We investigate Tukey functions from the ideal of all closed nowhere-dense subsets of 2ℕ. In particular, we answer an old question of Isbell and Fremlin by showing that this ideal is not Tukey reducible to the ideal of density zero subsets of ℕ. We also prove non-existence of various special types of Tukey reductions from the nowhere-dense ideal to analytic P-ideals. In connection with these results, we study families of clopen subsets of 2ℕ with the property that for each nowhere-dense subset of 2ℕ there is a set in not intersecting it. We call such families avoiding.

The four colour theorem states that the vertices of every planar graph can be coloured with at most four colours so that no two adjacent vertices receive the same colour. This theorem is famous for many reasons, including the fact that its original 1977 proof includes a non-trivial computer verification. Recently, a formal proof of the theorem was obtained with the equational logic program Coq [G. Gonthier, ‘Formal proof–the four color theorem’, Notices of Amer. Math. Soc. 55 (2008) no. 11, 1382–1393]. In this paper we describe an implementation of the computational method introduced by C. S. Calude and co-workers [Evaluating the complexity of mathematical problems. Part 1’, Complex Systems 18 (2009) 267–285; A new measure of the difficulty of problems’, J. Mult. Valued Logic Soft Comput. 12 (2006) 285–307] to evaluate the complexity of the four colour theorem. Our method uses a Diophantine equational representation of the theorem. We show that the four colour theorem is in the complexity class ℭU,4. For comparison, the Riemann hypothesis is in class ℭU,3 while Fermat’s last theorem is in class ℭU,1.

We prove that if K is an (infinite) stable field whose generic type has weight 1, then K is separably closed. We also obtain some partial results about stable groups and fields whose generic type has finite weight, as well as about strongly stable fields (where by definition all types have finite weight).

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.

Let X and Y be separable Banach spaces and denote by 𝒮𝒮(X,Y ) the subset of ℒ(X,Y ) consisting of all strictly singular operators. We study various ordinal ranks on the set 𝒮𝒮(X,Y ). Our main results are summarized as follows. Firstly, we define a new rank r𝒮 on 𝒮𝒮(X,Y ). We show that r𝒮 is a co-analytic rank and that it dominates the rank ϱ introduced by Androulakis, Dodos, Sirotkin and Troitsky [Israel J. Math.169 (2009), 221–250]. Secondly, for every 1≤p<+∞, we construct a Banach space Yp with an unconditional basis such that 𝒮𝒮(ℓp,Yp) is a co-analytic non-Borel subset of ℒ(ℓp,Yp) yet every strictly singular operator T:ℓp→Yp satisfies ϱ(T)≤2. This answers a question of Argyros.

Since any function f(x1, … , xm) from {0, 1}m in a finite field k can be uniquely written as a multilinear polynomial, we associate to it its inverse dual f*(x1, … , xm) expressing the coefficients of this canonical polynomial. We show that the unlikely hypothesis that the class P(k) of sequences of functions of polynomial complexity be closed by duality is equivalent to the well-known hypothesis P = #pP, where p is the characteristic of k.

In a first section we expose the result in the frame of classical Boolean calculus, that is, when k = ℤ/2ℤ. In a second section we treat the general case, introducing a notion of transformation whose duality is a special case; the transformations form a group isomorphic to GL2(k); among them, we distinguish the benign transformations, which have a weak effect on the complexity of functions; we show that, in this respect, all the non-benign transformations have the same power of harmfulness.

In the third section we consider functions from km into k, and in the last, after introducing #P = P to the landscape, we compare our results with those of Guillaume Malod, concerning the closure by ‘coefficient-function’ of various classes of complexity of sequences of polynomial defined in Valiant's way.

It is shown that a simple deduction engine can be developed for a propositional logic that follows the normal rules of classical logic in symbolic form, but the description of what is known about a proposition uses two numeric state variables that conveniently describe unknown and inconsistent, as well as true and false. Partly true and partly false can be included in deductions. The multi-valued logic is easily understood as the state variables relate directly to true and false. The deduction engine provides a convenient standard method for handling multiple or complicated logical relations. It is particularly convenient when the deduction can start with different propositions being given initial values of true or false. It extends Horn clause based deduction for propositional logic to arbitrary clauses. The logic system used has potential applications in many areas. A comparison with propositional logic makes the paper self-contained.

We prove that n-hypergraphs can be interpreted in e-free perfect PAC fields in particular in pseudofinite fields. We use methods of function field arithmetic, more precisely we construct generic polynomials with alternating groups as Galois groups over a function field.

A shift automorphism algebra is one satisfying the conditions of the shift automorphism theorem, and a shift automorphism variety is a variety generated by a shift automorphism algebra. In this paper, we show that every shift automorphism variety contains a countably infinite subdirectly irreducible algebra.

The arithmetic is interpreted in all the groups of Richard Thompson and Graham Higman, as well as in other groups of piecewise affine permutations of an interval which generalize the groups of Thompson and Higman. In particular, the elementary theories of all these groups are undecidable. Moreover, Thompson's group F and some of its generalizations interpret the arithmetic without parameters.

We apply the collapse techniques to Poizat's red differential field in order to obtain differentially closed fields of Morley rank ω·2 each equipped with an additive definable subgroup of rank ω. By means of the logarithmic derivative, we obtain a green field of rank ω·2 with a multiplicative definable divisible subgroup containing the field of constants, which is again definable in the reduct of the green field.

We consider a new subgroup In(G) in any group G of finite Morley rank. This definably characteristic subgroup is the smallest normal subgroup of G from which we can hope to build a geometry over the quotient group G/ In(G). We say that G is a geometric group if In(G) is trivial.

This paper is a discussion of a conjecture which states that every geometric group G of finite Morley rank is definably linear over a ring K1 ⊕…⊕ Kn where K1,…,Kn are some interpretable fields. This linearity conjecture seems to generalize the Cherlin–Zil'ber conjecture in a very large class of groups of finite Morley rank.

We show that, if this linearity conjecture is true, then there is a Rosenlicht theorem for groups of finite Morley rank, in the sense that the quotient group of any connected group of finite Morley rank by its hypercentre is definably linear.

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 prove that a finitely generated semigroup whose word problem is a one-counter language has a linear growth function. This provides us with a very strong restriction on the structure of such a semigroup, which, in particular, yields an elementary proof of a result of Herbst, that a group with a one-counter word problem is virtually cyclic. We prove also that the word problem of a group is an intersection of finitely many one-counter languages if and only if the group is virtually abelian.

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.

The classes in Valiant's theory are classes of polynomials defined by arithmetic circuits. We characterize them by different notions of tensor calculus, in the vein of Damm, Holzer and McKenzie. This characterization underlines in particular the role played by properties of parallelization in these classes. We also give a first natural complete sequence for the class VPnb, the analogue of the class P in this context.