We will describe the Ziegler spectrum over the ring of entire complex valued functions.

We provide algebraic conditions ensuring the decidability of the theory of modules over effectively given Prüfer (in particular Bézout) domains with infinite residue fields in terms of a suitable generalization of the prime radical relation. For Bézout domains these conditions are also necessary.

We introduce a notion of valued module which is suitable to study valued fields of positive characteristic. Then we built-up a robust theory of henselianity in the language of valued modules and prove Ax-Kochen Ershov type results.

We prove a direct image theorem stating that the direct image of a Galois formula by a morphism of difference schemes is equivalent to a Galois formula over fields with powers of Frobenius. As a consequence, we obtain an effective quantifier elimination procedure and a precise algebraic–geometric description of definable sets over fields with Frobenii in terms of twisted Galois formulas associated with finite Galois covers of difference schemes.

We show that any simple group of Morley rank 4 must be a bad group with no proper definable subgroups of rank larger than 1. We also give an application to groups acting on sets of Morley rank 2.

Regular groups and fields are common generalizations of minimal and quasi-minimal groups and fields, so the conjectures that minimal or quasi-minimal fields are algebraically closed have their common generalization to the conjecture that each regular field is algebraically closed. Standard arguments show that a generically stable regular field is algebraically closed. Let K be a regular field which is not generically stable and let p be its global generic type. We observe that if K has a finite extension L of degree n, then P(n) has unbounded orbit under the action of the multiplicative group of L.

Known to be true in the minimal context, it remains wide open whether regular, or even quasi-minimal, groups are abelian. We show that if it is not the case, then there is a counter-example with a unique nontrivial conjugacy class, and we notice that a classical group with one nontrivial conjugacy class is not quasi-minimal, because the centralizers of all elements are uncountable. Then, we construct a group of cardinality ω1 with only one nontrivial conjugacy class and such that the centralizers of all nontrivial elements are countable.

We show that R.J. Thompson’s groups F and T are bi-interpretable with the ring of the integers. From a result by A. Khélif, these groups are quasi-finitely axiomatizable and prime. So, the group T provides an example of a simple group which is quasi-finitely axiomatizable and prime. This answers questions posed by T. Altınel and A. Muranov in [2], and by A. Nies in [12].

We consider valued fields with a value-preserving automorphism and improve on model-theoretic results by Bélair, Macintyre and Scanlon on these objects by dropping assumptions on the residue difference field. In the equicharacteristic 0 case we describe the induced structure on the value group and the residue difference field.

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 to rational 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 to produce 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.

We establish an identification result of the projective special linear group of dimension 2 among 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.

L. van den Dries proved that the theory of n-valued rings has a model companion. We show here that this result is still true when the valuation rings are required to satisfy given inclusion relations (we restrict ourselves to the case of residual characteristic zero).