アブストラクト：
We will present some applications of set theoretic techniques to the study of models of
cardinality aleph_1 in various abstract elementary classes. The impetus for this work was the still-open
question of absoluteness of aleph_1-categoricity for the class of models of a fixed sentence of L_{omega_1, omega}.
Most of our results apply to the class of analytically presented AEC's, those whose restriction to countable models is analytic.
These classes have been studied for many years under various names. Time permitting, connections with Vaught's
Conjecture will also be discussed. This is joint work with Baldwin and Shelah.
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