It is proved that the following classes of finitely generated groups have $\Pi_1^1$-complete first-order theories: all finitely generated groups, the $n$-generated groups, and the strictly $n$-generated groups ($n\,{\geqslant}\,2$). Moreover, all those theories are distinct. Similar techniques show that quasi-finitely axiomatizable groups have a hyperarithmetical word problem, where a finitely generated group is quasi-finitely axiomatizable if it is the only finitely generated group satisfying an appropriate first-order sentence. The Turing degrees of word problems of quasi-finitely axiomatizable groups form a cofinal set in the Turing degrees of hyperarithmetical sets.

Assume that $G$ is a group covered by countably many sets $X_n,\,n\,{<}\,\omega$. It is proved that $G$ is generated in two or three steps by a small number of the sets $X_n$. These results are generalized for countable coverings of types and countable colourings of graphs.

The vanishing ideal of an arrangement of linear subspaces in a vector space is considered, and the paper investigates when this ideal can be generated by products of linear forms. A combinatorial construction (blocker duality) is introduced which yields such generators in cases with a great deal of combinatorial structure, and examples are presented that inspired the work. A construction is given which produces all elements of this type in the vanishing ideal of the arrangement. This leads to an algorithm for deciding if the ideal is generated by products of linear forms. Generic arrangements of points in ${\bf P}^2$ and lines in ${\bf P}^3$ are also considered.

Hall's theorem for bipartite graphs gives a necessary and sufficient condition for the existence of a matching in a given bipartite graph. Aharoni and Ziv have generalized the notion of matchability to a pair of possibly infinite matroids on the same set and given a condition that is sufficient for the matchability of a given pair $(\mathcal{M},\mathcal{W})$ of finitary matroids, where the matroid $\mathcal{M}$ is SCF (a sum of countably many matroids of finite rank). The condition of Aharoni and Ziv is not necessary for matchability. The paper gives a condition that is necessary for the existence of a matching for any pair of matroids (not necessarily finitary) and is sufficient for any pair $(\mathcal{M},\mathcal{W})$ of finitary matroids, where the matroid $\mathcal{M}$ is SCF.

The conjecture stated in an earlier paper by the author that there is a constant $\lambda$ (independent from both $n$ and $k$) such that $S(K_{n}^{d})\ge \lambda n^{d-1}$ holds for every $n\ge 2$ and $d\ge 2$, where $S(K_{n}^{d})$ is the length of the longest snake (cycle without chords) in the Cartesian product $K_{n}^{d}$ of $d$ copies of the complete graph $K_{n}$, is proved.

The classical Ramsey theorem states that every graph contains either a large clique or a large independent set. Here similar dichotomic phenomena are investigated in the context of finite metric spaces. Namely, statements are provided of the form ‘every finite metric space contains a large subspace that is nearly equilateral or far from being equilateral’. Two distinct interpretations are considered for being ‘far from equilateral’. Proximity among metric spaces is quantified through the metric distortion $\alpha$. Tight asymptotic answers are provided for these problems. In particular, it is shown that a phase transition occurs at $\alpha=2$.

It is proved that the 3-part of the class number of a quadratic field $\mathbb{Q}(\sqrt{D})$ is $O(|D|^{55/112 + \ep})$ in general and $O(|D|^{5/12+\ep})$ if $|D|$ has a divisor of size $|D|^{5/6}$. These bounds follow as results of nontrivial estimates for the number of solutions to the congruence $x^a \,{\con}\, y^b$ modulo $q$ in the ranges $x \,{\leqslant}\,X$ and $y\,{\leqslant}\, Y$, where $a,b$ are nonzero integers and $q$ is a square-free positive integer. Furthermore, it is shown that the number of elliptic curves over $\Q$ with conductor $N$ is $O(N^{55/112 + \ep})$ in general and $O(N^{5/12 + \ep})$ if $N$ has a divisor of size $N^{5/6}$. These results are the first improvements to the trivial bound $O(|D|^{1/2 + \ep})$ and the resulting bound $O(N^{1/2 + \ep})$ for the 3-part and the number of elliptic curves, respectively.

The question is addressed of when all pure-projective modules are direct sums of finitely presented modules. It is proved that this is the case over hereditary noetherian rings. Partial results are obtained for uniserial rings. Some of the methods are model-theoretic, and the techniques developed using these may be of interest in their own right.

