With probabilistic techniques, it is proved that an inner function in the little Bloch space can have every asymptotic radial behaviour on sets of Hausdorff dimension 1. This is applied to deduce some differentiability properties of positive singular smooth measures and a result on radial cluster set interpolation for inner functions.

Several differential geometric aspects of the space of representations of amenable Banach algebras are studied: differential structure, natural connection, geodesics, reductive structure, and so on. As an application we get a characterisation of amenable groups in terms of the existence of a reductive structure in that space.

Let X be a locally compact metric absolute neighbourhood retract for metric spaces, U ⊂ X be an open subset and f: U → X be a continuous map. The aim of the paper is to study the fixed point index of the map that f induces in the hyperspace of X. For any compact isolated invariant set, K ⊂ U, this fixed point index produces, in a very natural way, a Conley-type (integer valued) index for K. This index is computed and it is shown that it only depends on what is called the attracting part of K. The index is used to obtain a characterization of isolating neighbourhoods of compact invariant sets with non-empty attracting part. This index also provides a characterization of compact isolated minimal sets that are attractors.

It is proved that an excellent ring which is the homomorphic image of a regular ring of finite Krull dimension and having a resolution of singularities satisfies the uniform Artin–Rees property. Extensions of some results of Lipman, Sathaye and Teissier are also obtained.

We show the existence of surfaces of degree $d$ in ${\mathbb P}^3({\mathbb C})$ with approximately $(3j+2)/(6j(j+1))\,d^3$ singularities of type $A_j, 2\le j\le d-1$. The result is based on Chmutov's construction of nodal surfaces. For the proof we use plane trees related to the theory of Dessins d'Enfants.

Our examples improve the previously known lower bounds for the maximum number $\mu_{A_j}(d)$ of $A_j$-singularities on a surface of degree $d$ in most cases. We also give a generalization to higher dimensions which leads to new lower bounds even in the case of nodal hypersurfaces in ${\mathbb P}^n, n\geq5$.

To conclude, we work out in detail a classical idea of Segre which leads to some interesting examples, for example, to a sextic with 36 cusps.

Let A be a Banach algebra, and let E be a Banach A-bimodule. A linear map S:A→E is intertwining if the bilinear map

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is continuous, and a linear map D[ratio ]A→E is a derivation if δ1D=0, so that a derivation is an intertwining map. Derivations from A to E are not necessarily continuous.

The purpose of the present paper is to prove that the continuity of all intertwining maps from a Banach algebra A into each Banach A-bimodule follows from the fact that all derivations from A into each such bimodule are continuous; this resolves a question left open in [1, p. 36]. Indeed, we prove a somewhat stronger result involving left- (or right-) intertwining maps.

Let $f\in \mathbb{Z}[ x,y]$ be a reducible homogeneous polynomial of degree 3. We show that $f(x,y)$ has an even number of prime factors as often as an odd number of prime factors.

Let [gfr ] be a semisimple Lie algebra over an algebraically closed field [ ] of characteristic zero and G be its adjoint group. The notion of a principal nilpotent pair is a double counterpart of the notion of a regular (= principal) nilpotent element in [gfr ]. Roughly speaking, a principal nilpotent pair e = (e1, e2) consists of two commuting elements in [gfr ] that can independently be contracted to the origin and such that their simultaneous centralizer has the minimal possible dimension, that is, rk[gfr ]. The definition and the basic results are due to V. Ginzburg [3]. He showed that the theory of principal nilpotent pairs yields a refinement of well-known results by B. Kostant on regular nilpotent elements in [gfr ] and has interesting applications to representation theory. In particular, he proved that the number of G-orbits of principal nilpotent pairs is finite and gave a classification for [gfr ] = [sfr ][lfr ]([ ]). Trying to achieve a greater generality, Ginzburg also introduced a wider class of distinguished nilpotent pairs and, again, classified them for [sfr ][lfr ]([ ]). (The precise definitions for all notions related to nilpotent pairs are found in §1.)

It is proved that the associated variety of a Poisson prime ideal of the centre of a symplectic reflection algebra at parameter $t \,{=}\, 0$ is irreducible.

A few steps are made towards a representation theory of embeddability among uncountable graphs. A class of graphs is defined by forbidding some countable configurations which are related to the graph's end-structure. Using uncountable combinatorics, a representation theorem for embeddability in this class is proved, which asserts the existence of a surjective homomorphism from the relation of embeddability over isomorphism types of regular cardinality λ>[Nfr ]1 onto set inclusion over all subsets of reals or cardinality λ or less. As corollaries we obtain: (1) the complexity of the class in every regular uncountable λ>[Nfr ]1 is at least

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(2) a characterisation of graphs in the class for which some invariant of the graph has to be inherited by one of fewer than λ subgraphs whose union covers G. Some continuity properties of the homomorphism in the representation theorem are explored and are used to extend the first corollary to all singular cardinals below the first fixed point of second order. The first corollary shows that, contrary to what Shelah has shown for the class of all graphs, the relations of embeddability in the class under discussion is not independent of negations of the generalised continuum hypothesis.

