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.

Regenerative phenomena were introduced some forty years ago to address problems in the theory of continuous-time Markov processes. The early work in the theory left a number of difficult unsolved problems, in the classification of $p$-functions, oscillation and inequalities, the multiplicative theory, and the theory of unbounded semi-$p$-functions. Recent years have shown progress on all of these fronts, and this paper surveys these results, while drawing attention to significant problems that remain open.

This paper provides a survey of both old and new results about minimal surfaces and submanifolds.

The relative rank $\mbox{\rm rank}(S:U)$ of a subsemigroup $U$ of a semigroup $S$ is the minimum size of a set $V\subseteq S$ such that $U$ together with $V$ generates the whole of $S$. As a consequence of a result of Sierpiński, it follows that for $U\leq \mathcal{T}_{X}$, the monoid of all self-maps of an infinite set $X$, $\mbox{\rm rank}(\mathcal{T}_{X}:U)$ is either 0, 1 or 2, or uncountable. In this paper, the relative ranks $\mbox{\rm rank}(\mathcal{T}_{X}:\mathcal{O}_{X})$ are considered, where $X$ is a countably infinite partially ordered set and $\mathcal{O}_{X}$ is the endomorphism monoid of $X$. We show that $\mbox{\rm rank}(\mathcal{T}_{X}:\mathcal{O}_{X})\leq 2$ if and only if either: there exists at least one element in $X$ which is greater than, or less than, an infinite number of elements of $X$; or $X$ has $|X|$ connected components. Four examples are given of posets where the minimum number of members of $\mathcal{T}_{X}$ that need to be adjoined to $\mathcal{O}_{X}$ to form a generating set is, respectively, 0, 1, 2 and uncountable.

Twenty-five years ago, Conway and Norton published in this journal their remarkable paper ‘Monstrous Moonshine’, proposing a completely unexpected relationship between finite simple groups and modular functions. We review the progress made in broadening and understanding that relationship.

Tao has shown that hard summation (summing Fourier-type series by using terms in order of decreasing size of the Fourier coefficients) works for wavelets, but the present author has shown that it fails for classical Fourier series. This paper, which is intended for a general audience, exhibits the underlying ideas in the context of Haar and Walsh series, where many of the proofs simplify.

Power series $\sum_{n=0}^\infty f(n)x^n$ with non-zero convergence radius $R(f)$ are considered, and the arithmetical nature (that is, irrationality, or even transcendence) of the corresponding multivariate series $\smash{\sum_{i_1,\ldots,i_m\!=0}^\infty f(i_1+\ldots +i_m)x_1^{i_1}\cdot\ldots \cdot x_m^{i_m}}$ is studied if $x_1,\ldots,x_m$ and the sequence $(f(n))$ satisfy appropriate arithmetical conditions. It follows that such arithmetical results can be written down easily if linear independence results on the function $F(x)$, defined in $|x|\,{<}\,R(f)$ by the original one-dimensional power series, and possibly its derivatives at the points $x_\mu$ are known. Some typical applications are explicitly stated.

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 $5, 6, 7, \ldots.$

We determine the intersection theory on the Igusa compactification and the second Voronoi compactification of ${\cal A}_4$.

Let $K$ be the (real closed) field of Puiseux series in $t$ over ${\mathbf R}$ endowed with the natural linear order. Then the elements of the formal power series rings ${\mathbf R}[\![\xi_1,\dots,\xi_n]\!]$ converge $t$-adically on $[-t,t]^n$, and hence define functions $[-t,t]^n\to K$. Let ${\mathcal L}$ be the language of ordered fields, enriched with symbols for these functions. By Corollary 3.15, $K$ is o-minimal in ${\mathcal L}$. This result is obtained from a quantifier elimination theorem. The proofs use methods from non-Archimedean analysis.

In this paper we study several kinds of maximal almost disjoint families. In the main result of this paper we show that for successor cardinals $\kappa$, there is an unexpected connection between invariants $\frac{a}_{e}(\kappa), \frac{b} (\kappa)$ and a certain cardinal invariant $\frac{m}_{d}(\kappa^{+})$ on $\kappa^{+}$. As a corollary we get for example the following result. For a successor cardinal $\kappa$, even assuming that $\kappa^{<\kappa}=\kappa$ and $2^{\kappa}=\kappa^{+}$, the following is not provable in Zermelo–Fraenkel set theory. There is a $\kappa^{+}$-cc poset which does not collapse $\kappa$ and which forces $\frac{a} (\kappa)=\kappa^{+}<\frac{a}_{e}(\kappa)=\kappa^{++}=2^{\kappa}$. We also apply the ideas from the proofs of these results to study $\frac{a} =\frac{a} (\omega)$ and $\hbox{\rm non}(\cal{M})$.