It is proved that there does not exist a set of four positive integers with the property that the product of any two of its distinct elements plus their sum is a perfect square. This settles an old problem investigated by Diophantus and Euler.

Let $R$ be an artin algebra, and let mod-$R$ denote the category of finitely presented right $R$-modules. The radical ${\rm rad}={\rm rad}({\rm mod}\mbox{-}R)$ of this category and its finite powers play a major role in the representation theory of $R$. The intersection of these finite powers is denoted ${\rm rad}^\omega$, and the nilpotence of this ideal has been investigated, in $[{\bf 6}$, ${\bf 13}]$ for instance. In $[{/bf 17}]$, arbitrary transfinite powers, ${\rm rad}^\alpha$, of rad were defined and linked to the extent to which morphisms in ${\rm mod}\mbox{-}R$ may be factorised. In particular, it has been shown that if $R$ is an artin algebra, then the transfinite radical, ${\rm rad}^\infty $, the intersection of all ordinal powers of rad, is non-zero if and only if there is a ‘factorisable system’ of morphisms in rad and, in that case, the Krull–Gabriel dimension of ${\rm mod}\mbox{-}R$ equals $\infty$ (that is, is undefined). More precise results on the index of nilpotence of rad for artin algebras were proved in $[{\bf 14}$, ${/bf 20}$, ${/bf 24}\hbox{--}{/bf 26}]$.

Let $K$ be a function field of genus $g$ with a finite constant field ${\mathbb{F}}_q$. Choose a place $\infty$ of $K$ of degree $\delta$ and let ${\mathbb{C}}$ be the arithmetic Dedekind domain consisting of all elements of $K$ that are integral outside $\infty$. An explicit formula is given (in terms of $q$, $g$ and $\delta$) for the minimum index of a non-congruence subgroup in SL$_2({\mathcal{C}})$. It turns out that this index is always equal to the minimum index of an arbitrary proper subgroup in SL$_2({\mathcal{C}})$. The minimum index of a normal non-congruence subgroup is also determined.

Let \$C\$ be an irreducible, smooth, projective curve of genus \$g$\backslash$geqslant 3\$ over the complex field \$$\backslash$mathbb\{C\}\$. The curve \$C\$ is called \{$\backslash$em bielliptic\} if it admits a degree-two morphism \$$\backslash$pi$\backslash$colon C$\backslash$longrightarrow E\$ onto an elliptic curve \$E\$; such a morphism is called a \{$\backslash$em bielliptic structure\} on \$C\$. If \$C\$ is bielliptic and \$g$\backslash$geqslant 6\$, then the bielliptic structure on \$C\$ is unique, but if \$g=3,4,5\$, then this holds true only generically and there are curves carrying \$n>1\$ bielliptic structures. The sharp bounds \$n$\backslash$leqslant 21,10,5\$ exist for \$g=3,4,5\$ respectively. Let \$$\backslash$mathfrak\{M\}\_g\$ be the coarse moduli space of irreducible, smooth, projective curves of genus \$g=3,4,5\$. Denote by \$$\backslash$mathfrak\{B\}\_g{\^{}}n\$ the locus of points in \$$\backslash$mathfrak\{M\}\_g\$ representing curves carrying at least \$n\$ bielliptic structures. It is then natural to ask the following questions. Clearly \$$\backslash$mathfrak\{B\}\_g{\^{}}n$\backslash$subseteq $\backslash$mathfrak\{B\}\_g{\^{}}\{n-1\}\$; does \$$\backslash$mathfrak\{B\}\_g{\^{}}n$\backslash$neq$\backslash$mathfrak\{B\}\_g{\^{}}\{n-1\}\$ hold? What are the irreducible components of \$$\backslash$mathfrak\{B\}\_g{\^{}}n\$? Are the irreducible components of \$$\backslash$mathfrak\{B\}\_g{\^{}}n\$ rational? How do the irreducible components of \$$\backslash$mathfrak\{B\}\_g{\^{}}n\$ intersect each other? Let \$C$\backslash$in$\backslash$mathfrak\{B\}\_g{\^{}}2\$; how many non-isomorphic elliptic quotients can it have? Complete answers are given to the above questions in the case \$g=4\$.

