The paper considers pairs $(X,B)$ where $X$ is a normal projective surface over $\Bbb C$, and $B$ is a $\Bbb Q$-divisor whose coefficients are $1$ or $1-1/m$ for some natural number $m$. A log canonical singularity on such a pair is a quotient by a finite or infinite group, so if $(X,B)$ has log canonical singularities, the orbifold Euler number $e_{\rm orb}(X,B)$ can be defined. The main result is a Bogomolov-Miyaoka-Yau-type inequality which implies that if $(X,B)$ has log canonical singularities and $\kappa(X,K_X+B)\ge 0$ then $(K_X+B)^2\le 3e_{\rm orb}(X,B)$. The actual inequality proved is somewhat stronger and it also implies all the previously published versions of the Bogomolov-Miyaoka-Yau inequality. The proof involves the Log Minimal Model Program, $Q$-sheaves when $K_X+B$ is nef, and a study of the changes in the two sides of the inequality under a contraction. The paper also contains a further generalisation where the coefficients of $B$ can be arbitrary rational numbers in $[0,1]$, a different condition is imposed on the singularities and $K_X+B$ is required to be nef. Some applications of the inequalities are also given, for example, estimating the number of singularities or certain kinds of configurations of curves on surfaces.

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.

We study a $q$-deformation for the crossed product of the symmetric group $S_n$ with the Clifford algebra on $n$ anticommuting generators. We obtain a $q$-version of the projective analogue for the classical Young symmetrizer introduced by the second author. Our main tool is an analogue of the Hecke algebra of complex-valued functions on the group $\mbox{GL}_n$ over a $p$-adic field relative to the Iwahori subgroup.

Let $K$ be a non-archimedean local field of characteristic $0$ with residue class field not of characteristic $2$. In this note we will determine the minimal covolume $\mu(\mbox{SL}_2(K)/\Gamma)$ of a lattice $\Gamma$ in $\mbox{SL}_2(K)$. Furthermore, we describe all lattices of minimal covolume in terms of their associated graphs of groups and hence up to isomorphism.

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 $\Bbb H$.

We characterise those concealed-canonical algebras which arise as endomorphism rings of tilting modules, all of whose indecomposable summands have strictly positive rank, as those artin algebras whose module categories have a separating exact subcategory (that is, a separating tubular family of standard tubes).

This paper develops further the technique of shift automorphisms which arises from the tubular structure.

It is related to the characterisation of hereditary noetherian categories with a tilting object as the categories of coherent sheaves on a weighted projective line.

In this paper we define a large class of regularity conditions on a stratified set, which generalise the $(t)$ condition introduced by Thom in 1964. According to the first author's 1977 generalisation, a pair of adjacent strata $(X,Y)$ is said to be $(t^k )$-regular at $y_o$ in $Y$ if for every $C^k$-submanifold $S$ transverse to $Y$ at $y_o \in Y \cap S$, there is a neighbourhood of $y_o$ in which $S$ is transverse to $X$.

The very general microlocal conditions of this paper include $(t^k)$ for transversals of arbitrary h\‘olderian continuity and differentiability classes. Special cases of our conditions are equivalent to Verdier's $(w)$ condition, Whitney's $(a)$ condition, and their relative versions $(w_f)$ and $(a_f)$, respectively.

We analyse how the new conditions transform under pushforward and pullback by maps defining parametrised families of transversals to a stratum, and we deduce many consequences. In particular, whenever the pullback verifies $(w)$, the Verdier isotopy theorem (of which we prove a strengthened version in \S 4) implies stratified topological triviality of the family of transversals. This enables us to prove a conjecture made by the first author in 1981 about bounding the number of topological types of germs of transverse intersections with a subanalytic set, and also to give new and unified proofs, with stronger statements, of most of the known results in topological sufficiency theory for $V$-equivalence and right-equivalence.

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.

A systematic study is made of the isomorphic properties of the Banach space ${\cal C}_0(\Upsilon)$ of continuous functions, vanishing at infinity, on a tree $\Upsilon$, equipped with its natural locally compact topology. Necessary and sufficient conditions, expressed in terms of the combinatorial structure of $\Upsilon$, are obtained for ${\cal C}_0(\Upsilon)$ to possess equivalent norms with various good properties of smoothness and strict convexity. These characterizations, together with the construction of appropriate trees, lead to counter-examples refuting a number of conjectures about renormings. It is shown that the existence of a Fr\'echet-smooth renorming is not inherited by quotients, that strict convexifiability is not a three-space property and that neither the Kadec property nor the MLUR property implies the existence of an equivalent norm which is locally uniformly rotund. An example is also given of a space with a smooth norm but no equivalent strictly convex norm. Finally, it is shown that ${\cal C}_0(\Upsilon)$ always admits a ${\cal C}^\infty$ `bump-function', even in cases where no good norms exist.

The ideal space $\mbox{Id}(A)$ of a Banach algebra $A$ is studied as a bitopological space $(\mbox{Id}(A),\tau_u,\tau_n)$, where $\tau_u$ is the weakest topology for which all the norm functions $I\to\Vert a+I\Vert$ (with $a\in A$ and $I\in \mbox{Id}(A)$) are upper semi-continuous, and $\tau_n$ is the de Groot dual of $\tau_u$. When $A$ is separable, $\tau_n\vee\tau_u$ is either a compact, metrizable topology, or it is neither Hausdorff nor first countable. TAF-algebras are shown to exhibit the first type of behaviour. Applications to Banach bundles (which motivate the study), and to PI-Banach algebras, are given.

