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.

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 $\alpha=2$.

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.

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 $[{\bf 14}$, ${/bf 20}$, ${/bf 24}\hbox{--}{/bf 26}]$.

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 $G=G(q)$ 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.

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

When $T\,{:}\, X\,{\longrightarrow}\,X$ is a one-sided topologically mixing subshift of finite type and $\varphi\,{:}\,X\,{\longrightarrow}\,R$ is a continuous function, one can define the Ruelle operator ${\cal L}_\varphi\,{:}\,C(X)\,{\longrightarrow}\,C(X)$ on the space $C(X)$ of real-valued continuous functions on $X$. The dual operator ${\cal L}^*_\varphi$ always has a probability measure $\nu$ as an eigenvector corresponding to a positive eigenvalue (${\cal L}^*_\varphi\nu\,{=}\,\lambda\nu$ with $\lambda\,{>}\,0$). Necessary and sufficient conditions on such an eigenmeasure $\nu$ are obtained for $\varphi$ to belong to two important spaces of functions, $W(X,T)$ and ${\rm Bow} (X,T)$. For example, $\varphi\,{\in}\,{\rm Bow}(X,T)$ if and only if $\nu$ is a measure with a certain approximate product structure. This is used to apply results of Bradley to show that the natural extension of the unique equilibrium state $\mu_\varphi$ of $\varphi\,{\in}\,{\rm Bow}(X,T)$ has the weak Bernoulli property and hence is measure-theoretically isomorphic to a Bernoulli shift. It is also shown that the unique equilibrium state of a two-sided Bowen function has the weak Bernoulli property. The characterizations mentioned above are used in the case of $g$-measures to obtain results on the ‘reverse’ of a $g$-measure.

The main result ensures that the scalar problem $x''\,{=}\,f(x)$, $x(0)\,{=}\,x_0$, $x'(0)\,{=}\,x_1,$ has a nonconstant locally $W^{2,1}$ solution if and only if there exists a nontrivial interval $J$ such that $x_0 \in J$, $f \in L^1_{\rm loc}(J)$, $x_1^2+2\int_{x_0}^{y}{f(s)\,ds} > 0$ for almost all $y \in J$ and \[\frac{\max\{1,|f|\}}{ \sqrt{x_1^2+2\int_{x_0}^{\cdot}{f(s)\,ds}}} \in L^1_{\rm loc}(J).\] Necessary and sufficient conditions for local and global uniqueness and for existence of periodic solutions are also established.

The convergent rates for bounded solutions of Dirichlet problems of quasilinear elliptic (possibly degenerate) equations in slab-like domains are derived in terms of the convergent rates of the boundary data and the coefficients of the operator. The equations considered include the prescribed mean curvature equation. The results are proved by constructing a family of local barrier functions based on the structure of the operator and the convergent rate of the boundary data. The construction of local barriers is inspired by early work due to Finn and Serrin that is related to the minimal surface equation.

Characterizations of subnormality due to Trent are extended to finite systems of operators, thus generalizing results of Gãvruţã and Suciu (subnormal pairs) and Binzar and Pãunescu (subnormal triplets). Vector and operator multidimensional moment problems with jointly subnormal solutions are characterized entirely in terms of the initial data. Bounded as well as unbounded solutions are taken into consideration.

It is shown that the invariant subspace of the Bergman space $A^p$ of the unit disk, generated by either an $A^p$-interpolating sequence or a singular inner function with a single point mass on the unit circle, is complemented in $A^p$.

Given a hyperbolic invariant set of a diffeomorphism on a surface, it is proved that, if the holonomies are sufficiently smooth, then the diffeomorphism on the hyperbolic invariant set is rigid in the sense that it is C$^{1+}$ conjugate to a hyperbolic affine model.

Ideas from conformal field theory are applied to symplectic four-manifolds through the use of modular functors to ‘linearise’ Lefschetz fibrations. In Chern–Simons theory, this leads to the study of parabolic vector bundles of conformal blocks. Motivated by the Hard Lefschetz theorem, the author shows that the bundles of SU(2) conformal blocks associated to Kähler surfaces are Brill–Noether special, although the associated flat connexions may be irreducible if the surface is simply connected and not spin.

For ${\cal X}$ a stratification with suitable regularity, in particular for any Whitney stratification and, via regular embedding, for any abstract stratified set, time-dependent vector fields are used to prove an extension theorem for diffeomorphisms near the identity defined on strata of a given dimension. Then it is shown that after isotopy a stratified map $h\,{:}\,{\cal Z} \,{\longrightarrow}\, {\cal X}$ can be made transverse to a fixed stratified map $g\,{:}\, {\cal Y} \,{\longrightarrow}\, {\cal X}$.

Let $T=U(1)$ and $M$ be a Hamiltonian $T$-space with proper moment map $\mu\,{:}\,M\longrightarrow \rr$. When 0 is not a regular value of $\mu$, the symplectic quotient $X=\mu^{-1}(0)/T$ is a singular stratified space. A description is provided of the middle perversity intersection cohomology of $X$ as a subspace of the equivariant cohomology $H^*_T(\mu^{-1}(0))$. The approach is sheaf theoretic.