This paper concerns involutions and anti-automorphisms on finite-dimensional associative algebras over a field. It is based on a survey talk given at the workshop on ‘Quadratic forms, algebras with involution and algebraic K-theory’ at the University of Nottingham, 13–15 May 2004. This workshop was part of the EU network HPRN-CT-2002-00287 on ‘Algebraic $K$-theory, linear algebraic groups and related structures’ and followed the London Mathematical Society Midlands Regional Meeting in Nottingham on 12 May, 2004.

The contents of the article are as follows:

1. Introduction and historical remarks;

2. Some general results;

3. Central simple algebras;

4. Group algebras.

In his celebrated memoir, Morgan Ward's definition of elliptic divisibility sequences has the remarkable feature that it does not become at all clear until deep into the paper that there exist nontrivial examples of such sequences. Even then, Ward's proof of the coherence of his definition relies on displaying a sequence of values of quotients of Weierstraß $\sigma$-functions. We give a direct proof of coherence and show, rather more generally, that a sequence defined by a so-called Somos relation of width 4 is always also given by three-term Somos relations of all larger widths $5, 6, 7, \ldots.$

In this paper, we apply Moriwaki's arithmetic height functions to obtain an analogue of Silverman's specialization theorem for families of Abelian varieties over $K$, where $K$ is any field finitely generated over $\mathbb{Q}$.

The $\ell$-adic parabolic cohomology groups attached to noncongruence subgroups of $\mathop{\rm SL_2(\mathbb{Z})$ are finite-dimensional $\ell$-adic representations of $\mathop{\rm Gal}(\overline{/mathbb{K}})$ for some number field $K$. We exhibit examples (with $K=\overline{\mathbb{Q}}$) for which the primitive parts give Galois representations whose images are open subgroups of the full group of symplectic similitudes (of arbitrary dimension). The determination of the image of the Galois group relies on Katz's classification theorem for semisimple subalgebras of $\mathfrak{sl}_n$ containing a principal nilpotent element, for which we give a short conceptual proof, suggested by I. Grojnowski.

We prove that Schanuel's conjecture for the reals is equivalent to a uniform version of itself.

It is proved that a ring $R$ is right Artinian if and only if, for each countably generated right $R$-module $M$, there exists a finite subset $F$ of $M$ such that the annihilator of $M$ in $R$ equals the annihilator of $F$ in $R$.

Recently, J. Roe proved by a direct computation that the Baum–Connes assembly map can be expressed as a boundary map associated to a certain short exact sequence of C*-algebras. In this paper we deduce the same result as a corollary of the author's characterisation of the Baum–Connes assembly map.

This paper introduces the concept of orbit-homogeneity of permutation groups: a group $G$ is orbit-$t$-homogeneous if two sets of cardinality $t$ lie in the same orbit of $G$ whenever their intersections with each $G$-orbit have the same cardinality. For transitive groups, this coincides with the usual notion of $t$-homogeneity. This concept is also compatible with the idea of partition transitivity introduced by Martin and Sagan.

Further, this paper shows that any group generated by orbit-$t$-homogeneous subgroups is orbit-$t$-homogeneous, and that the condition becomes stronger as $t$ increases up to $\lfloor n/2\rfloor$, where $n$ is the degree. So any group $G$ has a unique maximal orbit-$t$-homogeneous subgroup $\Omega_t(G)$, and $\Omega_t(G)\le\Omega_{t-1}(G)$. Some structural results for orbit-$t$-homogeneous groups, and a number of examples, are also given.

We prove that certain universality properties of the partial sums force a power series to have Ostrowski-gaps. This has interesting consequences for some classes of universal functions.

A compact subset of the unit circle is an interpolation set for conformal mappings if and only if it has logarithmic capacity zero.

Semigroups (with zero) and inverse semigroups (with zero) are constructed from subshifts. The semigroups and inverse semigroups that are obtained by this construction are characterized and classified.

Let $G$ be a compactly generated, locally compact group of polynomial growth. Removing a restrictive technical condition from a previous work, we show that the weighted group algebra $L^{1}_{\omega}(G)$ is a symmetric Banach $*$-algebra if and only if the weight function $\omega $ satisfies the GRS-condition. This condition expresses in a precise technical sense that $\omega $ grows subexponentially.

We characterize composition operators on spaces of real analytic functions which at the same time have closed image and are open onto their images. Under some mild assumptions, we also characterize composition operators with closed range and composition operators open onto their images.

For each non-exact C*-algebra $A$ and infinite compact Hausdorff space $X$ there exists a continuous bundle ${\mathcal B}$ of C*-algebras on $X$ such that the minimal tensor product bundle $A \otimes {\mathcal B}$ is discontinuous. The bundle ${\mathcal B}$ can be chosen to be unital with constant simple fibre. When $X$ is metrizable, ${\mathcal B}$ can also be chosen to be separable. As a corollary, a C*-algebra $A$ is exact if and only if $A\otimes {\mathcal B}$ is continuous for all unital continuous C*-bundles ${\mathcal B}$ on a given infinite compact Hausdorff base space. The key to proving these results is showing that for a non-exact C*-algebra $A$ there exists a separable unital continuous C*-bundle ${\mathcal B}$ on [0,1] such that $A\otimes {\mathcal B}$ is continuous on [0,1] and discontinuous at 1, a counter-intuitive result. For a non-exact C*-algebra $A$ and separable C*-bundle ${\mathcal B}$ on [0,1], the set of points of discontinuity of $A\otimes{\mathcal B}$ in [0,1] can be of positive Lebesgue measure, and even of measure 1.

A Radon measure $\mu$ on ${mathbb R}^n$ is said to be $k$-monotone if $r\mapsto{\mu(B(x,r))}/{r^k}$ is a non-decreasing function on $(0,\infty)$ for every $x\in {\mathbb R}^n$. (If $\mu$ is the $k$-dimensional Hausdorff measure restricted to a $k$-dimensional minimal surface then this important property is expressed by the monotonicity formula.) We give an example of a 1 -monotone measure $\mu$ in ${\mathbb R}^2$ with non-unique and non-conical tangent measures at a point. Furthermore, we show that $\mu$ can be the one-dimensional Hausdorff measure restricted to a closed set $A\subset {\mathbb R}^2$.

We give a general method for expressing various characteristic numbers of the multiple-point manifolds of an immersion.

Given a first order elliptic partial differential equation we construct a solution which solves a given Riemann–Hilbert boundary value problem whose coefficients have singularities of the first kind at a finite number of some prescribed isolated points and are Holder–continuous outside those points while the free term has a finite number of integrable power singularities at some prescribed points. It is shown that the solution belongs to some weighted Sobolev space $W_{1, p}(D; \rho)$, where the weight function $\rho = \rho(z; \partial D)$ is the distance of the variable point $z$ from the boundary $\partial D$ raised to a certain power. The problem is solved by first reducing it to an analogous problem for holomorphic functions. The latter is then solved.