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 main result of this paper is the resolvent similarity criterion which says that linear growth of the resolvent towards the spectrum is sufficient for a Hilbert space contraction with finite rank defect operators and spectrum not covering the unit disc to be similar to a normal operator. Similar results are proved for operators having a spectral set bounded by a Dini-smooth Jordan curve; in particular, a dissipative operator with finite rank imaginary part is similar to a normal operator if and only if its resolvent grows linearly towards the spectrum. Relevant results on the insufficiency of linear resolvent growth not accompanied by smallness of defect operators are presented. Also it is proved that there is no restriction on the spectrum, other than finiteness, which together with linear resolvent growth implies similarity to a normal operator. The construction of corresponding examples depends on a characterization of well-known Ahlfors curves as curves of linear length growth with respect to linear fractional transformations.

A free semigroup algebra is the weak operator topology closed algebra generated by a set of isometries with pairwise orthogonal ranges. The most important example is the left regular free semigroup algebra generated by the left regular representation of the free semigroup on $n$ generators. This algebra is the appropriate non-commutative $n$-dimensional analogue of the analytic Toeplitz algebra. We develop a detailed picture of the invariant subspace structure analogous to Beurling's theorem and show that this algebra is hyper-reflexive with distance constant at most 51.

The free semigroup algebras, known as atomic, for which the range projections of words in the generators lie in an atomic masa are completely classified. This provides a complete classification for a large class of representations of the Cuntz C*-algebras $\mathcal{O}_n$. This allows us to describe completely the invariant subspace structure of these algebras, and thereby show that these algebras are all hyper-reflexive.

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 $q^{2n-4}$.

Let $k\ge 3$ be an integer. For $0<s<1$, let $D_s \subset {\Bbb R}^2$ be the set that is constructed iteratively as follows. Take a regular open $k$-gon with sides of unit length, attach regular open $k$-gons with sides of length $s$ to the middles of the edges, and so on. At each stage of the iteration the $k$-gons that are added are a factor $s$ smaller than the previous generation and are attached to the outer edges of the family grown so far. The set $D_s$ is defined to be the interior of the closure of the union of all the $k$-gons. It is easy to see that there must exist some $s_k>0$ such that no $k$-gons overlap if and only if $0<s\le s_k$. We derive an explicit formula for $s_k$.

The set $D_s$ is open, bounded, connected and has a fractal polygonal boundary. Let $E_{D_s}(t)$ denote the heat content of $D_s$ at time $t$ when $D_s$ initially has temperature $0$ and $\partial D_s$ is kept at temperature $1$. We derive the complete short-time expansion of $E_{D_s}(t)$ up to terms that are exponentially small in $1/t$. It turns out that there are three regimes, corresponding to $0<s<1/(k-1)$, $s=1/(k-1)$, and $1/(k-1)<s\le s_k$. For $s \neq 1/(k-1)$ the expansion has the form

$$ E_{D_s}(t)=p_s(\log t)t^{1-d_s/2}+A_st^{1/2}+Bt+O(e^{-r_s/t}), $$

where $p_s$ is a $\log (1/s^2)$-periodic function, $d_s=\log (k-1)/\log (1/s)$ is a similarity dimension, $A_s$ and $B$ are constants related to the edges and vertices, respectively, of $D_s$, and $r_s$ is an error exponent. For $s=1/(k-1)$, the $t^{1/2}$-term carries an additional $\log t$.

Suppose that $M$ is a hyperbolic 3-manifold which admits two Dehn fillings $M(r_1)$ and $M(r_2)$ such that $M(r_1)$ contains an essential annulus, and $M(r_2)$ contains an essential torus. It is known that $\Delta = \Delta (r_1, r_2) \leq 5.$ We will show that if $\Delta = 5$ then $M$ is the Whitehead sister link exterior, and if $\Delta = 4$ then $M$ is the exterior of either the Whitehead link or the 2-bridge link associated to the rational number $\frac{3}{10}$. There are infinitely many examples with $\Delta = 3$.

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].

The framework of post critically finite (p.c.f) self-similar fractals was introduced to capture the idea of a finitely ramified fractal, that is, a connected fractal set where any component can be disconnected by the removal of a finite number of points. These ramification points provide a sequence of graphs which approximate the fractal and allow a Laplace operator to be constructed as a suitable limit of discrete graph Laplacians. In this paper we obtain estimates on the heat kernel associated with the Laplacian on the fractal which are best possible up to constants. These are short time estimates for the Laplacian with respect to a natural measure and expressed in terms of an effective resistance metric. Previous results on fractals with spatial symmetry have obtained heat kernel estimates of a non-Gaussian form but which are of Aronson type. By considering a range of examples which are not spatially symmetric, we show that uniform Aronson type estimates do not hold in general on fractals.

We study the exit position $S_{T(r)} $ of a random walk $S_n$ from the interval $[-r, r]$, showing that the tightness of $\vert S_{T(r)}\vert / r$ is equivalent to a generalised kind of stochastic compactness of $S_n$ which we call $SC^\prime$.

This property is in turn equivalent to another kind of compactness property, which we call $SC^{\prime\prime}$, of the maximal sum ${S_n^\ast = \max_{1 \leq j \leq n}\vert S_j \vert}$.

The classes $SC^\prime$ and $SC^{\prime\prime}$, and a related class $SC_0$, which so far seem unexplored, are related to, but different from, the class of stochastically compact $S_n$ studied by Feller, and are similarly of interest in the study of the weak convergence properties of $S_n$ and $S_{T(r)}$.

We give equivalent characterisations of $SC'$ and $SC''$ in terms of the domination of $S_n$ and $S_n^*$ over their maximal increment, and also some analytic characterisations in terms of functionals of the underlying distribution. As a corollary we obtain an equivalence for the stochastic compactness of $\vert S_ {T(r)} \vert / r$.

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.