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.

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 $\hbox{\rm non}(\cal{M})$.

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 $K^22^{2K^2 - 2}|A|$.

A cyclotomic polynomial $\Phi(z)$ is said to be of order 3 if $n=pqr$ for three distinct odd primes $p$, $q$, and $r$. We establish the existence of an infinite family of such polynomials whose coefficients do not exceed 1 in modulus.

Let $m,\,g,\,q \in\mathbb{N}$ with $q\,{\geq}\, 2$ and $(m,q-1)=1$. For $n\in\mathbb{N}$, denote by $s_q(n)$ the sum of digits of $n$ in the $q$-ary digital expansion. Given a polynomial $f$ with integer coefficients, degree $d\ge 1$, and such that $f(\mathbb{N})\subset\mathbb{N}$, it is shown that there exists $C=C(f,m,q)>0$ such that for any $g\in\mathbb{Z}$, and all large $N$, \[|\{ 0\,{\leq}\, n\,{\leq}\, N : s_q(f(n))\md gm\}|\,{\geq}\, CN^{\min(1,2/d!)}\]. In the special case $m=q=2$ and $f(n)=n^2$, the value $C=1/20$ is admissible.

We consider the sequences of fractional parts $\{\xi \alpha^n\}, n \,{=}\,1,2,3,\dots,$ and of integer parts $[\xi \alpha^n], n \,{=}\,1,2,3,\dots,$ where $\xi$ is an arbitrary positive number and $\alpha\,{>}\,1$ is an algebraic number. We obtain an inequality for the difference between the largest and the smallest limit points of the first sequence. Such an inequality was earlier known for rational $\alpha$ only. It is also shown that for roots of some irreducible trinomials the sequence of integer parts contains infinitely many numbers divisible by either 2 or 3. This is proved, for instance, for $[\xi((\sqrt {13}-1)/2)^n], n\,{=}\,1,2,3,\dots.$ The fact that there are infinitely many composite numbers in the sequence of integer parts of powers was proved earlier for Pisot numbers, Salem numbers and the three rational numbers 3/2, 4/3, 5/4, but no such algebraic number having several conjugates outside the unit circle was known.

Let $C$ be an elliptic curve defined over $\mathbb{Q}$. We can associate two formal groups with $C$: the formal group $\^{C}(X,Y)$ determined by the formal completion of the Néron model of $C$ over $\mathbb{Z}$ along the zero section, and the formal group $F_L(X,Y)$ of the L-series attached to $l$-adic representations on $C$ of the absolute Galois group of $\mathbb{Q}$. Honda shows that $F_L(X, Y)$ is defined over $\mathbb{Z}$, and it is strongly isomorphic over $\mathbb{Z}$ to $\^{C}(X,Y)$. In this paper we give a generalization of the result of Honda to building blocks over finite abelian extensions of $\mathbb{Q}$. The difficulty is to define new matrix L-series of building blocks. Our generalization contains the generalization of Deninger and Nart to abelian varieties of $\rm{GL}_2$-type. It also contains the generalization of our previous paper to $\mathbb{Q}$-curves over quadratic fields.

The Borcherds isomorphism is proved to be Hecke equivariant if one considers multiplicative Hecke operators acting on the integral weight meromorphic modular forms. This answers a part of a question of Borcherds (see ‘Automorphic forms on ${\rm O}_{s+2,2}(R)$ and infinite products’, Invent. Math. 120 (1995) 161–213, 17.10), using his suggestion to define the multiplicative Hecke operators.

In this note it is shown that for $k$ a field, and for the four-dimensional algebra $\Lambda=k\langle x,y\rangle /\langle x^2,y^2,xy+qyx\rangle$ when $q^n\neq 1,0$ for all $n$, there exist a two-dimensional module $M$ and a family of two-dimensional modules $M_i$, $i=1,2,\ldots$, such that $\dim_k\Ext^i_\Lambda(M,M_j)=1$ for $i$ equal to 0, $j$ and $j+1$, and $\dim_k\Ext^i_\Lambda(M,M_j)=0$ otherwise. This is probably the most straightforward example giving a negative answer to a question raised by Maurice Auslander.

