Classification theory for abstract elementary classes

July 17 to July 21, 2006

at the

American Institute of Mathematics, Palo Alto, California

organized by

Rami Grossberg and Monica VanDieren

This workshop, sponsored by AIM and the NSF, 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). 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(Q) and generalized the categoricity conjecture:

Conjecture 2 (Categroricity 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.

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.

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