Classification theory for abstract elementary classes

July 17 to July 21, 2006

at the

American Institute of Mathematics, San Jose, California

organized by

Rami Grossberg and Monica VanDieren

Original Announcement

This workshop will focus on Shelah's categoricity conjecture for abstract elementary classes. Thirty years ago Saharon Shelah proposed a far reaching program of extending first-order classification theory for non-elementary classes. He proposed a conjecture as a test-case for measure of development of the theory:

Conjecture 1 (Categoricity for Lω1,ω). Let ψ be a sentence. If ψ is categorical in a cardinal >בω1 then ψ is categorical in all cardinals >בω1.

A few years later Shelah introduced the notion of Abstract Elementary Class (AEC) which is a semantic generalization of Lω1,ω(Q) and generalized the categoricity conjecture:

Conjecture 2 (Categoricity for AECs). Let K be an AEC. There exists a cardinal μ(K) such that categoricity in a cardinal greater than μ(K) implies categoricity in all cardinals greater than μ(K). Furthemore, μ(K) is conjectured to be the Hanf number of K.

Despite significant partial results by several people, even Conjecture 1 is still open. In recent years much progress was made on different aspects of Shelah's original program and several intriguing connections with classical fields of mathematics were discovered. This workshop will be dedicated to discussion of the present state of the conjectures as well as the broader program of developing a classification theory for AECs. Also several examples and applications will be discussed.

Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

Talk by Meeri Kesala.