We construct a counterexample to the conjugacy of maximal abelian diagonalizable subalgebras in extended affine Lie algebras.

We give a detailed proof of Siu’s theorem on extendibility of holomorphic vector bundles of rank larger than one, and prove a corresponding extension theorem for holomorphic sprays. We apply this result to study ellipticity properties of complements of compact subsets in Stein manifolds. In particular we show that the complement of a closed ball in ${\mathbb{C}}^n,n\ge 3$, is not subelliptic.

We study a certain improved fractional Sobolev–Poincaré inequality on domains, which can be considered as a fractional counterpart of the classical Sobolev–Poincaré inequality. We prove the equivalence of the corresponding weak and strong type inequalities; this leads to a simple proof of a strong type inequality on John domains. We also give necessary conditions for the validity of an improved fractional Sobolev–Poincaré inequality, in particular, we show that a domain of finite measure, satisfying this inequality and a ‘separation property’, is a John domain.

Inspired by the work of Z. Lu and G. Tian (Duke Math. J. 125:351–387, 2004) in the compact setting, in this paper we address the problem of studying the Szegö kernel of the disk bundle over a noncompact Kähler manifold. In particular we compute the Szegö kernel of the disk bundle over a Cartan–Hartogs domain based on a bounded symmetric domain. The main ingredients in our analysis are the fact that every Cartan–Hartogs domain can be viewed as an “iterated” disk bundle over its base and the ideas given in (Arezzo, Loi and Zuddas in Math. Z. 275:1207–1216, 2013) for the computation of the Szegö kernel of the disk bundle over an Hermitian symmetric space of compact type.

Let $A$ be a complex, commutative unital Banach algebra. We introduce two notions of exponential reducibility of Banach algebra tuples and present an analogue to the Corach-Suárez result on the connection between reducibility in $A$ and in $C(M(A))$. Our methods are of an analytical nature. Necessary and sufficient geometric/topological conditions are given for reducibility (respectively reducibility to the principal component of $U_n(A)$) whenever the spectrum of $A$ is homeomorphic to a subset of ${\mathbb{C}}^n$.

The main result of this paper, Theorem 1.5, gives explicit formulae for the kernels of the ergodic decomposition measures for infinite Pickrell measures on the space of infinite complex matrices. The kernels are obtained as the scaling limits of Christoffel-Uvarov deformations of Jacobi orthogonal polynomial ensembles.

Let $\mathbb{D}\subset \mathbb{C}$ be the unit disk and let ${\gamma }_1,{\gamma }_2:[0,T]\to \bar{\mathbb{D}}\setminus \{0\}$ be parametrizations of two slits ${\mathrm{\Gamma }}_1:=\gamma (0,T],{\mathrm{\Gamma }}_2:={\gamma }_2(0,T]$ such that ${\mathrm{\Gamma }}_1$ and ${\mathrm{\Gamma }}_2$ are disjoint.

Let $g_t$ be the unique normalized conformal mapping from $\mathbb{D}\setminus ({\gamma }_1[0,t]\cup {\gamma }_2[0,t])$ onto $\mathbb{D}$ with $g_t(0)=0$, $g_t^{\prime }(0)>0$. Furthermore, for $k=1,2$, denote by $h_{k;t}$ the unique normalized conformal mapping from $\mathbb{D}\setminus {\gamma }_k[0,t]$ onto $\mathbb{D}$ with $h_{k;t}(0)=0$, $h_{k;t}^{\prime }(0)>0$.

Loewner’s famous theorem (1923) can be stated in the following way: The function $t\mapsto h_{k;t}$ is differentiable at $t_0$ if and only if $t\mapsto log(h_{k;t}^{\prime }(0))$ is differentiable at $t_0$.

In this paper we compare the differentiability of $t\mapsto h_{k;t}$ with that of $t\mapsto g_t$. We show that the situation is more complicated in the case $t_0=0$ with ${\gamma }_1(0)={\gamma }_2(0)$.

Furthermore, we also look at this problem in the case of a multiply connected domain with its corresponding Komatu–Loewner equation.

In this paper we prove some results concerning stability of hypersurfaces in the four dimensional Euclidean space with zero scalar curvature. First we prove there is no complete stable hypersurface with zero scalar curvature, polynomial growth of integral of the mean curvature, and with the Gauss-Kronecker curvature bounded away from zero. We conclude this paper giving a sufficient condition for a regular domain to be stable in terms of the mean and the Gauss-Kronecker curvatures of the hypersurface and the radius of the smallest extrinsic ball which contains the domain.

The first author has associated in a natural way a profinite group to each irreducible subshift. The group in question was initially obtained as a maximal subgroup of a free profinite semigroup. In the case of minimal subshifts, the same group is shown in the present paper to also arise from geometric considerations involving the Rauzy graphs of the subshift. Indeed, the group is shown to be isomorphic to the inverse limit of the profinite completions of the fundamental groups of the Rauzy graphs of the subshift. A further result involving geometric arguments on Rauzy graphs is a criterion for freeness of the profinite group of a minimal subshift based on the Return Theorem of Berthé et al.

