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.

Introduction. We start by briefly describing the strong colouring problem for permutation groups that was investigated in [13] and [12]; this problem has its origins in the classification of maximal partial clones over a finite non-empty set [14, 15, 16]. We shall immediately reformulate this decision problem as a Constraint Satisfaction Problem (CSP) in order to exploit various universal algebraic tools to study its algorithmic complexity.

Introduction. For f ∈ ωω and an ultrafilter U on ω define f(U) = {X ⊆ ω: f−1(X) ∈ U}, and for f, g ∈ ωω we say that f = g mod U if there is X ∈ U such that f(n) = g(n) for n ∈ X.

We say that U is Hausdorff if for any two functions f, g ∈ ωω, if f(U) = g(U) then f = g mod U.

Let FtO be the collection of all finite-to-one functions f ∈ ωω. Recall that an ultrafilter U is a p-point if for every function f ∈ ωω either there is n such that f−1({n}) ∈ U or there exists g ∈ FtO such that f = g mod U. Similarly, U is Ramsey if for every function f ∈ ωω either there is n such that f−1({n}) ∈ U or there exists a one-to-one function g ∈ ωω such that f = g mod U.

In this paper we will assume that all ultrafilters U, and their images f(U) are non-principal.

It is worth mentioning that the following appears as an exercise in [7]. If f(U) = U then f = id mod U. Therefore, if U is not Hausdorff, then this is witnessed by two functions, both not one-to-one mod U. It follows from it that Ramsey ultrafilters are Hausdorff.