We provide an unconditional proof of the André–Oort conjecture for the coarse moduli space 𝒜2,1 of principally polarized abelian surfaces, following the strategy outlined by Pila–Zannier.

We prove the following theorems. Theorem 1: for any E-field with cyclic kernel, in particular ℂ or the Zilber fields, all real abelian algebraic numbers are pointwise definable. Theorem 2: for the Zilber fields, the only pointwise definable algebraic numbers are the real abelian numbers.

We devise a fairly general sufficient condition ensuring that the endomorphism monoid of a countably infinite ultrahomogeneous structure (i.e. a Fraïssé limit) embeds all countable semigroups. This approach not only provides us with a framework unifying the previous scattered results in this vein, but actually yields new applications for endomorphism monoids of the (rational) Urysohn space and the countable universal ultrahomogeneous semilattice.

According to the André–Oort conjecture, an algebraic curve in Y (1)n that is not equal to a special subvariety contains only finitely many points which correspond to ann-tuple of elliptic curves with complex multiplication. Pink’s conjecture generalizes the André–Oort conjecture to the extent that if the curve is not contained in a special subvariety of positive codimension, then it is expected to meet the union of all special subvarieties of codimension two in only finitely many points. We prove this for a large class of curves in Y (1)n. When restricting to special subvarieties of codimension two that are not strongly special we obtain finiteness for all curves defined over . Finally, we formulate and prove a variant of the Mordell–Lang conjecture for subvarieties of Y (1)n.

Given a set D of positive real numbers, let Xn(D) denote the graph with ℝn as the vertex set such that two points are joined if their distance is in D. Bukh conjectured in [Measurable sets with excluded distances. Geom. Funct. Anal.18 (2008), 668–697] that if D is algebraically independent, then Chr(Xn(D)), the chromatic number of Xn(D), is finite. Here we prove that Chr(Xn(D)) is countable and that, if n=2 , even the coloring number is countable. Furthermore, we prove that Chr (Y )is countable, where Y is the following graph on ℂn: let 𝔽 be a countable subfield of ℂ and let D⊆ℂ be algebraically independent over 𝔽; join a,b∈ℂn if there is some p(x,y)∈𝔽[x,y]such that p(x,x)is identically zero and p(a,b)≠0is algebraic over some d∈𝔽∪D.

We extract a quantitative variant of uniqueness from the usual hypotheses of the implicit function theorem. Not only does this lead to an a priori proof of continuity, but also to an alternative, full proof of the implicit function theorem. Additionally, we investigate implicit functions as a case of the unique existence paradigm with parameters.

We study the existence of some covers and envelopes in the chain complex category of R-modules. Let (𝒜,ℬ) be a cotorsion pair in R-Mod and let ℰ𝒜 stand for the class of all exact complexes with each term in 𝒜. We prove that (ℰ𝒜,ℰ𝒜⊥) is a perfect cotorsion pair whenever 𝒜 is closed under pure submodules, cokernels of pure monomorphisms and direct limits and so every complex has an ℰ𝒜-cover. As an application we show that every complex of R-modules over a right coherent ring R has an exact Gorenstein flat cover. In addition, the existence of -covers and -envelopes of special complexes is considered where and denote the classes of all complexes with each term in 𝒜 and ℬ, respectively.

We compare two influential ways of defining a generalized notion of space. The first, inspired by Gelfand duality, states that the category of ‘noncommutative spaces’ is the opposite of the category of C*-algebras. The second, loosely generalizing Stone duality, maintains that the category of ‘point-free spaces’ is the opposite of the category of frames (that is, complete lattices in which the meet distributes over arbitrary joins). Earlier work by the first three authors shows how a noncommutative C*-algebra gives rise to a commutative one internal to a certain sheaf topos. The latter, then, has a constructive Gelfand spectrum, also internal to the topos in question. After a brief review of this work, we compute the so-called external description of this internal spectrum, which in principle is a fibred point-free space in the familiar topos of sets and functions. However, we obtain the external spectrum as a fibred topological space in the usual sense. This leads to an explicit Gelfand transform, as well as to a topological reinterpretation of the Kochen–Specker theorem of quantum mechanics.

An interval in a combinatorial structure R is a set I of points that are related to every point in R∖I in the same way. A structure is simple if it has no proper intervals. Every combinatorial structure can be expressed as an inflation of a simple structure by structures of smaller sizes—this is called the substitution (or modular) decomposition. In this paper we prove several results of the following type: an arbitrary structure S of size n belonging to a class 𝒞 can be embedded into a simple structure from 𝒞 by adding at most f(n) elements. We prove such results when 𝒞 is the class of all tournaments, graphs, permutations, posets, digraphs, oriented graphs and general relational structures containing a relation of arity greater than two. The functions f(n) in these cases are 2, ⌈log 2(n+1)⌉, ⌈(n+1)/2⌉, ⌈(n+1)/2⌉, ⌈log 4(n+1)⌉, ⌈log 3(n+1)⌉ and 1, respectively. In each case these bounds are the best possible.

In Flaminio and Montagna [‘An algebraic approach to states on MV-algebras’, in: Fuzzy Logic 2, Proc. 5th EUSFLAT Conference, Ostrava, 11–14 September 2007 (ed. V. Novák) (Universitas Ostraviensis, Ostrava, 2007), Vol. II, pp. 201–206; ‘MV-algebras with internal states and probabilistic fuzzy logic’, Internat. J. Approx. Reason.50 (2009), 138–152], the authors introduced MV-algebras with an internal state, called state MV-algebras. (The letters MV stand for multi-valued.) In Di Nola and Dvurečenskij [‘State-morphism MV-algebras’, Ann. Pure Appl. Logic161 (2009), 161–173], a stronger version of state MV-algebras, called state-morphism MV-algebras, was defined. In this paper, we present the Loomis–Sikorski theorem for σ-complete MV-algebras with a σ-complete state-morphism-operator, showing that every such MV-algebra is aσ-homomorphic image of a tribe of functions with an internal state induced by a function where all the MV-operations are defined by points.

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.