We consider the connection of functional decompositions of rational functions over the real and complex numbers, and a question about curves in the complex plane which are invariant under a rational function.

We give a necessary and sufficient condition on a compact semitopological quantum semigroup which turns it into a compact quantum group. We give two applications of our results: a “noncommutative” version of Ellis joint continuity theorem for semitopological groups, a corollary to which is a new C∗-algebraic proof of the theorem for classical semitopological semigroup; we also investigate the question of the existence of the Haar state on a compact semitopological quantum semigroup and prove a “noncommutative” version of the converse Haar’s theorem.

In this paper, we develop the metric theory for the outer space of a free product of groups. This generalizes the theory of the outer space of a free group, and includes its relative versions. The outer space of a free product is made of $G$-trees with possibly non-trivial vertex stabilisers. The strategies are the same as in the classical case, with some technicalities arising from the presence of infinite-valence vertices.

We describe the Lipschitz metric and show how to compute it; we prove the existence of optimal maps; we describe geodesics represented by folding paths.

We show that train tracks representative of irreducible (hence hyperbolic) automorphisms exist and that their are metrically characterized as minimal displaced points, showing in particular that the set of train tracks is closed (in particular, answering to some questions raised in Axis in outer space (2011) concerning the axis bundle of irreducible automorphisms).

Finally, we include a proof of the existence of simplicial train tracks map without using Perron-Frobenius theory.

A direct corollary of this general viewpoint is an easy proof that relative train track maps exist in both the free group and free product case.

We propose to extend a conjecture of Bertram Huppert [Illinois J. Math. 44 (2000) 828–842] from finite non-Abelian simple groups to finite quasi-simple groups. Specifically, we conjecture that if a finite group $G$ and a finite quasi-simple group $H$ with ${\mathrm{Mult}}(H/\mathbf{Z}(H))$ cyclic have the same set of irreducible character degrees (not counting multiplicity), then $G$ is isomorphic to a central product of $H$ and an Abelian group. We present a pattern to approach this extended conjecture and, as a demonstration, we confirm it for the special linear groups in dimensions $2$ and $3$.

We discuss conformal metrics of curvature $1$ on tori and on the sphere, with four conic singularities whose angles are multiples of $\pi$. Besides some general results we study in detail the family of such symmetric metrics on the sphere, with angles $(\pi,3\pi,\pi,3\pi)$. As a consequence we find new Heun’s equations whose general solution is algebraic.

In this paper, we first consider boundedness properties of a family of operators generalizing the Hilbert operator in the upper triangle case. In the diagonal case, we give the exact norm of these operators under some restrictions on the parameters. Second, we consider boundedness properties of a family of positive Bergman-type operators of the upper-half plane. We give necessary and sufficient conditions on the parameters under which these operators are bounded in the upper triangle case.

Let $H$ be a separable Hilbert space, $A_{c}:\mathcal{D}_{c}\subset H\to H$ a densely defined unbounded operator, bounded from below, let $\mathcal{D}_{\min}$ be the domain of the closure of $A_{c}$ and $\mathcal{D}_{\max}$ that of the adjoint. Assume that $\mathcal{D}_{\max}$ with the graph norm is compactly contained in $H$ and that $\mathcal{D}_{\min}$ has finite positive codimension in $\mathcal{D}_{\max}$. Then the set of domains of selfadjoint extensions of $A_{c}$ has the structure of a finite-dimensional manifold $\mathfrak{SA}$ and the spectrum of each of its selfadjoint extensions is bounded from below. If $\zeta$ is strictly below the spectrum of $A$ with a given domain $\mathcal{D}_{0}\in\mathfrak{SA}$, then $\zeta$ is not in the spectrum of $A$ with domain $\mathcal{D}\in\mathfrak{SA}$ near $\mathcal{D}_{0}$. But $\mathfrak{SA}$ contains elements $\mathcal{D}_{0}$ with the property that for every neighborhood $U$ of $\mathcal{D}_{0}$ and every $\zeta\in\mathbb{R}$ there is $\mathcal{D}\in U$ such that $\operatorname{spec}(A_{\mathcal{D} })\cap(-\infty,\zeta)\neq\emptyset$. We characterize these “spectrally unstable” domains as being those satisfying a nontrivial relation with the domain of the Friedrichs extension of $A_{c}$.

We consider Diophantine equations of the kind $|F(x,y)|=m$, where $F(X,Y)\in\mathbb{Z}[X,Y]$ is a homogeneous polynomial of degree at least 3 that has non-zero discriminant, $m$ is a fixed positive integer and $x,y$ are relatively prime integer solutions. Our results improve upon previous theorems due to Bombieri and Schmidt and also Stewart. We further provide reasonable heuristics for conjectures of Schmidt and Stewart regarding such equations.

This paper presents a systematic study for structure of abstract Banach function $*$-algebras over homogeneous spaces of compact groups. Let $G$ be a compact group and $H$ be a closed subgroup of $G$. Let $\mu$ be the normalized $G$-invariant measure over the homogeneous space $G/H$ associated to the Weil’s formula and $1\leq p<\infty$. Then we introduce the notions of convolution and involution for the Banach function spaces $L^{p}(G/H,\mu)$.

Li and Shanmugalingam showed that annularly quasiconvex metric spaces endowed with a doubling measure preserve the property of supporting a $p$-Poincaré inequality under the sphericalization and flattening procedures. Because natural examples such as the real line or a broad class of metric trees are not annularly quasiconvex, our aim in the present paper is to study, under weaker hypotheses on the metric space, the preservation of $p$-Poincaré inequalites under those conformal deformations for sufficiently large $p$. We propose the hypotheses used in a previous paper by the same authors, where the preservation of $\infty$-Poincaré inequality has been studied under the assumption of radially star-like quasiconvexity (for sphericalization) and meridian-like quasiconvexity (for flattening). To finish, using the sphericalization procedure, we exhibit an example of a Cheeger differentiability space whose blow up at a particular point is not a PI space.

We give a sufficient condition for the strong symmetric version of Strassen’s additivity conjecture: the Waring rank of a sum of forms in independent variables is the sum of their ranks, and every Waring decomposition of the sum is a sum of decompositions of the summands. We give additional sufficient criteria for the additivity of Waring ranks and a sufficient criterion for additivity of cactus ranks and decompositions.

The goal of this paper is to introduce a new tool, called long turns, which is a useful addition to the train track technology for automorphisms of free groups, in that it allows one to control periodic INPs in a train track map and hence the index of the induced automorphism.

For any surface $\Sigma$ of genus $g\geq1$ and (essentially) any collection of positive integers $i_{1},i_{2},\ldots,i_{\ell}$ with $i_{1}+\cdots+i_{\ell}=4g-4$ Masur and Smillie (Comment. Math. Helv. 68 (1993) 289–307) have shown that there exists a pseudo-Anosov homeomorphism $h:\Sigma\to\Sigma$ with precisely $\ell$ singularities $S_{1},\ldots,S_{\ell}$ in its stable foliation $\mathcal{L}$, such that $\mathcal{L}$ has precisely $i_{k}+2$ separatrices raying out from each $S_{k}$.

In this paper, we prove the analogue of this result for automorphisms of a free group ${F}_{N}$, where “pseudo-Anosov homeomorphism” is replaced by “fully irreducible automorphism” and the Gauss–Bonnet equality $i_{1}+\cdots+i_{\ell}=4g-4$ is replaced by the index inequality $i_{1}+\cdots+i_{\ell}\leq2N-2$ from (Duke Math. J. 93 (1998) 425–452).