It is proved that if a Schur superalgebra is not semisimple, then it is neither cellular nor quasi-hereditary (Theorem 2), and it has infinite global dimension (Corollary 18). The algebra $S(m|n,r)$ with $m,n \ge 1$ is semisimple if and only if $p$, the characteristic of the ground field, is zero or greater than $r$, or when $m=n=1$ and $p$ does not divide $r$.

Let $\Gamma$ be a non-Euclidean crystallographic group. $\Gamma$ is said to be non-maximal if there exists a non-Euclidean crystallographic group $\Gamma'$ such that $\Gamma \le \Gamma'$ and the dimension of the Teichmüller space of $\Gamma$ equals the dimension of the Teichmüller space of $\Gamma'$. The full list of such pairs of groups is computed in the case when $\Gamma$ is non-normal in $\Gamma'$. The corresponding problem for Fuchsian groups was solved by Singerman.

Let ${\mathcal P}_n$ be the collection of all polynomials of degree at most $n$ with real coefficients. A subtle Bernstein-type extremal problem is solved by establishing the inequality \[\big\|U_n^{(m)}\big\|_{L_q({\mathbb{R}})} \leq \big(c^{1+1/q}m\big)^{m/2} n^{m/2} \|U_n\|_{L_q({\mathbb{R}})}\] for all $U_n \in \widetilde{G}_n\!$, $q \in (0,\infty]$, and $m=1,2, \ldots$, where $c$ is an absolute constant and $\[ \widetilde{G}_n :=\biggl\{f: f(t) = \sum_{j=1}^N {P_{m_j}(t) e^{-(t-\lambda_j)^2}}, \; \lambda_j \in {\mathbb{R}}, \; P_{m_j}\in\, {\mathcal P}_{m_j}\!, \; \sum_{j=1}^N{(m_j+1)} \leq n \biggr\}. \]$ Some related inequalities and direct and inverse theorems about the approximation by elements of $\widetilde{G}_n$ in $L_q({\mathbb{R}})$ are also discussed.

Let $A$ and $B$ be countable discrete groups, and let $\Gamma=A\ast B$ be their free product. We show that if $A$ and $B$ are uniformly embeddable into a uniformly convex Banach space, then so is $\Gamma$.

The Fredholm properties of Toeplitz operators $T_a$ on Hardy spaces $H^p$ ($1<p<\infty$) with continuous symbols $a$ are well understood. We consider $T_a$ acting on $H^1$, where the operator is bounded provided that $a$ belongs to the class of symbols given by Janson and Stegenga's result on the pointwise multipliers on $H^1$. A necessary and sufficient condition for $T_a$ to be a Fredholm operator is given when $a$ is continuous and satisfies a mild additional condition (much weaker than Hölder continuity). A formula for the index of $T_a$ is also derived. In addition, we study the case of matrix-valued symbols and Toeplitz operators on $\rm{BMO}_A$.

Using the twistorial approach and some previous results, we prove the conjecture that the dimension of the moduli space of harmonic maps of area $4 \pi d$ from the 2-sphere to the $2n$-sphere is $2 d+n^2$ for the particular case $n=3$.

Regular homotopy classes of immersions of homotopy $n$-spheres into $(n+q)$-space form an Abelian group under connected summation. The subgroup of immersions of those homotopy spheres which bound parallelizable manifolds is computed for $n=4k-1$ and arbitrary codimension $q$. A consequence of this computation is that in codimensions $q<2k+1$, the group of immersed homotopy $(4k-1)$-spheres is not the direct sum of the group of immersed standard spheres and the group of homotopy spheres. Also, our results shed light on Brieskorn's famous equations: for any exotic sphere (bounding a parallelizable manifold), a countable number of distinct equations give rise to codimension-two embeddings of this exotic sphere. It follows from our results that these embeddings lie in pairwise distinct regular homotopy classes, and that any regular homotopy class containing embeddings is representable by a map arising from such an equation.