We study the arithmetic of certain del Pezzo surfaces of degree 2. We produce examples of Brauer-Manin obstruction to the Hasse principle, coming from 2- and 4-torsion elements in the Brauer group.

The cohomology ring of the moduli space of stable holomorphic vector bundles of rank $n$ and degree $d$ over a Riemann surface of genus $g > 1$ has a standard set of generators when $n$ and $d$ are coprime. When $n = 2$ the relations between these generators are well understood, and in particular a conjecture of Mumford, that a certain set of relations is a complete set, is known to be true. In this article generalisations are given of Mumford's relations to the cases when $n > 2$ and also when the bundles are parabolic bundles, and these are shown to form complete sets of relations.

Let $F$ be an algebraically closed field of characteristic $p$, and let $H$ be a finite group. A natural and important problem in representation theory is to classify the pairs $(G, D)$, where $G$ is a subgroup of $H$, and $D$ is an irreducible $FH$-module that remains irreducible when restricted to $G$. For example, if $H$ is an almost simple group or a central extension of an almost simple group, then one encounters this problem in the attempt to determine the maximal subgroups of the finite classical groups. We solve the above problem for Schur's double covers $H = {\widehat S}_n$ and ${\widehat A}_n$ of the symmetric and alternating groups, and $G$ a Young-type subgroup. The answer is given in terms of the combinatorics of restricted, $p$-strict partitions; such partitions are used to parameterize the irreducible spin representations of ${\widehat S}_n$ and ${\widehat A}_n$. We exploit the connections, recently developed by Ariki, Grojnowski, Brundan, Kleshchev, and others, between modular branching rules, crystal graphs of affine Kac–Moody algebras, and affine Hecke algebras and related objects.

In this paper we describe a tensor space representation of the blob algebra over a ring allowing base change to every interesting (that is, non-semisimple) specialisation. We show that in quasihereditary specialisations this passes to a full tilting module. We consider the nature of the resultant Ringel dual algebra.

We estimate the $L^2$-norm of the $s$-dimensional Riesz transforms on some Cantor sets in ${\mathbb R}^d$. Towards this end, we show that the Riesz transforms truncated at different scales behave in a quasiorthogonal way. As an application, we obtain some precise numerical estimates for the Lipschitz harmonic capacity of these sets.

We develop the Fredholm theory for Toeplitz operators, with symbols in the C*-algebra $D = [SO, SAP]_{n, n}$ generated by all slowly oscillating (SO) and semi-almost periodic (SAP) $n\times n$ matrix functions, on the Hardy spaces $H^p_n$ (with $1 < p < \infty$) over the upper half-plane. Using limit operator techniques, we get necessary Fredholm conditions for any operator in the Banach algebra ${\rm alg}(S, D)$ of singular integral operators with coefficients in $D$ on the space $[L^p (\mathbb{R})]_n$. Applying the Allan–Douglas local principle and the theory of Toeplitz operators with SAP matrix symbols, we also establish Fredholm criteria for Toeplitz operators with matrix symbols $g \in D$ on the space $H^p_n$. An index formula for Fredholm Toeplitz operators with matrix symbols in $D$ is obtained on the basis of an appropriate approximation of slowly oscillating components of the symbols.

In this paper we classify families of square matrices up to the following natural equivalence. Thinking of these families as germs of smooth mappings from a manifold to the space of square matrices, we allow arbitrary smooth changes of co-ordinates in the source and pre- and post- multiply our family of matrices by (generally distinct) families of invertible matrices, all dependent on the same variables. We obtain a list of all the corresponding simple mappings (that is, those that do not involve adjacent moduli). This is a non-linear generalisation of the classical notion of linear systems of matrices. We also make a start on an understanding of the associated geometry.

We study the problem of existence of $\mathcal{F}$-structures (in the sense of Cheeger and Gromov, but not necessarily polarized) on compact complex surfaces. We give a complete classification of compact complex surfaces of Káhler type admitting $\mathcal{F}$-structures. In the non-Káhler case we give a complete classification modulo the gap in the classification of surfaces of class VII.

We then use these results to study the minimal entropy problem for compact complex surfaces: we prove that, modulo the gap in the classification of surfaces of class VII, all compact complex surfaces of Kodaira dimension less than or equal to 1 have minimal entropy zero and such a surface admits a smooth metric $g$ with zero topological entropy if and only if it is ${\mathbb C} P^2$, a ruled surface of genus 0 or 1, a Hopf surface, a complex torus, a Kodaira surface, a hyperelliptic surface or a Kodaira surface modulo a finite group. The key result we use to prove this is a new topological obstruction to the existence of metrics with vanishing topological entropy.

Finally we show that these results fit perfectly into Wall's study of geometric structures on compact complex surfaces. For instance, we show that the minimal entropy problem can be solved for a minimal compact Káhler surface $S$ of Kodaira dimension $- \infty$, 0 or 1 if and only if $S$ admits a geometric structure modelled on ${\mathbb C} P^2$, $S^2 \times S^2$, $S^2 \times {\mathbb E}^2$ or ${\mathbb E}^4$.

Using the Wiener–Poisson isomorphism, we show that if $(F_t)_{0 \leq t \leq 1}$ is a real, bounded, predictable process adapted to the filtration of a compensated Poisson process $(X_t)_{0 \leq t \leq 1}$, and if $\hat{M}_t$ is the operator corresponding to multiplication by $M_t = \int_0^t F_s dX_s$, then for any regular self-adjoint quantum semimartingale $J = (J_t)_{0 \leq t \leq 1}$, the essentially self-adjoint quantum semimartingale $(\hat{M}_t + J_t)_{0 \leq t \leq 1}$ satisfies the quantum Ito formula.

We also introduce a generalisation of the Poisson process to a measure space $(M, \mathcal{M} , \mu)$ as an isometry $I : L^2 (M, \mathcal{M}, \mu) \to L^2 (\Omega, \mathcal{F}, \mathbb{P})$ and give a new construction of the generalised Wiener–Poisson isomorphism ${\mathcal{W}_I} : \mathfrak{F}_+ (L^2(M)) \to L^2 (\Omega, \mathcal{F}, {\mathbb{P}} )$ using exponential vectors. Using C*-algebra theory, given any measure space we construct a canonical generalised Poisson process. Unlike other constructions, we make no a priori use of Poisson measures.