Logic Seminar - Isabel Muller - "Zil'bers Conjecture and Building-like Geometries"

Wed, 29/11/201711:00-13:00

In an attempt to classify the geometries arising in strongly minimal sets, Zil'ber conjectured them to split into three different types: Trivial geometries, vector space-like geometries and field-like geometries. Soon after, Hrushovski refuted this conjecture while introducing a new construction method, which has been modified and used a lot ever since. His counterexample to this trichotomy conjecture of Zil'ber was not one-based, whence it could neither carry a trivial geometry or one of vector space type, but on the other hand it forbade a specific point-line-plane configuration, which is always present in fields. Hrushovski called that property being CM-trivial. Later Pillay, with modifications by Evans, defined a whole hierarchy of new geometries, the ample hierarchy, on which base we find the non-one based structures (called 1-ample) and the non-CM-trivial ones (called 2-ample), while on the very top there are the fields, being n-ample for any natural number n.

Recently, Baudisch, Pizarro and Ziegler, and independently Tent, provided examples which prove that this ample hierarchy is indeed strict. While their examples were omega stable of infinite rank, it had remained open whether there are strongly minimal theories, which are at least 2-ample, but do not interpret a field. Under the supervision of Tent, we constructed a strongly minimal theory which is strictly 2-ample, using a modified Hrushovski construction method combined with ideas from the theory of buildings. 

In this talk, we will give an overview of Zil'bers trichotomy conjecture and its counter examples, outlining how our new geometry naturally plugs into the picture of existing counter examples and gives hope to be generalized for a construction of strictly n-ample, strongly minimal structures for arbitrary n.