In this paper, we solve a question of Simon Wassermann, whether the Calkin algebra can be written as a C*-tensor product of two infinite dimensional C*-algebras. More generally, we show that there is no surjective *-homomorphism from a SAW*-algebra onto C*-tensor product of two infinite dimensional C*-algebras.

Let R = k[x1,…,xn], where k is a field. The path ideal (of length t ≥ 2) of a directed graph G is the monomial ideal, denoted by It(G), whose generators correspond to the directed paths of length t in G. Let Cn be an n-cycle. We show that R/It(Cn) is Sr if and only if it is Cohen-Macaulay or $\lceil \frac{n}{n-t-1}\rceil\geq r+3$. In addition, we prove that R/It(Cn) is Gorenstein if and only if n = t or 2t + 1. Also, we determine all ordinary and symbolic powers of It(Cn) which are Cohen-Macaulay. Finally, we prove that It(Cn) has a linear resolution if and only if t ≥ n − 2.

We extend the local non-homogeneous Tb theorem of Nazarov, Treil and Volberg to the setting of singular integrals with operator-valued kernel that act on vector-valued functions. Here, ‘vector-valued’ means ‘taking values in a function lattice with the UMD (unconditional martingale differences) property’. A similar extension (but for general UMD spaces rather than UMD lattices) of Nazarov-Treil-Volberg's global non-homogeneous Tb theorem was achieved earlier by the first author, and it has found applications in the work of Mayboroda and Volberg on square-functions and rectifiability. Our local version requires several elaborations of the previous techniques, and raises new questions about the limits of the vector-valued theory.

We approach the analysis of the extent of the projectivity of modules from a fresh perspective as we introduce the notion of relative subprojectivity. A module M is said to be N-subprojective if for every epimorphism g : B → N and homomorphism f : M → N, there exists a homomorphism h : M → B such that gh = f. For a module M, the subprojectivity domain of M is defined to be the collection of all modules N such that M is N-subprojective. We consider, for every ring R, the subprojective profile of R, namely, the class of all subprojectivity domains for R modules. We show that the subprojective profile of R is a semi-lattice, and consider when this structure has coatoms or a smallest element. Modules whose subprojectivity domain is as smallest as possible will be called subprojectively poor (sp-poor) or projectively indigent (p-indigent), and those with co-atomic subprojectivy domain are said to be maximally subprojective. While we do not know if sp-poor modules and maximally subprojective modules exist over every ring, their existence is determined for various families. For example, we determine that artinian serial rings have sp-poor modules and attain the existence of maximally subprojective modules over the integers and for arbitrary V-rings. This work is a natural continuation to recent papers that have embraced the systematic study of the injective, projective and subinjective profiles of rings.

The proof of Theorem 3.2 in [1] (P. Tzermias, On the p-adic binomial series and a formal analogue of Hilbert's Theorem 90, Glasgow Math. J.47 (2005), 319–326) contains two opaque claims. The necessary clarifications are provided here.

We study the values of the zeta-function of the root system of type G2 at positive integer points. In our previous work we considered the case when all integers are even, but in the present paper we prove several theorems which include the situation when some of the integers are odd. The underlying reason why we may treat such cases, including odd integers, is also discussed.

We develop an essentially algebraic method to study biharmonic curves into an implicit surface. Although our method is rather general, it is especially suitable to study curves in surfaces defined by a polynomial equation: In particular, we use it to give a complete classification of biharmonic curves in real quadrics of the three-dimensional Euclidean space.

A mixed quasi-étale quotient is the quotient of the product of a curve of genus at least 2 with itself by the action of a group which exchanges the two factors and acts freely outside a finite subset. A mixed quasi-étale surface is the minimal resolution of its singularities. We produce an algorithm computing all mixed quasi-étale surfaces with given geometric genus, irregularity and self-intersection of the canonical class. We prove that all irregular mixed quasi-étale surfaces of general type are minimal. As an application, we classify all irregular mixed quasi-étale surfaces of general type with genus equal to the irregularity, and all the regular ones with K2 > 0, thus constructing new examples of surfaces of general type with χ = 1. We mention the first example of a minimal surface of general type with pg = q = 1 and Albanese fibre of genus bigger than K2.

Let G be a finite p-solvable group and P a Sylow p-subgroup of G. Suppose that $\gamma_{\ell (p-1)}(P)\subseteq \gamma_r(P)^{p^s}$ for ℓ(p−1) < r + s(p − 1), then the p-length is bounded by a function depending on ℓ.

We give several algorithms for finitely generated subgroups of the modular group PSL2(ℤ) given by sets of generators. First, we present an algorithm to check whether a finitely generated subgroup H has finite index in the full modular group. Then we discuss how to parametrise the right cosets of H in PSL2(ℤ), whether the index is finite or not. Further, we explain how an element in H can be written as a word in a given set of generators of H.

Inspired by Gowers' seminal paper (W. T. Gowers, Comb. Probab. Comput.17(3) (2008), 363–387, we will investigate quasi-randomness for profinite groups. We will obtain bounds for the minimal degree of non-trivial representations of SLk(ℤ/(pnℤ)) and Sp2k(ℤ/(pnℤ)). Our method also delivers a lower bound for the minimal degree of a faithful representation of these groups. Using the suitable machinery from functional analysis, we establish exponential lower and upper bounds for the supremal measure of a product-free measurable subset of the profinite groups SLk(ℤp) and Sp2k(ℤp). We also obtain analogous bounds for a special subgroup of the automorphism group of a regular tree.

The purpose of this paper is a further investigation on the 2-nilpotent multiplier, $\mathcal{M}$(2)(G), when G is a non-abelian p-group. Furthermore, taking G in the class of extra-special p-groups, we will get the explicit structure of $\mathcal{M}$(2)(G) and will classify 2-capable groups in that class.

This paper is devoted to determine the connectedness of the branch loci of the moduli space of non-orientable unbordered Klein surfaces. We obtain a result similar to Nielsen's in order to determine topological conjugacy of automorphisms of prime order on such surfaces. Using this result we prove that the branch locus is connected for surfaces of topological genus 4 and 5.

Given a finitely generated module over a commutative noetherian ring that satisfies certain reflexivity conditions, we show how failure of the semidualizing property for the module manifests in a disconnection of the prime spectrum of the ring.