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P. Zambrano (Bogota): Hrushovski Constructions and non-elementary classes

Freitag, 17.01.2014 10:45 im Raum 120.029
Mathematik und Informatik

B. Zilber conjectured that strongly minimal aleph_1 categorical sets are bi-interpretable with either a set without structure, or a linear space, or an algebraic closed field of a fixed characteristic. But E. Hrushovski gave an counterexample to this conjecture, constructing a new strongly minimal aleph_1-categorical set which is not bi-interpretable with any of the kind of structures given above (see [2]). The notion of Abstract Elementary Class (AEC, see [1]) corresponds to a generalization of the study of first order elementary classes together with the elementary substructure relation, in order to do a suitable model-theoretical study of non first order elementary classes of structures. Some examples of Hrushovski constructions fit in this setting, e.g. Zilber's pseudo-exponential fields [5] and Hrushovski fusions [2]. In this talk, we will talk about Hrushovski fusions as AECs [3,4]. References. [1] R. Grossberg. Classification theory for abstract elementary classes. Logic and Algebra, ed. Yi Zhang, Contemporary Mathematics, 302: 165--204, AMS, 2002. [2] E. Hrushovski. Strongly minimal expansions of algebraically closed fields. Israel J. of Math. 79: 129-151, 1992. [3] A. Villaveces and P. Zambrano. Hrushovski Constructions and Tame Abstract Elementary Classes. Preprint, 2009. Available at http://tinyurl.com/luzgs7s [4] P. Zambrano. Hrushovski constructions in non-elementary classes. Bol. Mat. 16(1): 33-56, 2009. [5] B. Zilber. Pseudo-exponentiation on algebraically closed fields of characteristic zero. Ann. Pure Appl. Logic, 132(1):67-95, 2005.



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