The uniserial $p$-adic space groups of coclass $r$ play a central role in the classification of the finite $p$-groups of coclass $r$. A practical algorithm to determine up to isomorphism the uniserial $p$-adic space groups of coclass $r$, where $p$ is an odd prime, is described. As an application, these groups are constructed or counted for some values on $p$ and $r$. For example, it is observed that there are 137 299 953 383 uniserial 3-adic space groups of coclass 4.

A sufficient criterion is found for certain permutation groups $G$ to have uncountable strong cofinality, that is, they cannot be expressed as the union of a countable, ascending chain $(H_i)_{i\in\o}$ of proper subsets $H_i$ such that $H_iH_i \subseteq H_{i+1}$ and $H_i\,{=}\,H_i^{-1}$. This is a strong form of uncountable cofinality for $G$, where each $H_i$ is a subgroup of $G$. This basic tool comes from a recent paper by Bergman on generating systems of the infinite symmetric groups, which is discussed in the introduction. The main result is a theorem which can be applied to various classical groups including the symmetric groups and homeomorphism groups of Cantor's discontinuum, the rationals, and the irrationals, respectively. They all have uncountable strong cofinality. Thus the result also unifies various known results about cofinalities. A notable example is the group BSym ($\Q$) of all bounded permutations of the rationals $\Q$ which has uncountable cofinality but countable strong cofinality.

Let $\textbf{F}\,{=}\,(F_1, \ldots, F_m)$ be an $m$-tuple of primitive positive binary quadratic forms and let $U_{\textbf{F}}(x)$ be the number of integers not exceeding $x$ that can be represented simultaneously by all the forms $F_j$, $j = 1, \ldots, m$. Sharp upper and lower bounds for $U_{\textbf{F}}(x)$ are given uniformly in the discriminants of the quadratic forms.

As an application, a problem of Erdős is considered. Let $V(x)$ be the number of integers not exceeding $x$ that are representable as a sum of two squareful numbers. Then $V(x) = x(\log x)^{-\alpha+o(1)}$ with $\alpha\,{=}\,1-2^{-1/3}\,{=}\,0.206\ldots$.

Let $G=G(q)$ be a finite almost simple exceptional group of Lie type over the field of $q$ elements, where $q=p^a$ and $p$ is prime. The main result of the paper determines all maximal subgroups $M$ of $G(q)$ such that $M$ is an almost simple group which is also of Lie type in characteristic $p$, under the condition that ${\rm rank}(M) > {1\over 2}{\rm rank}(G)$. The conclusion is that either $M$ is a subgroup of maximal rank, or it is of the same type as $G$ over a subfield of $\F_q$, or $(G,M)$ is one of $(E_6^\e(q),F_4(q))$, $(E_6^\e(q),C_4(q))$, $(E_7(q),\,^3\!D_4(q))$. This completes work of the first author with Saxl and Testerman, in which the same conclusion was obtained under some extra assumptions.

The class of co-context-free groups is studied. A co-context-free group is defined as one whose co-word problem (the complement of its word problem) is context-free. This class is larger than the subclass of context-free groups, being closed under the taking of finite direct products, restricted standard wreath products with context-free top groups, and passing to finitely generated subgroups and finite index overgroups. No other examples of co-context-free groups are known. It is proved that the only examples amongst polycyclic groups or the Baumslag–Solitar groups are virtually abelian. This is done by proving that languages with certain purely arithmetical properties cannot be context-free; this result may be of independent interest.

Several important cases of vector bundles with extra structure (such as Higgs bundles and triples) may be regarded as examples of twisted representations of a finite quiver in the category of sheaves of modules on a variety/manifold/ringed space. It is shown that the category of such representations is an abelian category with enough injectives by the construction of an explicit injective resolution. Using this explicit resolution, a long exact sequence is found that computes the Ext groups in this new category in terms of the Ext groups in the old category. The quiver formulation is directly reflected in the form of the long exact sequence. It is also shown that under suitable circumstances, the Ext groups are isomorphic to certain hypercohomology groups.

A study is made of groups with finitely many derived groups of subgroups or of infinite subgroups. These groups are classified completely in the locally graded case. In the general case, detailed structural information about groups in each class is found.

It is proved that the Nottingham group $\cal N_{{\mathbb F}_p}$ is finitely presented as a pro-$p$ group for $p\,{>}\,2$.