Valuations are among the fundamental structures of number theory and of algebraic geometry. This was recognized early by model theorists, with gratifying results: Robinson's description of algebraically closed valued fields as the model completion of the theory of valued fields; the Ax-Kochen, Ershov study of Henselian fields of large residue characteristic with the application to Artin's conjecture work of Denef and others on integration; and work of Macintyre, Delon, Prestel, Roquette, Kuhlmann, and others on p-adic fields and positive characteristic. The model theory of valued fields is thus one of the most established and deepest areas of the subject.

However, precisely because of the complexity of valued fields, much of the work centers on quantifier elimination and basic properties of formulas. Few tools are available for a more structural model-theoretic analysis. This contrasts with the situation for the classical model complete theories, of algebraically closed and real closed fields, where stability theory and o-minimality make possible a study of the category of definable sets. Consider for instance the statement that fields interpretable over ℂ are finite or algebraically closed. Quantifier elimination by itself is of little use in proving this statement. One uses instead the notion of ω-stability; it is preserved under interpretation, implies a chain condition on definable subgroups, and, by a theorem of Macintyre, ω-stable fields are algebraically closed. With more analysis, using notions such as generic types, one can show that indeed every interpretable field is finite or definably isomorphic to ℂ itself.

Introduction. The general context of this article is an investigation of foundational frameworks for constructive mathematics in the style of Bishop. In the early seventies, prompted by Bishop's monograph Foundations of Constructive Analysis [9], various formal systems emerged to provide a clear and precise basis for the newborn mathematical style. Completely new systems were introduced like Feferman's explicit mathematics [16] and Martin–Löf's type theory [23, 24]. In addition, various systems in the form of the Zermelo–Fraenkel axiomatization but based on intuitionistic logic were put forward for example by Friedman [20, 21], Myhill [27] and Aczel [2, 4, 5]. The latter author was motivated by Myhill's Constructive set theory [28] in which a system of sets and functions was introduced adhering to a (generalized) predicative approach. Constructive set theory thus avoided the use of unbounded forms of separation and of the full axiom of powerset. Aczel brought these ideas in a system expressed in the language of Zermelo–Fraenkel set theory and also provided a natural interpretation for it in a version of Martin–Löf's type theory with a universe of small types and an inductive type built over it.

We give here a brief preview of stability theory, as it underpins stable domination. We also introduce some of the model-theoretic notation used later. Familiarity with the basic notions of logic (languages, formulas, structures, theories, types, compactness) is assumed, but we explain the model theoretic notions beginning with saturation, algebraic closure, imaginaries. We have in mind a reader who is familiar with o-minimality or some model theory of valued fields, but has not worked with notions from stability. Sources include Shelah's Classification Theory as well as books by Baldwin, Buechler, Pillay and Poizat. There is also a broader introduction by Hart intended partly for non-model theorists, and an introduction to stability theory intended for a wider audience in. Most of the stability theoretic results below should be attributed to Shelah. Our treatment will mostly follow Pillay.

Stability theory is a large body of abstract model theory developed in the 1970s and 1980s by Shelah and others, but having its roots in Morley's 1965 Categoricity Theorem: if a complete theory in a countable language is categorical in some uncountable power, then it is categorical in all uncountable powers. Shelah formulated a radical generalization of Morley's theorem, weakening the categoricity assumption from one isomorphism type to any number less than the set-theoretic maximum. The conclusion is that all models of the theory, in any power, are classifiable by a small tree of numerical dimensions.

Much late 19th and early 20th century work in logic was in a 2nd order framework; infinitary logics in the modern sense were foreshadowed by Schroeder and Pierce before being formalized in modern terms in Poland during the late 20's. First order logicwas only singled out as the ‘natural’ language to formalize mathematics as such authors as Tarski, Robinson, and Malcev developed the fundamental tools and applied model theory in the study of algebra. Serious work extending the model theory of the 50's to various infinitary logics blossomed during the 1960's and 70's with substantial work on logics such as Lω1, ω and Lω1, ω(Q). At the same time Shelah's work on stable theories completed the switch in focus in first order model theory from study of the logic to the study of complete first order theories As Shelah in [44, 46] sought to bring this same classification theory standpoint to infinitary logic, he introduced a total switch to a semantic standpoint. Instead of studying theories in a logic, one studies the class of models defined by a theory.

Introduction. Several applications of model-theoretic methods in the theory of cohomology have appeared recently, most probably influenced by ideas of Macintyre. In his programmatic paper [10], he shows that, after expanding the language of rings by certain sorts and predicates for cohomology, the axioms of Weil cohomology theories are first order and one can form new cohomology theories as ultraproducts of already existing ones.

