Given the Diophantine equation a2+b2+c2=3abc, a solution triple of natural numbers (a, b, c) can be arranged in ascending order so that a[les ]b[les ]c. Then, given the largest element c, one can ask whether this uniquely determines the triple. This is referred to as the Markoff conjecture. The paper proves that, if c is prime, then there is indeed only one triple that solves the equation with c as the largest element. The proof uses only standard algebraic number theory, but it was prompted by geometric considerations.

For any given real number $\alpha$ with bounded partial quotients, we construct explicitly continuum many real numbers $\beta$ with bounded partial quotients for which the pair $(\alpha, \beta)$ satisfies a strong form of the Littlewood conjecture. Our proof is elementary and rests on the basic theory of continued fractions.

In this paper we study quaternion-Kähler manifolds endowed with an isometric S1-action. We consider the corresponding moment map μ and prove that the only compact quaternion-Kähler manifold with positive scalar curvature which admits an isometric circle action free on μ−1(0) is the quaternionic projective space ℍPn.

We present the classification of all real hypersurfaces in complex hyperbolic space $\mathbb{C}H^{n}$, $n \geq 3$, with three distinct constant principal curvatures.

We introduce two related families of polynomials, easily computable by simple recursions into which any Kazhdan–Lusztig (and inverse Kazhdan–Lusztig) polynomial of any Coxeter group can be expanded linearly, and we give combinatorial interpretations to the coefficients in these expansions. This yields a combinatorial rule for computing the Kazhdan–Lusztig polynomials in terms of paths in a directed graph, and a completely combinatorial reformulation of the nonnegativity conjecture [15, p. 166].

The moduli theory of Calabi–Yau threefolds is investigated, and using Griffiths' work on the period map, some finiteness results are derived. In particular, a prediction of Morrison's cone conjecture is confirmed.

The first nontrivial DCE (2-computably enumerable) Turing approximation to the class of computably enumerable degrees is obtained. This depends on the following extension of the splitting theorem for the DCE degrees. For any DCE degree ${\bf a}$ and any computably enumerable degree ${\bf b}$ , if ${\bf b} < {\bf a}$ , then there are DCE degrees ${\bf x_0}, {\bf x_1}$ such that ${\bf b} < {\bf x_0}, {\bf x_1} < {\bf a}$ and ${\bf a} = {\bf x_0} \lor {\bf x_1}$ . The construction is unusual in that it is incompatible with upper cone avoidance.

Fundamental solutions to second-order parabolic and elliptic equations with lower-order terms are studied. New sufficient conditions on the coefficients of the lower-order terms are given that guarantee the validity of two-sided Gaussian bounds on the fundamental solution to the parabolic equation, locally and globally in time. As an application, the problem of existence and non-existence of positive solutions to a class of semi-linear equations is studied.

Commutative rings for which every ideal can be written as a finite product of invertible and prime ideals are characterized.

Let [Gscr ] be a family of functions analytic in a domain D in the complex plane. It is proved that [Gscr ] is a normal family, provided that for each f∈[Gscr ], there exists k = k(f) > 1 such that the kth iterate fk has no repulsive fixpoint in D. A new proof of a result of Bergweiler and Terglane concerning the dynamics of entire functions is also given.

There has been interest recently concerning when a left ordered group is locally indicable. Chiswell and Kropholler proved that every left ordered solvable-by-finite group is locally indicable, while Bergman gave examples of left ordered groups which are not locally indicable. This paper proves that every left ordered elementary amenable group is locally indicable. Every solvable-by-finite group is elementary amenable, and every elementary amenable group is amenable. The author leaves it as an open problem as to whether every left ordered amenable group is locally indicable.

Recently Bergelson and Leibman proved an extension to van der Waerden's theorem involving polynomials. They also generalised the Hales–Jewett theorem in a similar way. The paper presents short and purely combinatorial proofs of those results.

Ideas from string theory and quantum field theory have been the motivation for new invariants of knots and 3-dimensional manifolds which have been constructed from complex algebraic structures such as Hopf algebras [17, 22], monoidal categories with additional structure [24], and modular functors [14, 23]. These constructions are closely related. Here we take a unifying categorical approach based on a natural 2-dimensional generalisation of a topological field theory in the sense of Atiyah [1], and show that the axioms defining these complex algebraic structures are a consequence of the underlying geometry of surfaces.

A group $G$ is said to have the Bergman property (the property of uniformity of finite width) if given any generating $X$ with $X=X^{-1}$ of $G$, we have that $G=X^k$ for some natural $k$, that is, every element of $G$ is a product of at most $k$ elements of $X$. We prove that the automorphism group $\operatorname{Aut}(N)$ of any infinitely generated free nilpotent group $N$ has the Bergman property. Also, we obtain a partial answer to a question posed by Bergman by establishing that the automorphism group of a free group of countably infinite rank is a group of uniformly finite width.