Let $p$ be a prime for which the congruence group $\Gamma_0(p)^*$ is of genus zero, and let $j_p^*$ be the corresponding Hauptmodul. Let $f$ be a nearly holomorphic modular form of weight 1/2 on $\Gamma_0(4p)$ which satisfies some congruence condition on its Fourier coefficients. We interpret $f$ as a vector valued modular form. Applying Borcherds lifting of vector valued modular forms we construct infinite products associated to $j_p^*$ and relate them to Zagier's trace formula for the singular values of $j_p^*$.

For any $n \geq 3$, let $F \in \mathbb{Z}[X_0,\ldots,X_n]$ be a form of degree $d\geq 5$ that defines a non-singular hypersurface $X \subset \mathbb{P}^{n}$. The main result in this paper is a proof of the fact that the number $N(F;B)$ of $\mathbb{Q}$-rational points on $X$ which have height at most $B$ satisfies $N(F;B)=O_{d,\varepsilon,n}(B^{n-1+\varepsilon}),$ for any $\varepsilon >0$. The implied constant in this estimate depends at most upon $d$, $\varepsilon$ and $n$. New estimates are also obtained for the number of representations of a positive integer as the sum of three $d$th powers, and for the paucity of integer solutions to equal sums of like polynomials.

We study the extent to which sets $A \subseteq \mathbb{Z}/N\mathbb{Z}, N$ prime, resemble sets of integers from the additive point of view (‘up to Freiman isomorphism’). We give a direct proof of a result of Freiman, namely that if $|A + A| \leq K|A|$ and $|A| < c(K)N$, then $A$ is Freiman isomorphic to a set of integers. Because we avoid appealing to Freiman's structure theorem, we obtain a reasonable bound: we can take $\;c(K) \geq (32K)^{-12K^2}$. As a byproduct of our argument we obtain a sharpening of the second author's result on sets with small sumset in torsion groups. For example, if $A \subseteq \mathbb{F}_2^n$, and if $|A + A| \leq K|A|$, then $A$ is contained in a coset of a subspace of size no more than $K^22^{2K^2 - 2}|A|$.

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 $\mathbb{Q}$.

We prove tight closure analogues of results of Watanabe about chains and families of integrally closed ideals.

The statement of the numerical values $\kappa$ and $z_0$ on page 167 of [1, Section 3] contained an error. The values that were actually used were (to nine decimal places): $$\begin{eqnarray*} \kappa &=& 2 \omega_1 -\int_1^\infty (4t^3-4t+1)^{- {1}/{2}}\,dt \\[4pt] &=&1.859\,185\,431, \\[6pt] z_0&=&2\omega_3+\int_{-1}^\infty (4t^3-4t+1)^{- {1}/{2}}\,dt \\[4pt] &=&0.204\,680\,500+1.225\,694\,691i, \end{eqnarray*}$$ these being shifted, by the periods $2\omega_1$ and $2\omega_3$ respectively, compared with the values given in [1] (with $\omega_1=1.496\,729\,323$ and $\omega_3=1.225\,694\,691i$). With $\tau_0=\tau_1 =1$ and $\sigma (z)$ denoting the sigma function $\sigma (z;g_2,g_3)$ with invariants $g_2=4$, $g_3=-1$ associated with the elliptic curve given by equation (3.2), these values of $\kappa$ and $z_0$ yield $$\begin{eqnarray*} \sigma (\kappa )&=&1.555\,836\,426, \\[4pt] A=\frac{\tau_0}{\si (z_0 )} &=& 0.112\,724\,016-0.824\,911\,686i, \\[4pt] B=\frac{\si (\ka )\si (z_0 )\,\tau_1} {\si (z_0+\ka )\,\tau_0}&=&0.215\,971\,963+0.616\,028\,193i. \end{eqnarray*}$$ and the latter three values all agree with those stated in the paper (apart from rounding down the last digit in the imaginary part of $A$).

It is a well-known problem of Von Neumann to discover whether the countable chain condition and weak distributivity of a complete Boolean algebra imply that it carries a strictly positive probability measure. It was shown recently by Balcar, Jech and Pazák, and by Veličković, that it is consistent with ZFC, modulo the consistency of a supercompact cardinal, that every ccc weakly distributive complete Boolean algebra carries a contiuous strictly positive submeasure – that is, it is a Maharam algebra. We use some ideas of Gitik and Shelah and implications from the inner model theory to show that some large cardinal assumptions are necessary for this result.

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.

A description is given of serial rings whose maximal quotient rings are quasi-Frobenius (QF). Every serial ring is a factor of a serial ring whose maximal quotient ring is a QF-ring. This result is used to give a new, conceptual proof for the selfduality of serial rings, emphasising the importance of weakly symmetric rings.