Logic seminar - Omer Mermelstein - "Template structures for the class of Hrushovski ab initio geometries"

Wed, 13/12/201711:00-13:00
Math 209
Zilber's trichotomy conjecture, in modern formulation, distinguishes three flavours of geometries of strongly minimal sets --- disintegrated/trivial, modular, and the geometry of an ACF. Each of these three flavours has a classic ``template'' --- a set with no structure, a projective space over a prime field, and an algebraically closed field, respectively. The class of ab initio constructions with which Hrushovski refuted the conjecture features a new flavour of geometries --- non-modular, yet prohibiting any algebraic structure.
The purpose of this talk is to propose ``template'' structures for the class of (CM-trivial) ab initio Hrushovski constructions. After presenting the standard construction, by generalizing Hrushovski's predimension function, we show that the \emph{geometries} associated to certain Hrushovski constructions are ab initio constructions themselves. If time permits, we elaborate on how these template structures may generate the class of geometries of ab initio constructions under the Hrushovski fusion operation.