In 1933 Henri Cartan proved a fundamental theorem in Nevanlinna theory, namely a generalization of Nevanlinna's second fundamental theorem. Cartan's theorem works very well for certain kinds of problems. Unfortunately, it seems that Cartan's theorem, its proof, and its usefulness, are not as widely known as they deserve to be. To help give wider exposure to Cartan's theorem, the simple and general forms of the theorem are stated here. A proof of the general form is given, as well as several applications of the theorem.

Hodge theory for a smooth algebraic curve includes both the Hodge structure (period matrix) on cohomology and the use of that Hodge structure to study the geometry of the curve, via the Jacobian variety. Hodge extended the theory of the period matrix to smooth algebraic varieties of any dimension, defining in general a Hodge structure on the cohomology of the variety. He gave a few applications to the geometry of the variety, but these did not attain the richness of the Jacobian variety. In recent years, Hodge theory has been successfully extended to arbitrary varieties, and to families of varieties. In this expository paper, some of these developments are reviewed, with special emphasis on instances where these extensions can be used to study the geometry – especially the algebraic cycles – on the variety.

Let $G$ be a group that has a presentation with $n$ generators and $m$ relators, where $m\,{<}\,n$, and let $S$ be an arbitrary generating set for $G$. Then some subset of $n-m$ elements of $S$ freely generates a free subgroup of $G$. This leads to the proof of a conjecture of Gromov on growth of groups.

Upper bounds are obtained for the heat content and the torsional rigidity of a complete Riemannian manifold $M$, assuming a Hardy inequality for the Dirichlet Laplacian on $M$.

Let $(W,S)$ be a finite Coxeter system, and let $J\,{\subseteq}\, S$. Any $w\,{\in}\,W$ has a unique factorization $w\,{=}\,w^Jw_J$, where $w_J$ belongs to the parabolic subgroup $W_J$ generated by $J$, and $w^J$ is of minimal length in the coset $wW_J$. It is shown here that $w_I\,{=}\,w^J$ if and only if $w^I\,{=}\,w_J$, for all $I,J\,{\subseteq}\,S$. Furthermore, a similar symmetry property in arbitrary $(W_I,W_J)$-double cosets is conjectured, which links this result to the Solomon descent algebra of $W$.

It is shown that a Carter subgroup of a non-soluble unitary group can only be the normalizer of a Sylow 2-subgroup. This result, combined with previous results, implies that no finite simple group can be a minimal counterexample to the conjugacy conjecture of Carter subgroups in a finite group.The third author was partially supported by the University of Padova, the RFBR, grants 02-01-00495 and 02-01-06226, Ministry of Education RF, E00-1.0-77, ‘Universities of Russia’ UR.04.01.031.

Let $G$ be a classical algebraic group, $X$ a maximal rank reductive subgroup and $P$ a parabolic subgroup. This paper classifies when $X\backslash G/P$ is finite. Finiteness is proven using geometric arguments about the action of $X$ on subspaces of the natural module for $G$. Infiniteness is proven using a dimension criterion that involves root systems.

For an irreducible, crystallographic root system $\Phi$ in a Euclidean space $V$ and a positive integer $m$, the arrangement of hyperplanes in $V$ given by the affine equations $(\alpha, x)\,{=}\,k$, for $\alpha\,{\in}\,\Phi$ and $k\,{=}\,0, 1,\dots,m$, is denoted here by ${\mathcal A}_{\Phi}^m$. The characteristic polynomial of ${\mathcal A}_{\Phi}^m$ is related in the paper to that of the Coxeter arrangement ${\mathcal A}_{\Phi}$ (corresponding to $m\,{=}\,0$), and the number of regions into which the fundamental chamber of ${\mathcal A}_{\Phi}$ is dissected by the hyperplanes of ${\mathcal A}_{\Phi}^m$ is deduced to be equal to the product $\prod_{i=1}^{\ell} ({e_i\,{+}\,m h\,{+}\,1})/({e_i\,{+}\,1})$, where $e_1, e_2,\dots,e_\ell$ are the exponents of $\Phi$ and $h$ is the Coxeter number. A similar formula for the number of bounded regions follows. Applications to the enumeration of antichains in the root poset of $\Phi$ are included.

This paper investigates how to relate the syzygy periodicity of a self-injective algebra $A$ to its Auslander–Reiten periodicity. Moreover, a characterization is provided of the Auslander–Reiten bounded $A$–$A$-bimodules that are periodic.

