# Adi Jarden: Non-Forking as a Tool for Classification of AECs

Infinite Combinatorics Seminar (BIU)

14/June/2015, 14:00-16:00,
Room 201, Building 216, Bar-Ilan University

Title: Non-Forking as a Tool for Classification of AECs

Abstract: Abstract Elementary Classes (AECs in short) are classes of models for a fixed vocabulary with a partial order $\preceq$, satisfying some properties of elementary classes (classes of first order theories with the relation of being an elementary submodel). While many results about the classification of elementary classes are well-known, about AECs much less is known.

The main tool for classification of elementary classes is the non-forking’ notion. Using this tool, Shelah proved the categoricity conjecture for first order theories. The study of non-forking notions in AECs is the main issue of Shelah’s book on non-elementary classes of models.

In the following two talks, we will discuss non-forking notions:

1. examples of non-forking notions
2. the properties of a non-forking notion
3. a non-forking notion of sequences (and dimension) and
4. a way to derive a non-forking relation on models from a non-forking relation on elements.

Using a non-forking relation on models, we define a partial order, $\preceq^{NF}$ on models. Our main result in the paper Tameness, Uniqueness Triples and Amalgamation’, is that the partial orders $\preceq$ and $\preceq^{NF}$ are equivalent (under reasonable hypotheses). We will show the main steps of the proof.

The equivalence between the relations $\preceq$ and $\preceq^{NF}$ is an important step towards proving the categoricity conjecture for AECs.