In the paper we study finite covers of countable $\omega$-categorical structures by means of $\omega$-categorical expansions of bases. It is shown that the cohomology approach of P. Cameron to two-graphs can be applied to covers of some wide class of structures. For example, it works for some reducts of orderings and hypergraphs. The finite covers of the countable highly homogeneous structures and the reducts of the countable random graph are classified.

In this paper we obtain uniform estimates for the lattice pointproblem in the hyperbolic plane $\Bbb H$ under theassumption that the action is by a Fuchsian group $\Gamma$ which is co-finite. We fix a point $w$ from $\Bbb H$ and set $N_t(z,w)$ equal to the number of translates of $w$ by the group$\Gamma$ which lie in a geodesic ball of radius$t$ centred at a point $z$ of $\Bbb H$. The behaviour of$N_t(z,w)$ is then examined when $t$ is large and $z$ is allowed to vary over $\Bbb H$. We show that the finite quantity

$\limsup_{t\rightarrow\infty}\sup_{z\in\Bbb H} N_t (z,w) / \cosh ^2(t/2)$

depends crucially on the point $w$, and indeed can become arbitrarily large with $w$. On the other hand, for the average of this quotient we derive the estimate

\[\sup_{z\in\Bbb H} \ \frac{1}{t}\int_0^t \frac{N_{\tau}(z,w)}{\cosh ^2(\tau /2)} \, d\tau\ = \ \frac{4\pi}{| \Gamma _w | \mbox{vol}(F)} \,+ \, O\!\left(\frac{\log t}{t}\right)\]

as $t\rightarrow \infty$, where the implied constant is an explicit function of $w$. In this formula, $\mbox{vol}(F)$ is the hyperbolic volume of a Dirichletfundamental domain $F$ for $\Gamma$, and $|\Gamma _w |$ denotes the number of elements from $\Gamma$ fixing $w$. This estimate is then combined with a recent sampling theorem of K. Seip to obtain an inequality which decides whether or not the orbit $\Gamma . w$ forms a set of interpolation for a given weighted Bergman space in $\Bbb H$.

It is shown that the theta functions of level $n$ on the principally polarised Prym varieties of an algebraic curve are dual to the sections of the orthogonal theta line bundle on the moduli space of Spin($n$)-bundles over the curve. As a by-product of our computations, we also note that when $n$ is odd, the Pfaffian line bundle on moduli space has a basis of sections labelled by the even theta characteristics of the curve.

The paper studies convex bodies and star bodies in $\Bbb R^n$ by using Radon transforms on Grassmann manifolds, $p$-cosine transforms on the unit sphere, and convolutions on the rotation group of $\Bbb R^n$. It presents dual mixed volume characterizations of $i$-intersection bodies and $L_p$-balls which are related to certain volume inequalities for cross sections of convex bodies. It considers approximations of special convex bodies by analytic bodies and various finite sums of ellipsoids which preserve special geometric properties. Convolution techniques are used to derive formulas for mixed volumes, mixed surface measures, and $p$-cosine transforms. They are also used to prove characterizations of geometric functionals, such as surface area and dual quermassintegrals.

The low-dimensional projective irreducible representations in cross characteristics of the projective special linear group $\mbox{PSL}_{n}(q)$ are investigated.

If $n \geq 3$ and $(n,q) \neq (3,2)$, $(3,4)$, $(4,2)$, $(4,3)$, all such representations of the first degree (which is $(q^{n}-q)/(q-1) - \kappa$ with $\kappa = 0$ or $1$) and the second degree (which is $(q^{n}-1)/(q-1)$) come from Weil representations. We show that the gap between the second and the third degree is roughly $q^{2n-4}$.

In this paper, we consider Kawanaka's generalized Gelfand-Greav characters associated with the various unipotent classes of a finite group of Lie type $G(q)$ (where $q$ is a power of a prime~$p$). Under the assumption that $p$ is large enough, Lusztig [{\em Advances in Math.} 94 (1992) 139-179] has expressed these characters in terms of characteristic functions of certain character sheaves on $G$, where the resulting formulae contain as unknown quantities certain fourth roots of unity. It is one purpose of this paper to determine these roots of unity explicitly, in the case where the centre of $G$ is connected. We then use these results to study the restriction of character sheaves to the unipotent variety of~$G$. Our motivation was to find a more conceptual approach to the properties listed by Lusztig at the end of the introduction of [{\em J. Algebra} 104 (1986) 146-194].

In this paper we obtain a classification of those subgroups of the finite general linear group $\mbox{GL}_d (q)$ with orders divisible by a primitive prime divisor of $q^e - 1$ for some $e > \frac{1}{2}d$. In the course of the analysis, we obtain new results on modular representations of finite almost simple groups. In particular, in the last section, we obtain substantial extensions of the results of Landazuri and Seitz on small cross-characteristic representations of some of the finite classical groups.

This article addresses the question of which non-empty, compact, proper subsets $E$ of the extended complex plane $\widehat{\mathbb C}$ have the feature that, for some $K$ in $[1,\infty)$, the family of$K$-quasiconformal self-mappings of $\widehat{\mathbb C}$ which leave $E$ invariant acts transitively on the set $E\times E^c$, where $E^c$ is the complement of $E$ in $\widehat{\mathbb C}$. The main result in the paper asserts that the class of sets with this property comprises all one- and two-point subsets of $\widehat{\mathbb C}$, all quasicircles in $\widehat{\mathbb C}$ and all images of the Cantor ternary set under quasiconformal self-mappings of $\widehat{\mathbb C}$. It is shown that the third category includes the limit set of any non-cyclic, finitely generated Schottky group.