it is shown that the natural map from the mapping class groups of surfaces to the automorphism groups of free groups induces an infinite loop map on the classifying spaces of the stable groups after plus construction. the proof uses automorphisms of free groups with boundaries which play the role of mapping class groups of surfaces with several boundary components.

three families of pairs of curves are presented; each pair consists of geometrically non-isomorphic curves whose jacobians are isomorphic to one another as unpolarized abelian varieties. each family is parametrized by an open subset of ${\mathbb p}^1$. the first family consists of pairs of genus-2 curves whose equations are given by simple expressions in the parameter; the curves in this family have reducible jacobians. the second family also consists of pairs of genus-2 curves, but generically the curves in this family have absolutely simple jacobians. the third family consists of pairs of genus-3 curves, one member of each pair being a hyperelliptic curve and the other a plane quartic. examples from these families show that in general it is impossible to tell from the jacobian of a genus-2 curve over $\mathbb q$ whether or not the curve has rational points – or indeed whether or not it has real points. the families are constructed using methods that depend on earlier joint work with franck leprévost and bjorn poonen, and on peter bending's explicit description of the curves of genus 2 whose jacobians have real multiplication by $\mathbb z[\sqrt{2}]$.

The classical Ramsey theorem states that every graph contains either a large clique or a large independent set. Here similar dichotomic phenomena are investigated in the context of finite metric spaces. Namely, statements are provided of the form ‘every finite metric space contains a large subspace that is nearly equilateral or far from being equilateral’. Two distinct interpretations are considered for being ‘far from equilateral’. Proximity among metric spaces is quantified through the metric distortion $\alpha$. Tight asymptotic answers are provided for these problems. In particular, it is shown that a phase transition occurs at $\alpha=2$.

The aim of the paper is to establish a complete scale of isomorphisms for elliptic pseudodifferential boundary value problems generated by the operators of the Boutet de Monvel algebra. The result is an extension of the corresponding known results on elliptic differential boundary value problems. Because for any elliptic pseudodifferential boundary value problem there exists a parametrix belonging to the Boutet de Monvel algebra, the proof presented is much shorter than the known proof for differential problems. Systems that are elliptic in the sense of Douglis and Nirenberg are also considered.

One of the consequences of the uniformization theorem of Koebe and Poincaré is that any smooth complex algebraic curve $C$ of genus $g > 1$ is conformally equivalent to ${\bb H}/G$ , where $G \subset \hbox{PSL}_2({\bb R})$ is a Fuchsian group and is naturally endowed with a hyperbolic metric. Conversely, any compact hyperbolic surface is isomorphic to an algebraic curve. Hence any curve of genus $g > 1$ may be described in two ways, either by an equation or by a Fuchsian group.

The quantum group Uq(SLn) introduced by Drinfel'd [2] and Jimbo [5] is a Hopf algebra which is naturally paired with Oq(sln), the coordinate ring of quantum SLn. When q is not a root of unity, the finite dimensional representation theory of Uq(sln) is essentially the same as that of U(sln). Furthermore, it is known that Uq(sln) is essentially a quasitriangular Hopf algebra [2]. When q is a root of unity the situation changes dramatically, and the representation theory of U(sln) is no longer effective. Moreover, Uq(sln) is not quasitriangular. In this case one can consider the quotient Hopf algebra Uq(sln)′, introduced by Lusztig [7], which is a finite dimensional Hopf algebra with a nice representation theory. It is well known that Uq(sl2)′ is quasitriangular. Finite dimensional quasitriangular Hopf algebras are important for the study of knot invariants [11, 12]. Thus, a natural question is: when is Uq(sln)′ quasitriangular? The somewhat unexpected answer is given in Theorem 3.7: it depends sharply on the greatest common divisor of n and the order of q1/2. For these Hopf algebras we classify all the possible R-matrices and give necessary and sufficient conditions for them to be minimal quasitriangular. These conditions depend again on n and the order of q1/2. In the process we describe the groups of Hopf automorphisms of Uq(sln)′ and Oq(SLn)′, where Oq(SLn)′ is a finite dimensional quotient Hopf algebra of Qq(SLn) proved to be the dual of Uq(sln)′ in [15].

The main result ensures that the scalar problem $x''\,{=}\,f(x)$, $x(0)\,{=}\,x_0$, $x'(0)\,{=}\,x_1,$ has a nonconstant locally $W^{2,1}$ solution if and only if there exists a nontrivial interval $J$ such that $x_0 \in J$, $f \in L^1_{\rm loc}(J)$, $x_1^2+2\int_{x_0}^{y}{f(s)\,ds} > 0$ for almost all $y \in J$ and \[\frac{\max\{1,|f|\}}{ \sqrt{x_1^2+2\int_{x_0}^{\cdot}{f(s)\,ds}}} \in L^1_{\rm loc}(J).\] Necessary and sufficient conditions for local and global uniqueness and for existence of periodic solutions are also established.

