A curvature inequality is established for contractive commuting tuples of operators $\mathbf{T}$ in the Cowen–Douglas class $B_n(\Omega )$ of rank $n$ defined on some bounded domain $\Omega$ in ${\mathbb{C}}^m$. Properties of the extremal operators (that is, the operators which achieve equality) are investigated. Specifically, a substantial part of a well-known question due to R. G. Douglas involving these extremal operators, in the case of the unit disc, is answered.

We start from a discrete random variable, $\mathbf{O}$, defined on $(0,1)$ and taking on $2^{M+1}$ values with equal probability—any member of a certain family whose simplest member is the Rademacher random variable (with domain $(0,1)$), whose constant value on $(0,1/2)$ is $-1$. We create (via left-shifts) independent copies, ${\mathbf{X}}_i$, of $\mathbf{O}$ and let ${\mathbf{S}}_n:={\sum }_{i=1}^n{X}_i$. We let ${\mathbf{S}}_n^{\ast }$ be the quantile of ${\mathbf{S}}_n$. If $\mathbf{O}$ is Rademacher, the sequence $\left\{{\mathbf{S}}_n\right\}$ is the equiprobable random walk on $\mathbb{Z}$ with domain $(0,1)$. In the general case, ${\mathbf{S}}_n$ follows a multinomial distribution and as $\mathbf{O}$ varies over the family, the resulting family of multinomial distributions is sufficiently rich to capture the full generality of situations where the Central Limit Theorem applies.

The ${\mathbf{X}}_1,\dots ,{\mathbf{X}}_n$ provide a representation of ${\mathbf{S}}_n$ that is strong in that their sum is equal to ${\mathbf{S}}_n$ pointwise. They represent ${\mathbf{S}}_n^{\ast }$ only in distribution. Are there strong representations of ${\mathbf{S}}_n^{\ast }$? We establish the affirmative answer, and our proof gives a canonical bijection between, on the one hand, the set of all strong representations with the additional property of being trim and, on the other hand, the set of permutations, ${\pi }_n$, of $\{0,\dots ,2^{n(M+1)}-1\}$, with the property that we call admissibility. Passing to sequences, $\left\{{\pi }_n\right\}$, of admissible permutations, these provide a complete classification of trim, strong triangular array representations of the sequence $\left\{{\mathbf{S}}_n^{\ast }\right\}$. We explicitly construct two sequences of admissible permutations which are polynomial time computable, relative to a function ${\tau }_1^{\mathbf{O}}$ which embodies the complexity of $\mathbf{O}$ itself. The trim, strong triangular array representation corresponding to the second of these is as close as possible to the representation of $\left\{{\mathbf{S}}_n\right\}$ provided by the ${\mathbf{X}}_i$.

We prove a version of the Jenkins–Serrin theorem for the existence of constant mean curvature graphs over bounded domains with infinite boundary data in ${Sol}_3$. Moreover, we construct examples of admissible domains where the results may be applied.

We characterize when the Zariski space $Zar\left(K\right|D)$ (where $D$ is an integral domain, $K$ is a field containing $D$, and $D$ is integrally closed in $K$) and the set ${Zar}_{\min }\left(L\right|D)$ of its minimal elements are Noetherian spaces.

This article investigates an explicit description of the Baum–Connes assembly map of the wreath product $\Gamma =F\wr {\mathbb{F}}_n={\oplus }_{{\mathbb{F}}_n}F⋊{\mathbb{F}}_n$, where $F$ is a finite and ${\mathbb{F}}_n$ is the free group on $n$ generators. In order to do so, we take Davis–Lück’s approach to the topological side which allows computations by means of spectral sequences. Besides describing explicitly the K-groups and their generators, we present a concrete 2-dimensional model for the classifying space $\underline{\mathrm{E}}\Gamma$. As a result of our computations, we obtain that ${\mathrm{K}}_0\left({\mathrm{C}}_{\mathrm{r}}^{\ast }\right(\Gamma \left)\right)$ is the free abelian group of countable rank with a basis consisting of projections in ${\mathrm{C}}_{\mathrm{r}}^{\ast }\left({\oplus }_{{\mathbb{F}}_n}F\right)$, and ${\mathrm{K}}_1\left({\mathrm{C}}_{\mathrm{r}}^{\ast }\right(\Gamma \left)\right)$ is the free abelian group of rank $n$ with a basis represented by the unitaries coming from the free group.

This paper extends Auslander–Reiten duality in two directions. As an application, we obtain various criteria for freeness of modules over local rings in terms of vanishing of Ext modules, which recover a lot of known results on the Auslander–Reiten conjecture.