Let $V$ be a complete nonsingular projective surface defined over an algebraic number field $k$, such that the Picard variety of $V$ is trivial and Pic($\bar{V}$) is torsion-free. Since our main interest is in necessary conditions for $V(k)$ not to be empty, we shall further assume that $V(k_v)$ is non-empty for every completion $k_v$ of $k$. We do not assume that $\bar{V}$ is rational, and indeed the case which primarily interests us is when $V$ is a K3 surface. Our objective is to describe effective ways of computing the Brauer–Manin obstructions to the existence of points on $V$ defined over $k$. Write \[ \hbox{Br}^0 (V) = \{\hbox{Ker}(\hbox{Br}(V)\longrightarrow\hbox{Br}(\bar{V}))\}/\{{\rm Im}({\rm Br}(k)\longrightarrow\hbox{Br}(V))\};\] then it is known that $\hbox{Br}^0(V)$ is isomorphic to ${\rm H}^1(k, \hbox{Pic}(\bar{V}))$, though to the best of our knowledge there is in general no known algorithm for computing either $\hbox{Br}^0(V)$ or this isomorphism. It is generally believed that $\hbox{Br}^0(V)$ contains all the information about the Brauer group Br$(V)$ which is useful in this context. Most of this paper is concerned with computing groups isomorphic to $\hbox{Br}^0(V)$, and with describing in terms of these groups the Brauer–Manin obstructions coming from elements of $\hbox{Br}^0(V)$.

In an earlier paper, we solved LeVeque's problem, to establish a central limit theorem for the number of solutions of the diophantine inequality \[ \left\vert x-\frac{p}{q}\right\vert\leq\frac{f(\log q)}{q^2} \] in unknowns $p,q$ with $q\,{>}\,0$, where $f$ is a function satisfying special assumptions and $x$ is chosen randomly in the unit interval. Here, we are interested in the almost sure behavior of the solution set. In particular, we obtain a generalized law of the iterated logarithm and we prove a result that gives strong evidence that the law of the iterated logarithm with the standard norming sequence (suggested by the central limit theorem) holds as well. Both results have to be compared with a theorem of W. M. Schmidt; they imply an inverse to Schmidt's theorem and a strong law of large numbers with an error term that is essentially better than the one provided by Schmidt's result.

We prove that from a topological point of view most numbers fail to be normal in a spectacular way. For an integer $N \geq 2$ and $x \in [0, 1]$, let $x = \sum^\infty_{n=1} \varepsilon_{N,n}(x)/N^n$, where $\varepsilon_{N, n}(x) \in \{0, 1,\ldots, N - 1\}$ for all $n$, denote the unique non-terminating $N$-adic expansion of $x$. For a positive integer $n$ and a finite string ${\bf i} = i_1\cdots i_k$ with entries $i_j \in \{0, 1,\ldots, N - 1\}$, we write $$ \Pi_N (x,{\bf i};n) = \frac{|\{1 \le i \le n | \varepsilon_{N,i}(x) = i_1, \ldots, \varepsilon_{N,i+k-1} (x) = i_k\}|}{n} $$ for the frequency of the string ${\bf i}$ among the first $n$ digits in the $N$-adic expansion of $x$, and let $\Pi_N^k(x;n) \,{=}\, (\Pi_N(x,{\bf i};n))_{\bf i}$ denote the vector of frequencies $\Pi_N(x,{\bf i};n)$ of all strings ${\bf i} = i_1\cdots i_k$ of length $k$ with entries $i_j \in \{0, 1,\ldots, N-1\}$. We say that a number $x$ is extremely non-normal if each shift invariant probability vector in ${\mathbb R}^{N^k}$ is an accumulation point of the sequence $(\Pi_N^k(x;n))_n$simultaneously for all $k$ and all bases $N$, and we denote the set of extremely non-normal numbers by ${\mathbb E}$, i.e. \begin{eqnarray*} &&{\mathbb E} = \bigcap_N \bigcap_k \big\{x \in [0,1]|\, {\rm each}\, {\bf p} \in \Gamma^k_N\\ &&\quad\hbox{is an accumulation point of the sequence } (\Pi_N (x, {\bf i};n))_n\big\}, \end{eqnarray*} where $\Gamma^k_N$ denotes the simplex of shift invariant probability vectors in ${\mathbb R}^{N^k}$. Our main result says that ${\mathbb E}$ is a residual set, i.e. the complement $[0,1]\backslash {\mathbb E}$ is of the first category. Hence, from a topological point of view, a typical number in [0, 1] is as far away from being normal as possible. This result significantly strengthens results by Maxfield and Schmidt. We also determine the Hausdorff dimension and the packing dimension of ${\mathbb E}$.

