This paper ties together much of the model theory of the last 50 years. Shelah's attempts to generalize the Morley theorem beyond first order logic led to the notion of excellence, which is a key to the structure theory of uncountable models. The notion of Abstract Elementary Class arose naturally in attempting to prove the categoricity theorem for L ω 1 ,ω (Q). More recently, Zilber has attempted to identify canonical mathematical structures as those whose theory (in an appropriate logic) (...) is categorical in all powers. Zilber's trichotomy conjecture for first order categorical structures was refuted by Hrushovski, by the introducion of a special kind of Abstract Elementary Class. Zilber uses a powerful and essentailly infinitary variant on these techniques to investigate complex exponentiation. This not only demonstrates the relevance of Shelah's model theoretic investigations to mainstream mathematics but produces new results and conjectures in algebraic geometry. (shrink)

A large homogeneous model M is strongly minimal, if any definable subset is either bounded or has bounded complement. In this case is a pregeometry, where bcl denotes the bounded closure operation. In this paper, we show that if M is a large homogeneous strongly minimal structure and is non-trivial and locally modular, then M interprets a group. In addition, we give a description of such groups.

We suggest a method of finding a notion of type to abstract elementary classes and determine under what assumption on these types the class has a well-behaved homogeneous and universal "monster" model, where homogeneous and universal are defined relative to our notion of type.