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.

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 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.

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.

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 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 $\vert S_ {T(r)} \vert / r$.