Let $D$ be a Dedekind scheme with the characteristic of all residue fields not equal to 2. To every tame cover $C\to D$ with only odd ramification we associate a second Stiefel-Whitney class in the second cohomology with mod 2 coefficients of a certain tame orbicurve $[D]$ associated to $D$. This class is then related to the pull-back of the second Stiefel-Whitney class of the push-forward of the line bundle of half of the ramification divisor. This shows (indirectly) that our Stiefel-Whitney class is the pull-back of a sum of cohomology classes considered by Esnault, Kahn and Viehweg in ‘Coverings with odd ramification and Stiefel-Whitney classes’. Perhaps more importantly, in the case of a proper and smooth curve over an algebraically closed field, our Stiefel-Whitney class is shown to be the pull-back of an invariant considered by Serre in ‘Revêtements à ramification impaire et thêta-caractéristiques’, and in this case our arguments give a new proof of the main result of that article.

Starting with a commutative ring $R$ and an ideal $I$, it is possible to define a family of rings $R{(I)}_{a,b}$, with $a,b\in R$, as quotients of the Rees algebra ${\oplus }_{n\ge 0}{I}^n{t}^n$; among the rings appearing in this family we find Nagata’s idealization and amalgamated duplication. Many properties of these rings depend only on $R$ and $I$ and not on $a$, $b$; in this paper we show that the Gorenstein and the almost Gorenstein properties are independent of $a$, $b$. More precisely, we characterize when the rings in the family are Gorenstein, complete intersection, or almost Gorenstein and we find a formula for the type.

We construct relative and global Euler sequences of a module. We apply it to prove some acyclicity results of the Koszul complex of a module and to compute the cohomology of the sheaves of (relative and absolute) differential $p$-forms of a projective bundle. In particular we generalize Bott’s formula for the projective space to a projective bundle over a scheme of characteristic zero.

Given a compact set of real numbers, a random $C^{m+\alpha }$-diffeomorphism is constructed such that the image of any measure concentrated on the set and satisfying a certain condition involving a real number $s$, almost surely has Fourier dimension greater than or equal to $s/(m+\alpha )$. This is used to show that every Borel subset of the real numbers of Hausdorff dimension $s$ is $C^{m+\alpha }$-equivalent to a set of Fourier dimension greater than or equal to $s/(m+\alpha )$. In particular every Borel set is diffeomorphic to a Salem set, and the Fourier dimension is not invariant under $C^m$-diffeomorphisms for any $m$.

This paper is devoted to exploiting the restrictions of Riesz–Morrey potentials on either unbounded or bounded domains in Euclidean spaces.

Consider a complete abelian category which has an injective cogenerator. If its derived category is left-complete we show that the dual of this derived category satisfies Brown representability. In particular, this is true for the derived category of an abelian AB$4^{\ast }$-$n$ category and for the derived category of quasi-coherent sheaves over a nice enough scheme, including the projective finitely dimensional space.

We prove that there exists uncountably many pairwise disjoint open subsets of the Gelfand space of the measure algebra on any locally compact non-discrete abelian group which shows that this space is not separable (in fact, we prove this assertion for the ideal $M_0(G)$ consisting of measures with Fourier-Stieltjes transforms vanishing at infinity which is a stronger statement). As a corollary, we obtain that the spectras of elements in the algebra of measures cannot be recovered from the image of one countable subset of the Gelfand space under Gelfand transform, common for all elements in the algebra.

We provide a refinement of the Poincaré inequality on the torus ${\mathbb{T}}^d$: there exists a set $\mathcal{B}\subset {\mathbb{T}}^d$ of directions such that for every $\alpha \in \mathcal{B}$ there is a $c_{\alpha }>0$ with $${\parallel \mathrm{\nabla }f\parallel }_{L^2({\mathbb{T}}^d)}^{d-1}{\parallel \langle \mathrm{\nabla }f,\alpha \rangle \parallel }_{L^2({\mathbb{T}}^d)}\ge c_{\alpha }{\parallel f\parallel }_{L^2({\mathbb{T}}^d)}^d\text{for all}f\in H^1\left({\mathbb{T}}^d\right)\text{with mean 0.}$$ The derivative $\langle \mathrm{\nabla }f,\alpha \rangle$ does not detect any oscillation in directions orthogonal to $\alpha$, however, for certain $\alpha$ the geodesic flow in direction $\alpha$ is sufficiently mixing to compensate for that defect. On the two-dimensional torus ${\mathbb{T}}^2$ the inequality holds for $\alpha =(1,\sqrt{2})$ but is not true for $\alpha =(1,e)$. Similar results should hold at a great level of generality on very general domains.

The paper presents a study of Fuglede’s $p$-module of systems of measures in condensers in polarizable Carnot groups. In particular, we calculate the $p$-module of measures in spherical ring domains, find the extremal measures, and finally, extend a theorem by Rodin to these groups.