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.