Graham Higman proved that a finitely generated group can be embedded in a finitely presented group if and only if it has a recursively enumerable set of defining relations. The analogue for lattice-ordered groups is considered here. Clearly, the finitely generated lattice-ordered groups that can be $\ell$-embedded in finitely presented lattice-ordered groups must have recursively enumerable sets of defining relations. The converse direction is proved for a special class of lattice-ordered groups.

THEOREM. Every finitely generated Abelian lattice-ordered group that has finite rank and a recursively enumerable set of defining relations can be$\ell$-embedded in a finitely presented lattice-ordered group.

If $\xi$ is a real number, let $D(\xi)$ be the Abelian rank 2 group $\Z^2$ with order $(m,n)>0$ if and only if $m+n\xi>0$.

COROLLARY. $D(\xi)$can be$\ell$-embedded in a finitely presented lattice-ordered group if and only if$\xi$is a recursive real number.

Thus an algebraic characterisation of recursive real numbers is obtained. In particular, $\pi$ is ‘$\ell$-algebraic’ in that it can be captured by finitely many relations in this language.

We consider the Sturm–Liouville equation \renewcommand{\theequation}{1.\arabic{equation}} \begin{equation} y^{\prime\prime}(x)+\{\lambda - q(x)\}y(x) = 0\quad (0 \le x < \infty) \end{equation} with a boundary condition at $x = 0$ which can be either the Dirichlet condition \begin{equation} y(0) = 0 \end{equation} or the Neumann condition \begin{equation} y^\prime(0) = 0. \end{equation} As usual, $\lambda$ is the complex spectral parameter with $0 \le \arg \lambda < 2\pi$ , and the potential $q$ is real-valued and locally integrable in $[0, \infty)$ .

Fix an integer $N\ge 2$. For a positive integer $n\in\Bbb N$, let $n=d_{0}(n)+d_{1}(n)N+d_{2}(n)N^{2}+\ldots+d_{\gamma(n)}(n)N^{\gamma(n)}$ where $d_{i}(n)\in\{0,1,2,\ldots,N-1\}$ and $d_{\gamma(n)}(n)\not=0$ denote the $N$-ary expansion of $n$. For a probability vector $\bold p=(p_{0},\ldots,p_{N-1})$ and $r>0$, the $r$ approximative discrete Besicovitch–Eggleston set $B_{r}(\bold p)$ is defined by $$B_{r}(\bold p) = \bigg\{n\in\Bbb N\,{|}\,\bigg\|\,\frac{|\{0\le k\le\gamma(n)\,{|}\,d_{k}(n)=i\}|}{\gamma(n)\,{+}\,1} - p_{i} \bigg| \le r \,\,\text{for all $i$} \bigg\}\,,$$ that is, $B_{r}(\bold p)$ is the set of positive integers $n$ such that the frequency of the digit $i$ in the $N$-ary expansion of $n$ differs from $p_{i}$ by less than $r$ for all $i\in\{0,1,2,\ldots,N-1\}$. Three natural fractional dimensions of subsets $E$ of ${\mathbb N}$ are defined, namely, the lower fractional dimension $\underline\dim(E)$, the upper fractional dimension $\overline\dim(E)$ and the exponent of convergence $\delta(E)$, and the dimensions of various subsets of ${\mathbb N}$ defined in terms of the frequencies of the digits in the $N$-ary expansion of the positive integers are studied. In particular, the dimensions of $B_{r}(\bold p)$ are computed (in the limit as $r\searrow 0$). Let ${\bold p}=(p_{0},\ldots,p_{N-1})$ be a probability vector. Then $$\lim_{r\searrow 0}\,\,\underline{\dim}(B_{r}({\bold p})) = \lim_{r\searrow 0}\,\,\overline{\dim}(B_{r}({\bold p})) = \lim_{r\searrow 0}\,\,\delta(B_{r}({\bold p})) = -\frac{\sum_{i}p_{i}\log p_{i}}{\log N}\,.$$ This result provides a natural discrete analogue of a classical result due to Besicovitch and Eggleston on the Hausdorff dimension of certain sets of non-normal numbers.

Several applications to the theory of normal numbers are given.

A coherent system ([Escr ], V) consists of a holomorphic bundle plus a linear subspace of its space of holomorphic sections. Motivated by the usual notion in geometric invariant theory, a notion of slope stability can be defined for such objects. In the paper it is shown that stability in this sense is equivalent to the existence of solutions to a certain set of gauge theoretic equations. One of the equations is essentially the vortex equation (that is, the Hermitian–Einstein equation with an additional zeroth order term), and the other is an orthonormality condition on a frame for the subspace V⊂H0([Escr ]).

Given a family of sets/vectors of the same cardinality/dimension you get the shadow by deleting one element/coordinate from a set/vector in all possible ways. You find the family with the smallest shadow by ordering all sets/vectors. The set case was solved by Kruskal (1963), and Katona (1966), and has many applications. We study two orderings which solve the 0, 1 vector case, and give the shadow size.

