A language-theoretic definition of word hyperbolic semigroup is given, which coincides with the original definition for a group. Word problems of semigroups are also considered.

In this paper we describe the relationship between characters of finitely generated torsion-free nilpotent groups of class 2 and their completions. In particular we give a detailed description of this connection for the discrete Heisenberg group.

Let $C$ be a curve of genus $g$ and $E$ a generic (semistable) vector bundle of rank $r$ and degree $d$. Fix a rank $r'\,{<}\,r$ and a degree $d'$ for subsheaves $E'$ of $E$. If $r'd-rd'=r'(r-r')(g-1)$, the number of such subbundles is finite. We shall denote this number by $m(r,d,r',g)$.

We show that a non-elementary Möbius group acting in higher dimensions is discrete if and only if its elementary subgroup whose elements fix every limit point is finite and every two-generator subgroup is discrete. This result gives an improvement over earlier results of Abikoff and Haas [1], Fang and Nai [5], Martin [11] and the author [9]. We also give an example to show that the first condition is necessary when dimension $n\ge 3$.

We give a closed formula for the multivariable Conway potential function of any graph link in a homology sphere. As corollaries, we answer three questions by Walter Neumann [8] about graph links.

We use the tangle model to study the action of the site-specific recombinase Gin, an enzyme that can introduce topological changes on circular DNA molecules. Gin and its bound DNA are modelled as a 2-string tangle which undergoes changes during recombination, thereby changing the topology of the DNA substrate. We show that the tangles involved in the analysis are all rational tangles. This technique allows us to prove that, under the model's assumptions, there is a unique topological description of the enzymatic action. The Gin system is one of the few to date where tangle analysis can be carried out systematically and rigorously, yielding a single, biologically reasonable solution.

We prove that every closed, connected contact 3-manifold can be obtained from $S^3$ with its standard contact structure by contact (${\pm}1$)-surgery along a Legendrian link. As a corollary, we derive a result of Etnyre and Honda about symplectic cobordisms (in a slightly stronger form).

The moduli space ${\frak N}^{c}_{g,1}$ of non-orientable surfaces of genus $g \geq 0$ with $c \geq 0$ distinguished points and one boundary curve is described via a model ${\frak P}(h,c)$: the space of configurations of $h=g+c+1$ pairs of parallel slits in $\mathbb{C}$. Based on this model, we prove that ${\frak N}^{c}_{g,1}$ is a non-orientable manifold, and we compute its homology for $h \leq 3$ with ${\mathbb Z}_2$-coefficients.

We give an elementary proof of the rational Hurewicz theorem and compute the rational cohomology groups of Eilenberg–MacLane spaces and the rational homotopy groups of spheres. Instead of using the Serre spectral sequence, we only assume the classical Hurewicz theorem, and give a short proof of the rational Gysin and Wang long exact sequences, which are applied inductively to the path fibration of Eilenberg–MacLane spaces.

We compute the cobordism rings of the classifying spaces of a certain class of Chevalley groups. In the particular case of the general linear group, we prove that it is isomorphic to the Chow ring.

Let $Q$ be any II$_1$–factor. It is shown that any standard lattice ${\euscript G}$ can be realized as the standard invariant of a free product of (several) rescalings of $Q$. In particular, if $Q$ has fundamental group equal to the positive reals and if $P$ is the free product of infinitely many copies of $Q$, then $P$ has subfactors giving rise to all possible standard invariants. Similarly, given a II$_1$–subfactor $N\subset M$, it is shown there are subfactors ${\hat N}\subset{\hat M}$ having the same standard invariant as $N\subset M$ but where ${\hat M}$, respectively ${\hat N}$, is the free product of $M$, respectively $N$, with rescalings of $Q$.

A Kolmogorov-type inequality is derived from a Jackson-type inequality and a Bernstein-type inequality. Several new results follow as corollaries.

We investigate inclusion indices for general function spaces, not necessarily symmetric. Using them, we estimate the grade of proximity between two spaces $E\,{\hookrightarrow}\, F$ when we have certain information on the inclusion. The results are based on ideas from interpolation theory.

We investigate, for a locally compact group $G$, the operator amenability of the Fourier–Stieltjes algebra $B(G)$ and of the reduced Fourier–Stieltjes algebra $B_r(G)$. The natural conjecture is that any of these algebras is operator amenable if and only if $G$ is compact. We partially prove this conjecture with mere operator amenability replaced by operator $C$-amenability for some constant $C\,{<}\, 5$. In the process, we obtain a new decomposition of $B(G)$, which can be interpreted as the non-commutative counterpart of the decomposition of $M(G)$ into the discrete and the continuous measures. We further introduce a variant of operator amenability – called operator Connes-amenability – which also takes the dual space structure on $B(G)$ and $B_r(G)$ into account. We show that $B_r(G)$ is operator Connes-amenable if and only if G is amenable. Surprisingly, $B({\mathbb F}_2)$ is operator Connes-amenable although ${\mathbb F}_2$, the free group in two generators, fails to be amenable.

Consider $e_n = \inf \exp \int \log \parallel x - f(x) \parallel dP(x)$, where $p$ is a probability measure on $\real^d$ and the infimum is taken over all measurable maps $f{:}\ \real^d \rightarrow \real^d$ with $| f(\real^d)| \leq n$. We study solutions $f$ of this minimization problem. For absolutely continuous distributions and for self-similar distributions we derive the exact rates of convergence to zero of the $n$th quantization error $e_n$ as $n \rightarrow \infty$. We establish a relationship between the quantization dimension that rules the rates and the Hausdorff dimension of $P$.

We consider singular block operator problems of the type arising in the study of stability of the Ekman boundary layer. The essential spectrum is located, and an analysis of the $L^2$ solutions of a related first order system of differential equations allows the development of a Titchmarsh–Weyl coefficient $M(\lambda)$. This, in turn, permits a rigorous analysis of the convergence of approximations to the spectrum arising from regular problems. Numerical results illustrate the theory.