We consider bihomogeneous polynomials on complex Euclidean spaces that are positive outside the origin and obtain effective estimates on certain modifications needed to turn them into squares of norms of vector-valued polynomials on complex Euclidean space. The corresponding results for hypersurfaces in complex Euclidean spaces are also proved. The results can be considered as Hermitian analogues of Hilbert's seventeenth problem on representing a positive definite quadratic form on $\mathbb{R}^n$ as a sum of squares of rational functions. They can also be regarded as effective estimates on the power of a Hermitian line bundle required for isometric projective embedding. Further applications are discussed.

Introducing the framework of pseudo-motivic homology, the paper finishes the proof that the Brauer–Manin obstruction is the only obstruction to the local–global principle for zero-cycles on a Severi–Brauer fibration of squarefree index over a smooth projective curve over a number field, provided that the Tate–Shafarevich group of the Jacobian of the base curve is finite. More precisely, for such a variety the Chow group of global zero-cycles is dense in the subgroup of collections of local cycles that are orthogonal to the (cohomological) Brauer group of the variety.

The distribution of binomial coefficients in residue classes modulo prime powers and with respect to the p-adic valuation is studied. For this purpose, general asymptotic results for arithmetic functions depending on blocks of digits with respect to q-ary expansions are established.

Certain linear operators from a Banach algebra $A$ into a Banach $A$-bimodule $X$, which are called approximately local derivations, are studied. It is shown that when $A$ is a ${\rm C^*}$-algebra, a Banach algebra generated by idempotents, a semisimple annihilator Banach algebra, or the group algebra of a SIN or a totally disconnected group, bounded approximately local derivations from $A$ into $X$ are derivations. This, in particular, extends a result of B. E. Johnson that ‘local derivations on ${\rm C^*}$-algebras are derivations’ and provides an alternative proof of it.

Büchi's problem asked whether there exists an integer $M$ such that the surface defined by a system of equations of the form

$$x_{n}^2+x_{n-2}^2=2x_{n-1}^2+2,\quad n=2,\dotsc, M-1,$$

has no integer points other than those that satisfy $\pm x_n=\pm x_0+n$ (the $\pm$ signs are independent). If answered positively, it would imply that there is no algorithm which decides, given an arbitrary system $Q=(q_1,\dotsc,q_r)$ of integral quadratic forms and an arbitrary $r$-tuple $B=(b_1,\dotsc,b_r)$ of integers, whether $Q$ represents $B$ (see T. Pheidas and X. Vidaux, Fund. Math. 185 (2005) 171–194). Thus it would imply the following strengthening of the negative answer to Hilbert's tenth problem: the positive-existential theory of the rational integers in the language of addition and a predicate for the property ‘$x$ is a square’ would be undecidable. Despite some progress, including a conditional positive answer (depending on conjectures of Lang), Büchi's problem remains open.

In this paper we prove the following:

1. an analogue of Büchi's problem in rings of polynomials of characteristic either 0 or $p\geq17$ and for fields of rational functions of characteristic 0; and

2. an analogue of Büchi's problem in fields of rational functions of characteristic $p\geq19$, but only for sequences that satisfy a certain additional hypothesis.

Let $F$ be a field of characteristic either 0 or at least 17 and let $t$ be a variable. Let $L_{t}$ be the first order language which contains symbols for 0 and 1, a symbol for addition, a symbol for the property ‘$x$ is a square’ and symbols for multiplication by each element of the image of $\mathbb{Z}[t]$ in $F[t]$. Let $R$ be a subring of $F(t)$, containing the natural image of $\mathbb{Z}[t]$ in $F(t)$. Assume that one of the following is true:

1. $R\subset F[t]$;

2. the characteristic of $F$ is either 0 or $p\geq19$.

An ovoid in a 3-dimensional projective geometry PG(3, q) over the field GF(q), where q is a prime power, is a set of q2+1 points no three of which are collinear. Because of their connections with other combinatorial structures ovoids are of interest to mathematicians in a variety of fields; for from an ovoid one can construct an inversive plane [3], a generalised quadrangle [11], and if q is even, a translation plane [13]. In fact the only known finite inversive planes are those arising from ovoids in projective spaces (see [2]). Moreover, there are only two classes of ovoids known, namely the elliptic quadric and, for q even and not a square, the Tits ovoids; these will be described in the next section. If the field order q is odd, then it was shown by Barlotti and Panella (see [3, 1.4.50]) that the only ovoids are the elliptic quadrics. This paper contains a geometrical characterisation of the two known classes of ovoids.