We give a local estimate of the shape of Schottky space at certain boundary points. This is done by studying the geometric limits of sequences of cyclic groups and applying the results to the generators of Schottky groups. We also obtain information on discrete, non-faithful representations close to the cusps.

Given a surface $S$ and an integer $r\,{\ge}\,1$, there is a variety $X_{r-1}$ parametrizing all clusters of $r$ proper and infinitely near points of $S$. We study the geometry of the varieties $X_r$, showing that for every Enriques diagram ${\bf D}$ of $r$ vertices the subset ${\mathit{Cl}}({\bf D})\,{\subset}\,X_{r-1}$ of the clusters with Enriques diagram ${\bf D}$ is locally closed. We study also the relative positions of the subvarieties ${\mathit{Cl}}({\bf D})$, showing that they do not form a stratification and giving criteria for adjacencies between them.

Let $L$ be an unsplittable, prime, oriented, alternating link type in $S^3$. Let $D$ be a reduced alternating diagram representing $L$. We define the Murasugi atoms of $D$ as the oriented link types represented by the prime factors of the Murasugi special components of $D$. We prove (an invariance theorem) that the collection of Murasugi atoms depends only on $L$ and not on $D$. This has the following corollary. Let $L$ be as above and assume that $L$ is achiral. Write its HOMFLY polynomial as $P_{L}(v,z)\,{=}\,\sum_{m}^{M} b_{j}(v) z^j$. Then $b_{M}(v)\,{=}{\pm}\, \beta(v) \beta(v^{-1})$ for some polynomial $\beta(v) \in\mathbb{Z}[v, v^{-1}]$. As a consequence, the leading coefficient of the Conway polynomial of $L$ is a square (up to sign).

The notion of almost alternating links was introduced by C. Adams et al ([1]). We give a sufficient condition for an almost alternating link diagram to represent a non-splittable link. This solves a question asked in [1, 3]. A partial solution for special almost alternating links has been obtained by M. Hirasawa ([5]). As an application, Theorem 2.2 gives us a simple finite algorithm to decide if a given almost alternating link diagram represents a splittable link without increasing the number of crossings of diagrams in the process. Moreover, we show that almost alternating links with more than two components are non-trivial. In Section 2, we state these results in detail. To prove our theorem, we use a technique invented by W. Menasco (see [6, 7]), this is reviewed in Section 3. We also apply “the charge and discharge method” to our graph-theoretic argument, which was used to prove the four color theorem in [4], we explain this in Section 4.

A classical E-infinity operad is formed by the bar construction of the symmetric groups. Such an operad has been introduced by M. Barratt and P. Eccles in the context of simplicial sets in order to have an analogue of the Milnor FK-construction for infinite loop spaces. The purpose of this paper is to prove that the associative algebra structure on the normalized cochain complex of a simplicial set extends to the structure of an algebra over the Barratt–Eccles operad. We also prove that differential graded algebras over the Barratt–Eccles operad form a closed model category. Similar results hold for the normalized Hochschild cochain complex of an associative algebra. More precisely, the Hochschild cochain complex is acted on by a suboperad of the Barratt–Eccles operad which is equivalent to the classical little squares operad.

In the metastable range $3 \leq p < q < 2p - 3$, and modulo the action of the group of homotopy spheres, we classify diffeomorphism types of smooth manifolds that are homotopy equivalent to the products $S^{p} \times S^{q}$ of spheres.

We prove the toral rank conjecture of Halperin in some new cases. Our results apply to certain elliptic spaces that have a two-stage Sullivan minimal model, and are obtained by combining new lower bounds for the dimension of the cohomology and new upper bounds for the toral rank. The paper concludes with examples and suggestions for future work.

We propose a method of constructing examples in operator ergodic theory which unifies and extends some previously known examples. It also allows us to answer several questions that have been open for some time (including a question of Allan [1]).

Consider a system of diagonal equations \begin{equation}\sum_{j=1}^sa_{ij}x_j^k=0\quad (1\le i\le r),\end{equation} satisfying the property that the (fixed) integral coefficient matrix $(a_{ij})$ contains no singular $r\times r$ submatrix. A recent paper of the authors [3] establishes that whenever $k\ge 3$ and $s>(3r+1)2^{k-2}$, then the expected asymptotic formula holds for the number $N(P)$ of integral solutions ${\bf x}$ of ($1{\cdot}1$) with $|x_i|\le P$$(1\le i\le s)$.

Spacetimes with collisionless matter evolving from data on a compact Cauchy surface with hyperbolic symmetry are shown to be timelike and null geodesically complete in the expanding direction, provided the data satisfy a certain size restriction.

We study the distribution of closed geodesics in homology on a finite area hyperbolic surface. We obtain an estimate which is uniform as the homology class varies, refining an asymptotic formula due to C. Epstein.