The values taken by $\Gamma_0(n)$-modular functions at elliptic points of order 2 for the Fricke group $\Gamma_0(n)^\dagger$ that lie outside $\Gamma_0(n)$ are studied. In the case of a principal modulus (‘Hauptmodul’) for $\Gamma_0(n)$ or $\Gamma_0(n)^\dagger$, the class fields generated by these values are determined.

A proof is given to show that the operator Hilbert space $\operatorname{OH}$ does not embed completely isomorphically into the predual of a semi-finite von Neumann algebra. This complements Junge's recent result, which admits such an embedding in the non-semi-finite case.

It is well known that no non-trivial Killing vector field exists on a compact Riemannian manifold of negative Ricci curvature; analogously, no non-trivial harmonic one-form exists on a compact manifold of positive Ricci curvature. One can consider the following, more general, problem. By reducing the assumption on the Ricci curvature to one on the scalar curvature, such vanishing theorems cannot hold in general. This raises the question: “What information can we obtain from the existence of non-trivial Killing vector fields (or, respectively, harmonic one-forms)?” This paper gives answers to this problem; the results obtained are optimal.

The metric entropy of absolute convex hulls of sets in Hilbert spaces is studied for the general case when the metric entropy of the sets is arbitrary. Under some regularity assumptions, the results are sharp.

It is proved that if $R$ is an associative ring that is cotorsion as a left module over itself, and $J$ is the Jacobson radical of $R$, then the quotient ring $R/J$ is a left self-injective von Neumann regular ring and idempotents lift modulo $J$. In particular, if $R$ is indecomposable, then it is a local ring.

The paper introduces a new grading on the preprojective algebra of an arbitrary locally finite quiver. Viewing the algebra as a left module over the path algebra, the author uses the grading to give an explicit geometric construction of a canonical collection of exact sequences of its submodules. If a vertex of the quiver is a source, the above submodules behave nicely with respect to the corresponding reflection functor. It follows that when the quiver is finite and without oriented cycles, the canonical exact sequences are the almost split sequences with preprojective terms, and the indecomposable direct summands of the submodules are the non-isomorphic indecomposable preprojective modules. The proof extends that given by Gelfand and Ponomarev in the case when the finite quiver is a tree.

The author recently proved that 1-cotilting modules are pure-injective; however, the problem of determining whether n-cotilting modules are pure-injective remained open. In this paper, necessary and sufficient conditions are given for the pure-injectivity of n-cotilting modules.

A subset $A$ of the integers is said to be sum-free if there do not exist elements $x,y,z \,{\in}\, A$ with $x \,{+}\, y \,{=}\, z$. It is shown that the number of sum-free subsets of $\{1,\ldots,N\}$ is $O(2^{N/2})$, confirming a well-known conjecture of Cameron and Erdős.

The authors present a technique of deriving basic hypergeometric identities from specializations using fewer parameters, by using the classical Cauchy identity on the expansion of the power of $x$ in terms of the $q$-binomial coefficients. This method is referred to as ‘Cauchy augmentation’. Despite its simple appearance, the Cauchy identity plays a key role in parameter augmentation. For example, one can reach the $q$-Gauss summation formula from the Euler identity by using the Cauchy augmentation twice. This idea also applies to Jackson's $_2\phi_1$ to $_3\phi_1$ transformation formula. Moreover, a transformation formula analogous to Jackson's formula is obtained.

The class of nilpotent groups of class $c$ and prime exponent $p\,{>}\,c$ with additional predicates $P_c\,{\subseteq}\,P_{c-1}\,{\subseteq}\,\ldots\,{\subseteq}\,P_1$ for suitable subgroups has the amalgamation property. Hence the Fraïssé limit $D$ of the finite groups of this class exists. $\langle 1\rangle\,{\subseteq}\,P_c(D)\,{\subseteq}\,\ldots\,{\subseteq}\,P_2(D)\,{\subseteq}\,P_1(D)\,{=}\,D$ is the lower and the upper central series of $D$. In this extended language, $D$ is ultrahomogeneous. The elementary theory of $D$ allows the elimination of quantifiers and it is $\aleph_0$-categorical. For $c\,{=}\,2$ this was proved by Baudisch in Bull. London Math. Soc. 33 (2001) 513–519.

This paper describes – in terms of potential-theoretic properties – the necessary and sufficient conditions required for a bounded domain satisfying the capacity density condition to be, respectively: a John domain, a uniform domain and an inner uniform domain.