We discuss dividing lines (in model theory) and some test questions, mainly the universality spectrum. So there is much on conjectures, problems and old results, mainly of the author and also on some recent results.

§1. Introduction. With Hrushovski's proof of the function field Mordell-Lang conjecture [16] the relevance of geometric stability theory to diophantine geometry first came to light. A gulf between logicians and number theorists allowed for contradictory reactions. It has been asserted that Hrushovski's proof was simply an algebraic argument masked in the language of model theory. Another camp held that this theorem was merely a clever one-off. Still others regarded the argument as magical and asked whether such sorcery could unlock the (...) secrets of a wide coterie of number theoretic problems.In the intervening years each of these prejudices has been revealed as false though such attitudes are still common. The methods pioneered in [16] have been extended and applied to a number of other problems. At their best, these methods have been integrated into the general methods for solving diophantine problems. Moreover, the newer work suggests limits to the application of model theory to diophantine geometry. For example, all such known applications are connected with commutative algebraic groups. This need not be an intrinsic restriction, but its removal requires serious advances in the model theory of fields. (shrink)

Non-$n$-ampleness as defined by Pillay [20] and Evans [5] is preserved under analysability. Generalizing this to a more general notion of $\Sigma$-ampleness, this gives an immediate proof for all simple theories of a weakened version of the Canonical Base Property (CBP) proven by Chatzidakis [4] for types of finite SU-rank. This is then applied to the special case of groups.

A type analysable in one-based types in a simple theory is itself one-based.

Generalizing and unifying the known theorems for difference and differential fields, it is shown that for every finite free algebra scheme.

This paper began as a generalization of a part of the author's PhD thesis about ACFA and ended up with a characterization of groups definable in T A . The thesis concerns minimal formulae of the form x ∈ A ∧ σ(x) = f(x) for an algebraic curve A and a dominant rational function f: A → σ(A). These are shown to be uniform in the Zilber trichotomy, and the pairs (A, f) that fall into each of the three cases (...) are characterized. These characterizations are definable in families. This paper covers approximately half of the thesis, namely those parts of it which can be made purely model-theoretic by moving from ACFA, the model companion of the class of algebraically closed fields with an endomorphism, to T A , the model companion of the class of models of an arbitrary totally-transcendental theory T with an injective endomorphism, if this model-companion exists. A T A analog of the characterization of groups definable in ACFA is obtained in the process. The full characterization of the cases of the Zilber trichotomy in the thesis is obtained from these intermediate results with heavy use of algebraic geometry. (shrink)

I see model theory as becoming increasingly detached from set theory, and the Tarskian notion of set-theoretic model being no longer central to model theory. In much of modern mathematics, the set-theoretic component is of minor interest, and basic notions are geometric or category-theoretic. In algebraic geometry, schemes or algebraic spaces are the basic notions, with the older “sets of points in affine or projective space” no more than restrictive special cases. The basic notions may be given sheaf-theoretically, or functorially. (...) To understand in depth the historically important affine cases, one does best to work with more general schemes. The resulting relativization and “transfer of structure” is incomparably more flexible and powerful than anything yet known in “set-theoretic model theory”.It seems to me now uncontroversial to see the fine structure of definitions as becoming the central concern of model theory, to the extent that one can easily imagine the subject being called “Definability Theory” in the near future.Tarski's set-theoretic foundational formulations are still favoured by the majority of model-theorists, and evolution towards a more suggestive language has been perplexingly slow. None of the main texts uses in any nontrivial way the language of category theory, far less sheaf theory or topos theory. Given that the most notable interactions of model theory with geometry are in areas of geometry where the language of sheaves is almost indispensable, this is a curious situation, and I find it hard to imagine that it will not change soon, and rapidly. (shrink)

In this paper, we introduce an AEC framework for studying fields with commuting automorphisms. Fields with commuting automorphisms are closely related to difference fields. Some authors define a difference ring (or field) as a ring (or field) together with several commuting endomorphisms, while others only study one endomorphism. Z. Chatzidakis and E. Hrushovski have studied in depth the model theory of ACFA, the model companion of difference fields with one automorphism. Our fields with commuting automorphisms generalize this setting. We have (...) several automorphisms and they are required to commute. Hrushovski has proved that in the case of fields with two or more commuting automorphisms, the existentially closed models do not necessarily form a first order model class. In the present paper, we introduce FCA-classes, an AEC framework for studying the existentially closed models of the theory of fields with commuting automorphisms. We prove that an FCA-class has AP and JEP and thus a monster model, that Galois types coincide with existential types in existentially closed models, that the class is homogeneous, and that there is a version of type amalgamation theorem that allows to combine three types under certain conditions. Finally, we use these results to show that our monster model is a simple homogeneous structure in the sense of S. Buechler and O. Lessman (this is a non-elementary analogue for the classification theoretic notion of a simple first order theory). (shrink)

We study algebraic and model-theoretic properties of existentially closed fields with an action of a fixed finite group. Such fields turn out to be pseudo-algebraically closed in a rather strong sense. We place this work in a more general context of the model theory of fields with a group scheme action.

We examine topological dynamical systems on the Cantor set from the point of view of the continuous model theory of commutative C*-algebras. After some general remarks, we focus our attention on the generic homeomorphism of the Cantor set, as constructed by Akin, Glasner, and Weiss. We show that this homeomorphism is the prime model of its theory. We also show that the notion of “generic” used by Akin, Glasner, and Weiss is distinct from the notion of “generic” encountered in Fraïssé (...) theory. (shrink)

We study simplicity and stability in some large strongly homogeneous expansions of Hilbert spaces. Our approach to simplicity is that of Buechler and Lessmann 69). All structures we consider are shown to have built-in canonical bases.

We give a reduction of the function field Mordell–Lang conjecture to the function field Manin–Mumford conjecture, for abelian varieties, in all characteristics, via model theory, but avoiding recourse to the dichotomy theorems for Zariski geometries. Additional ingredients include the “Theorem of the Kernel”, and a result of Wagner on commutative groups of finite Morley rank without proper infinite definable subgroups. In positive characteristic, where the main interest lies, there is one more crucial ingredient: “quantifier-elimination” for the corresponding [Formula: see text] (...) where [Formula: see text] is a saturated separably closed field. (shrink)