In order to construct a counterexample to Zilber's conjecture—that a strongly minimal set has a degenerate, affine or field-like geometry—Ehud Hrushovski invented an amalgamation technique which has yielded all the exotic geometries so far. We shall present a framework for this construction in the language of standard geometric stability and show how some of the recent constructions fit into this setting. We also ask some fundamental questions concerning this method.

The notion of an angular function has been introduced by Zilber as one possible way of connecting non-commutative geometry with two ‘counterexamples’ from model theory: the non-classical Zariski curves of Hrushovski and Zilber, and Poizat's field with green points. This article discusses some questions of Zilber relating to existentially closed structures in the class of algebraically closed fields with an angular function.

We construct the free fusion of two geometric thories over a common ω-categorical strongly minimal reduct. If the two theories are supersimple of rank 1 (and satisfy an additional hypothesis true in particular for stable theories or trivial reduct), the completions of the free fusion are supersimple of rank at most ω.

We draw a connection between the model-theoretic notions of modularity (or one-basedness), orthogonality and internality, as applied to difference fields, and questions of descent in in algebraic dynamics. In particular we prove in any dimension a strong dynamical version of Northcott's theorem for function fields, answering a question of Szpiro and Tucker and generalizing a theorem of Baker's for the projective line.

The paper comes in three parts. This first part contains an exposition some of the main results of the model theory of difference fields, and their immediate connection to questions of descent in algebraic dynamics. We present the model-theoretic notion of internality in a context that does not require a universal domain with quantifier-elimination. We also note a version of canonical heights that applies well beyond polarized algebraic dynamics. Part II sharpens the structure theory to arbitrary base fields and constructible maps where in part I we emphasize finite base change and correspondences. Part III will include precise structure theorems related to the Galois theory considered here, and will enable a sharpening of the descent results for non-modular dynamics.

This second part of the paper strengthens the descent theory described in the first part torational maps and arbitrary base fields. We obtain in particular a decomposition of any difference field extension into a tower of finite, field-internal and one-based difference field extensions. This is needed in order to obtain the ‘dynamical Northcott’ Theorem 1.11 of Part I in sharp form.

While the classification project for the simple groups of finite Morley rank is unlikely toproduce a classification of the simple groups of finite Morley rank, the enterprise has already arrived at a considerably closer approximation to that ideal goal than could have been realistically anticipated, with a mix of results of several flavors, some classificatory and others more structural, which can be combined when the stars are suitably aligned to produce results at a level of generality which, in parallel areas of group theory, would normally require either some additional geometric structure, or an explicit classification. And Bruno Poizat is generally awesome, though sometimes he goes too far.

Dichotomies in various conjectures from algebraic geometry are in fact occurrences of the dichotomy among Zariski structures. This is what Hrushovski showed and which enabled him to solve, positively, the geometric Mordell–Lang conjecture in positive characteristic. Are we able now to avoid this use of Zariski structures? Pillay and Ziegler have given a direct proof that works for semi-abelian varieties they called ‘very thin’, which include the ordinary abelian varieties. But it does not apply in all generality: we describe here an abelian variety which is not very thin. More generally, we consider from a model-theoretical point of view several questions about the fields of definition of semi-abelian varieties.

We prove that if M0 is a model of a simple theory, and p(x) is a complete type of Cantor–Bendixon rank 1 over M0, then p is stationary and regular. As a consequence we obtain another proof that any countable model M0 of a countable complete simple theory T has infinitely many countable elementary extensions up to M0-isomorphism. The latter extends earlier results of the author in the stable case, and is a special case of a recent result of Tanovic.

We establish an identification result of the projective special linear group of dimension 2among a certain class of groups the Morley rank of which is finite.

This paper is a brief survey of recent results and some open problems related to linear groups of finite Morley rank, an area of research where Bruno Poizat's impact is very prominent. As a sign of respect to his strongly expressed views that mathematics has to be done, written and pulished only in the native tongue of the immediate author–the scribe, in effect–of the text, I insist on writing my paper in Russian, even if the results presented belong to a small but multilingual community of researchers of American, British, French, German, Kazakh, Russian, Turkish origin. To emphasise even further the linguistic subtleties involved, I use British spelling in the English fragments of my text.

We construct a bad field in characteristic zero. That is, we construct an algebraically closed field which carries a notion of dimension analogous to Zariski-dimension, with an infinite proper multiplicative subgroup of dimension one, and such that the field itself has dimension two. This answers a longstanding open question by Zilber.

Countable homogeneous models are ‘simple’ objects from a model theoretic point of view. From a recursion theoretic point of view they can be complex. For instance the elementary theory of such a model might be undecidable, or the set of complete types might be recursively complex. Unfortunately even if neither of these conditions holds, such a model still can be undecidable. This paper investigates countable homogeneous models with respect to a weaker notion of decidability called almost decidable. It is shown that for theories that have only countably many type spectra, any countable homogeneous model of such a theory that has a Σ2 type spectrum is almost decidable.

It is shown that no functor F exists from the category of sets with injections, to the category of algebraically closed fields of given characteristic, with monomorphisms, having the properties that for all sets A. F(A) is an algebraically closed field having transcendence base A and for all injections f. F(f) extends f. There does exist such a functor from the category of linearly-ordered sets with order monomorphisms.

An application to model-theory using the same methods is given showing that while the theory of algebraically closed fields is ω-stable, its Skolemization is not stable in any power.

In Section 1 below I describe two measures of the complexity of a binary relation. J The theorem says that these two measures never disagree very much. Both measures of complexity arose in connection with Saharon Shelah's notion [5] of a stable firstt order theory; Shelah showed in effect that one measure is finite, if, and only if, the other is finite too. This follows trivially from the theorem below. I confess my main aim was not to get the extra information which the theorem provides, but to eliminate Shelah's use of uncountable cardinals, which seemed strangely heavy machinery for proving a purely finitary result. Section 2 below explains the modeltheoretic setting.