Series: Penn State Logic Seminar Date: Tuesday, March 19, 2002 Time: 2:30 - 3:45 PM Place: 113 McAllister Building Speaker: Monica Van Dieren, Carnegie Mellon University, Mathematics Title: Towards a Version of Morley's Theorem for Abstract Elementary Classes Abstract: Morley and Shelah's solutions to Los' Categoricity Conjecture shaped modern day first-order model theory through the introduction of such concepts as prime models, superstable theories and rank functions. In the mid-seventies, Shelah posed an analog of Morley's Theorem for $L_{\omega_1,\omega}$. Later this conjecture was generalized to a categoricity conjecture for abstract elementary classes (AECs), namely Shelah's Categoricity Conjecture. This conjecture serves as the main test question for the development of a model theory for non-first order logics (particularly for AECs). Similar to a solution to Los' Conjecture, a solution to Shelah's Categoricity Conjecture may provide the machinery necessary for the development of a rich model theory for AECs. Despite over 500 pages of partial results, Shelah's Categoricity Conjecture remains open. One of the central concepts that arises in partial solutions towards the conjecture is the amalagamation property. In this talk I will introduce abstract elementary classes, discuss the history of the categoricity conjecture with respect to the amalgamation property and briefly discuss my solution to a related conjecture of Shelah and Villaveces.