The paper shows the existence of several new families of noncompact constant mean curvature surfaces: (i) singly punctured surfaces of arbitrary genus $g\,{\geq}\,1$, (ii) doubly punctured tori, and (iii) doubly periodic surfaces with Delaunay ends.

three families of pairs of curves are presented; each pair consists of geometrically non-isomorphic curves whose jacobians are isomorphic to one another as unpolarized abelian varieties. each family is parametrized by an open subset of ${\mathbb p}^1$. the first family consists of pairs of genus-2 curves whose equations are given by simple expressions in the parameter; the curves in this family have reducible jacobians. the second family also consists of pairs of genus-2 curves, but generically the curves in this family have absolutely simple jacobians. the third family consists of pairs of genus-3 curves, one member of each pair being a hyperelliptic curve and the other a plane quartic. examples from these families show that in general it is impossible to tell from the jacobian of a genus-2 curve over $\mathbb q$ whether or not the curve has rational points – or indeed whether or not it has real points. the families are constructed using methods that depend on earlier joint work with franck leprévost and bjorn poonen, and on peter bending's explicit description of the curves of genus 2 whose jacobians have real multiplication by $\mathbb z[\sqrt{2}]$.

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 $\alpha=2$.

A geometric property is introduced, and it is proved that Banach spaces with this property must carry weakly amenable algebras of approximable operators. These results are also applied to particular examples of Banach spaces, including the James spaces and the Tsirelson space.

Our object of study is the natural tower which, for any given map f[ratio ]A→B and each space X, starts with the localization of X with respect to f and converges to X itself. These towers can be used to produce approximations to localization with respect to any generalized homology theory E∗, yielding, for example, an analogue of Quillen's plus-construction for E∗. We discuss in detail the case of ordinary homology with coefficients in ℤ/p or ℤ[1/p]. Our main tool is a comparison theorem for nullification functors (that is, localizations with respect to maps of the form f[ratio ]A→pt), which allows us, among other things, to generalize Neisendorfer's observation that p-completion of simply-connected spaces coincides with nullification with respect to a Moore space M(ℤ[1/p], 1).

Suppose that [Pscr ] is a distribution of N points in the unit square U=[0, 1]2. For every x=(x1, x2)∈U, let B(x)=[0, x1]×[0, x2] denote the aligned rectangle containing all points y=(y1, y2)∈U satisfying 0[les ]y1[les ]x1 and 0[les ]y2[les ]x2. Denote by Z[[Pscr ]; B(x)] the number of points of [Pscr ] that lie in B(x), and consider the discrepancy function

D[[Pscr ]; B(x)]=Z[[Pscr ]; B(x)]−Nμ(B(x)),

where μ denotes the usual area measure.

The quantum group Uq(SLn) introduced by Drinfel'd [2] and Jimbo [5] is a Hopf algebra which is naturally paired with Oq(sln), the coordinate ring of quantum SLn. When q is not a root of unity, the finite dimensional representation theory of Uq(sln) is essentially the same as that of U(sln). Furthermore, it is known that Uq(sln) is essentially a quasitriangular Hopf algebra [2]. When q is a root of unity the situation changes dramatically, and the representation theory of U(sln) is no longer effective. Moreover, Uq(sln) is not quasitriangular. In this case one can consider the quotient Hopf algebra Uq(sln)′, introduced by Lusztig [7], which is a finite dimensional Hopf algebra with a nice representation theory. It is well known that Uq(sl2)′ is quasitriangular. Finite dimensional quasitriangular Hopf algebras are important for the study of knot invariants [11, 12]. Thus, a natural question is: when is Uq(sln)′ quasitriangular? The somewhat unexpected answer is given in Theorem 3.7: it depends sharply on the greatest common divisor of n and the order of q1/2. For these Hopf algebras we classify all the possible R-matrices and give necessary and sufficient conditions for them to be minimal quasitriangular. These conditions depend again on n and the order of q1/2. In the process we describe the groups of Hopf automorphisms of Uq(sln)′ and Oq(SLn)′, where Oq(SLn)′ is a finite dimensional quotient Hopf algebra of Qq(SLn) proved to be the dual of Uq(sln)′ in [15].

If σ is an automorphism and δ is a σ-derivation of a ring R, then the subring of invariants is the set R(δ)={r∈R[mid ]δ(r)=0}. The main result of this paper is ‘let R be a semiprime ring with an algebraic σ-derivation δ such that R(δ) is central; then R is commutative’. This theorem generalizes results on the invariants of automorphisms and derivations and is proved by reducing down to the special cases of automorphisms and derivations.

Let 2 [les ] m [les ] n. The paper gives necessary and sufficient conditions on the parameters s1, s2, …, sm, p1, p2, …, pm such that the Jacobian determinant extends to a bounded operator from [Hscr ]s1p1 × [Hscr ]s2p2 × … × [Hscr ]smpm into [Sscr ]′. Here all spaces are defined on ℝn or on domains Ω⊂ℝn. In almost all cases the regularity of the Jacobian determinant is calculated exactly.

One of the consequences of the uniformization theorem of Koebe and Poincaré is that any smooth complex algebraic curve $C$ of genus $g > 1$ is conformally equivalent to ${\bb H}/G$ , where $G \subset \hbox{PSL}_2({\bb R})$ is a Fuchsian group and is naturally endowed with a hyperbolic metric. Conversely, any compact hyperbolic surface is isomorphic to an algebraic curve. Hence any curve of genus $g > 1$ may be described in two ways, either by an equation or by a Fuchsian group.