Hrushovski-type structres, most notably to ones supporting a structure of an algebraically closed field.

If time allows we will discuss briefly: (a) The obstacles in generalizing these results to more complicated structures (e.g. the fusion of two strongly minimal Zariski geometries). (b) The nature of a possible analytic prototype in which our interpretation of the collapse could be described in analytic terms.