Various kinds of minimal 3-spheres with constant sectional curvature in the complex projective space $\mathbb{C}^n$ are studied.

The heat content asymptotics on a Riemannian manifold with smoooth boundary defined by Dirichlet, Neumann, transmittal and transmission boundary conditions are studied.

In the last decade much research has been done on questions related to circle packings, which are collections of circles in the plane with prescribed patterns of tangencies encoded by simplicial 2-complexes. The interest in circle packings and their connections with analytic functions was initiated by W. Thurston in 1985 when he suggested a method for approximation of conformal mappings using circle packings [20]. This method was shown to work by B. Rodin & D. Sullivan [15] later that year (see also [17]). Since then several results have been published (for example, [3, 4, 5, 10, 11, 14]) which seem to indicate the possibility of creating a discrete analog of the theory of analytic functions via circle packings. However, most of these results deal with univalent circle packings, that is, genuine packings of circles. Influenced by the idea of [3] that circle packings might be used to construct discrete parallels of analytic functions, we started in [7] to investigate non-univalent circle packings, particularly branched packings. We introduced there the notion of discrete Blaschke products and proved a branched version of the finite Riemann mapping theorem (see [20, 15]). We feel though that to have a solid foundation for a circle packing analog of the theory of analytic functions one needs something more. Since complex polynomials form a fundamental and very important class of analytic functions, it is natural to try to establish circle packing counterparts of these mappings. This is exactly the main theme of our paper.

Let P be a polyhedron in H3 of finite volume such that the group Γ(P) generated by reflections in the faces of P is a discrete subgroup of Isom H3. Let Γ+(P) denote the subgroup of index 2 consisting entirely of orientation-preserving isometries so that Γ+(P) is a Kleinian group of finite covolume. Γ+(P) is called a polyhedral group.

As discussed in [12] and [13] for example (see §2 below), associated to a Kleinian group Γ of finite covolume is a pair (AΓ, kΓ) which is an invariant of the commensurability class of Γ; kΓ is a number field called the invariant trace-field, and AΓ is a quaternion algebra over kΓ. It has been of some interest recently (cf. [13, 16]) to identify the invariant trace-field and quaternion algebra associated to a Kleinian group Γ of finite covolume since these are closely related to the geometry and topology of H3/Γ.

In this paper we give a method for identifying these in the case of polyhedral groups avoiding trace calculations. This extends the work in [15] and [11] on arithmetic polyhedral groups. In §6 we compute the invariant trace-field and quaternion algebra of a family of polyhedral groups arising from certain triangular prisms, and in §7 we give an application of this calculation to construct closed hyperbolic 3-manifolds with ‘non-integral trace’.

Let $p$ be a prime and let $q = p^a$, where $a$ is a positive integer. Let $G = G({\mathbb{F}}_q)$ be a Chevalley group over $\mathbb{ F}_q$, with associated system of roots $\Phi$ and Weyl group $W$. Steinberg showed in 1957 that $G$ has an irreducible complex representation whose degree equals the $p$-part of $|G|$ [11]. This representation, now known as the Steinberg representation, has remarkable properties, which reflect the structure of $G$, and there have been many research papers devoted to its study. The module constructed in [11] is in fact a right ideal in the integral group ring $\mathbb{ Z}G$ of $G$, and is thus a $\mathbb{ Z}G$-lattice, which we propose to call the Steinberg lattice of $G$. It should be noted that lattices not integrally isomorphic to the Steinberg lattice may also afford the Steinberg representation, and such lattices may differ considerably in their properties compared with the Steinberg lattice.

The commonly known properties of continuous measures such as the symmetry of the A∞ condition, its equivalence to the reverse Hölder inequality, the left-openness of the Ap condition, etc., are no longer necessarily true when the underlying measure is allowed to have atoms. The measures that preserve these properties are called good measures. The class of good measures is investigated and various criteria for a measure to belong to this class are presented.

A theorem of Macbeath asserts that μ(A+B)[ges ]min(1, μ(A)+μ(B)) for any subsets A and B of a finite-dimensional torus. We conjecture that, when the obvious exceptions are excluded, a stronger inequality

formula here

holds, and we prove this conjecture under some technical restrictions.

For a discrete group G there are two well known completions. The first is the Malcev (or unipotent) completion. This is a prounipotent group [Uscr ], defined over ℚ, together with a homomorphism ψ : G → [Uscr ] that is universal among maps from G into prounipotent ℚ-groups. To construct [Uscr ], it suffices for us to consider the case where G is nilpotent; the general case is handled by taking the inverse limit of the Malcev completions of the G/ΓrG, where Γ[bull ]G denotes the lower central series of G. If G is abelian, then [Uscr ] = G [otimes ] ℚ. We review this construction in Section 2.