A theory of bifurcation equivalence for forced symmetry breaking bifurcation problems is developed. We classify $(O(2),1)$ problems of corank 2 of low codimension and discuss examples of bifurcation problems leading to such symmetry breaking.

Suppose $N$ is a nice subgroup of the primary abelian group $G$ and $A=G/N$. The paper discusses various contexts in which $G$ satisfying some property implies that $A$ also satisfies the property, or visa versa, especially when $N$ is countable. For example, if $n$ is a positive integer, $G$ has length not exceeding $\omega_1$ and $N$ is countable, then $G$ is $n$-summable if $A$ is $n$-summable. When $A$ is separable and $N$ is countable, we discuss the condition that any such $G$ decomposes into the direct sum of a countable and a separable group, and we show that it is undecidable in ZFC whether this condition implies that $A$ must be a direct sum of cyclics. We also relate these considerations to the study of nice bases for primary abelian groups.

We give a $\operatorname{div}$-$\operatorname{curl}$ type lemma for the wedge product of closed differential forms on ${\mathbb R}^n$ when they have coefficients respectively in a Hardy space and $L^\infty$ or $\mathit{BMO}$. In this latter case, the wedge product belongs to an appropriate Hardy-Orlicz space.

Motivated by a question of B. Wajnryb we construct embeddings of the braid group in mapping class groups of surfaces, which are not geometric in the sense that the images of standard generators are not Dehn twists. Our construction uses non-orientable surfaces and the fact that the mapping class group of such a surface embeds via lifting of homeomorphisms in the mapping class group of its orientable double cove

In this article we study the interplay of the theory of classical Dirichlet series in one complex variable with recent development on monomial expansions of holomorphic functions in infinitely many variables. For a given Dirichlet series we obtain new strips of convergence in the complex plane related to Bohr’s classical strips of uniform but non absolute convergence.

We give a new proof of the classification of normal singular surface germs admitting non-invertible holomorphic self-maps and due to J. Wahl. We then draw an analogy between the birational classification of singular holomorphic foliations on surfaces, and the dynamics of holomorphic maps. Following this analogy, we introduce the notion of minimal holomorphic model for holomorphic maps. We give sufficient conditions which ensure the uniqueness of such a model.

We show that the baker's map is a finite product of transpositions (particularly pleasant involutions), and conclude from this that an existing very short proof of the simplicity of Thompson's group $V$ applies with equal brevity to the higher dimensional Thompson groups $nV$.

We consider the relationship in the variable Lebesgue space $L^{p(\cdot)}(\Omega)$ between convergence in norm, convergence in modular, and convergence in measure, for both bounded and unbounded exponent functions.

Let ${\mathcal H}$ be a complex Hilbert space. We study the relationships between the angles between closed subspaces of ${\mathcal H}$, the oblique projections associated to non direct decompositions of ${\mathcal H}$ and a notion of compatibility between a positive (semidefinite) operator $A$ acting on ${\mathcal H}$ and a closed subspace ${\mathcal S}$ of ${\mathcal H}$. It turns out that the compatibility is ruled by the values of the Dixmier angle between the orthogonal complement ${\mathcal S}^\perp$ of ${\mathcal S}$ and the closure of $A{\mathcal S}$. We show that every redundant decomposition ${\mathcal H}={\mathcal S}+{\mathcal M}^\perp$ (where redundant means that ${\mathcal S}\cap{\mathcal M}^\perp$ is not trivial) occurs in the presence of a certain compatibility. We also show applications of these results to some signal processing problems (consistent reconstruction) and to abstract splines problems which come from approximation theory.

A new class of metric measure spaces is introduced and studied. This class generalises the well-established doubling metric measure spaces as well as the spaces $(\mathbb{R}^n,\mu)$ with $\mu(B(x,r))\leq Cr^d$, in which non-doubling harmonic analysis has recently been developed. It seems to be a promising framework for an abstract extension of this theory. Tolsa's space of regularised BMO functions is defined in this new setting, and the John-Nirenberg inequality is proven.

A band is a semigroup whose elements are idempotents. The paper is motivated by some open problems concerning linear representations of bands. A counterexample to a conjecture stated in F. Cedó and J. Okniński, "Faithful linear representations of bands," Publ. Mat. 53 (1) (2009), 119–140 is given. On the other hand, it is proved that there exist linear bands with two components such that the associated commutative $K$-algebra constructed in Cedó and Okniński cannot be embedded into any Noetherian ring, even in the case where this algebra is reduced.