Elisabeth Bouscaren

Proof of the Mordell-Lang conjecture for function fields

In this chapter we present Hrushovski’s model-theoretic proof of the “relative Mordell-Lang conjecture” (“The Mordell-Lang Conjecture for function fields” [Hr 96]).

An Introduction to Independence and Local Modularity

This brief survey has a very modest goal. I hope to present an introduction to some of the common features which lay in the background of at least three of the main topics of this workshop: the spectrum function for countable theories [8], smoothly approximated structures [6], the model theory of fields with an automorphism and its application to the Manin-Mumford Conjecture [23, 4].

Martin’s Conjecture forω-stable theories

InA proof of Vaught’s Conjecture for ω-stable theories, S. Shelah, L. Harrington and M. Makkai show thatω-stable theories satisfy Vaught’s Conjecture. By using their results and pushing the analysis one step further, we show thatω-stable theories also satisfy Martin’s Conjecture.

Elementary pairs of models

Abstract We give a survey of some recent results and some remaining questions concerning the model theory of elementary pairs of models of a complete first-order theory, in the language with a new predicate for the small model of the pair.

Introduction to model theory

In this informal presentation we introduce some of the main definitions and results which form the basis of model theory. We have chosen an approach adapted to the particular subject of this book. For proofs and formal definitions as well as for all that we have here purposely omitted, we suggest [Ho] or [Po 85] both rather close in spirit to the point of view adopted here. For a more classical approach, see [ChKe].

S-Homogeneity and Automorphism Groups

We consider the question of when, given a subset A of M , the setwise stabilizer of the group of automorphisms induces a closed subgroup on Sym( A ). We define s-homogeneity to be the analogue of homogeneity relative to strong embeddings and show that any subset of a countable, s-homogeneous, ω -stable structure induces a closed subgroup and contrast this with a number of negative results. We also show that for ω -stable structures s-homogeneity is preserved under naming countably many constants, but under slightly weaker conditions it can be lost by naming a single point.

D.O.P and N-Tuples of Models

Abstract We show that, for T a complete superstable theory, if the theory of pairs of models of T is stable, then, for all n, the theory of n-tuples of models of T is stable. In order to prove this, we generalize to arbitrary n-tuples of models a previous result of ours: the theory of pairs of T is stable if and only if it is superstable if and only if T does not have the Dimensional Order Property [2].