The present author has remarked that, although the cohomologies with torsion coefficients do not satisfy the axioms for a Weil cohomology theory individually, they do so “on average”, and one can obtain cohomologies with coefficients in pseudofinite fields of characteristic zero by taking ultraproducts [13]. He shows that this “pseudofinite cohomology” is at least as good as the l-adic theory when dealing with issues around the Weil conjectures. In parallel, Brunjes and Serpe developed the theory of nonstandard sheaves systematically, and they even show that the pseudofinite cohomology is better behaved than the l-dic one, the former being a derived functor cohomology [1].

The purpose of this short note is to clarify which aspects and invariants of the theory of (étale) constructible sheaves and cohomologies are definable in the language of rings. It should serve as a bridge between the algebraic-geometric and model-theoretic language and should encourage model-theorists to use the sophisticated techniques already developed by geometers. We show that, in case one needs to consider an invariant defined in terms of constructible sheaves over a finite or a pseudofinite ground field, there is a good chance that it is definable.

I intend in this article to introduce nonspecialists to the fact that there are deep results in contemporary universal algebra. The first four sections give the context in which universal algebra has something to say, and describe some of the basic results upon which much of the work in the field is built. Section 5 covers the highlights of tame congruence theory, a sophisticated point of view from which to analyze locally finite algebras. Section 6 describes some of the field's “big” results and open problems concerning finite algebras, notably the undecidability of certain finite axiomatizability problems and related problems, and the so-called “RS problem,” currently the most important open problem in the field.

This article presents a personal view of current universal algebra, one which is limited both by my ignorance of large parts of the field as well as the likely interests of the intended audience and the need to keep the article focused. For example, I do not mention natural duality theory, one of the most vigorous subdisciplines of the field, nor algebraic logic or work that is motivated by and serves computer science. Therefore the views expressed in this article should not be taken to be a comprehensive statement of “what universal algebra is.”

The text is frequently imprecise and proofs, when present, are merely sketched. Credit is not always given where due. My aim is to give the reader an impression of the field. Resources for further reading are provided in the final section.

introduction. Parametrized p-adic integrals are being better understood recently by the work of Denef in an algebraic context and the work of the author in an analytic context (and, by the theory of motivic integrals, but we will not treat uniformity questions here). In this paper we give a contribution to the theory of p-adic integrals, based on the Cell Decomposition Theorem of [3]. In [3], a framework of constructible functions which is stable under integration is introduced. Here we extend the notion of constructible functions such that parametrized (possibly twisted) local zeta functions, which are p-adic integrals of quite a general form, are included, and we prove that it is stable under p-adic integration. We control the possible order of poles of the zeta functions, uniform in the parameters.

Introduction. A variety of sensorimotor coordination mechanisms are being successfully modelled on the basis of continuous dynamical system approaches (Beer [1997]; Steinhage and Bergener [2000]; Turvey and Carello [1995]). This work is invoked as empirical support for a sweeping “dynamicist” thesis: intelligent and adaptive biological behaviours are to be ultimately accounted for in terms of continuous (dynamical) systems; properly computational investigations make approximate simulation tools available for dynamical theories but play no essential theoretical role (Port and Gelder [1995]). Similar claims about mathematical theorizing in cognitive ethology and biology at large can be found in (Beer [1995], Steels [1995]) and (Longo [2003]), respectively. These claims are critically examined here, in the light of theoretical models of simple sensorimotor adaptive behaviours. These case studies are particularly suited to our purposes, for continuous dynamical approaches are supposedly at their best in the modelling of sensorimotor coordination mechanisms. Computational approaches, we submit, are being fruitfully pursued there too.

As developed in, stability theory is based on the notion of an invariant type, more specifically a definable type, and the closely related theory of independence of substructures. We will review the definitions in Chapter 2 below; suffice it to recall here that an (absolutely) invariant type gives a recipe yielding, for any substructure A of any model of T, a type p│A, in a way that respects elementary maps between substructures; in general one relativizes to a set C of parameters, and considers only A containing C. Stability arose in response to questions in pure model theory, but has also provided effective tools for the analysis of algebraic and geometric structures. The theories of algebraically and differentially closed fields are stable, and the stability-theoretic analysis of types in these theories provides considerable information about algebraic and differential-algebraic varieties. The model companion of the theory of fields with an automorphism is not quite stable, but satisfies the related hypothesis of simplicity; in an adapted form, the theory of independence remains valid and has served well in applications to difference fields and definable sets over them. On the other hand, such tools have played a rather limited role, so far, in o-minimality and its applications to real geometry.

Where do valued fields lie? Classically, local fields are viewed as closely analogous to the real numbers. We take a “geometric” point of view however, in the sense of Weil, and adopt the model completion as the setting for our study.