We propose the notion of partial resolution of a ring, which is by definition the endomorphism ring of a certain generator of the given ring. We prove that the singularity category of the partial resolution is a quotient of the singularity category of the given ring. Consequences and examples are given.

In two earlier papers with the same title, we studied connections between Voronoi’s formula in the divisor problem and Atkinson’s formula for the mean square of Riemann’s zeta-function. Now we consider this correspondence in terms of segments of sums appearing in these formulae and show that a certain arithmetic conjecture concerning the divisor function implies best possible bounds for the classical error terms Δ(x) and E(T).

Nous montrons que le morphisme de Frobenius quantique construit par Lusztig dans le cadre des algèbres enveloppantes quantiques $U_{\mathcal{B}}$ spécialisées en une racine de l’unité admet un scindage multiplicatif (non unitaire). Nous utilisons pour ce faire une base de la partie torique de la petite algèbre quantique constituée d’idempotents orthogonaux deux à deux et de somme 1 et faisons de même dans le cas «modulaire» pour l’algèbre des distributions d’un groupe algébrique semi-simple.

We show that the quantum Frobenius morphism constructed by Lusztig in the setting of the quantum enveloping algebra $U_{\mathcal{B}}$ specialized at a root of unity admits a (nonunital) multiplicative splitting. We construct the splitting using a basis of the toral part of the small quantum algebra consisting of pairwise orthogonal idempotents summing up to 1, and likewise in the modular case of the algebra of distributions for a semisimple algebraic group.

We determine all helix surfaces with parallel mean curvature vector field which are not minimal or pseudo-umbilical in spaces of type $M^{n}(c)\times\mathbb{R}$, where Mⁿ(c) is a simply connected n-dimensional manifold with constant sectional curvature c.

We study the smoothness of the Siciak–Zaharjuta extremal function associated to a convex body in $\mathbb{R}^{2}$. We also prove a formula relating the complex equilibrium measure of a convex body in $\mathbb{R}^{n}$ (n≥2) to that of its Robin indicatrix. The main tool we use is extremal ellipses.

We will prove Sarnak’s conjecture on Möbius disjointness for continuous interval maps of zero entropy and also for orientation-preserving circle homeomorphisms by reducing these result to a well-known theorem of Davenport from 1937.

We construct Fatou–Bieberbach domains in $\mathbb{C}^{n}$ for n>1 which contain a given compact set K and at the same time avoid a totally real affine subspace L of dimension < n, provided that K∪L is polynomially convex. By using this result, we show that the domain $\mathbb{C}^{n}\setminus\mathbb{R}^{k}$ for 1≤k< n enjoys the basic Oka property with approximation for maps from any Stein manifold of dimension < n.

We characterize mappings S_i and T_i, not necessarily linear, from sets $\mathcal {J}_{i}$, i=1,2, onto multiplicative subsets of function algebras, subject to the following conditions on the peripheral spectra of their products: σ_π(S₁(a)S₂(b))⊂σ_π(T₁(a)T₂(b)) and σ_π(S₁(a)S₂(b))∩σ_π(T₁(a)T₂(b))≠∅, $a\in \mathcal {J}_{1}$, $b\in \mathcal {J}_{2}$. As a direct consequence we obtain a large number of previous results about mappings subject to various spectral conditions.

We show that the standard graded cover of the well-known category $\mathcal{O}$ of Bernstein–Gelfand–Gelfand can be characterized by its compatibility with the action of the center of the enveloping algebra.

We introduce and study the notion of singular hermitian metrics on holomorphic vector bundles, following Berndtsson and Păun. We define what it means for such a metric to be positively and negatively curved in the sense of Griffiths and investigate the assumptions needed in order to locally define the curvature Θ^h as a matrix of currents. We then proceed to show that such metrics can be regularised in such a way that the corresponding curvature tensors converge weakly to Θ^h. Finally we define what it means for h to be strictly negatively curved in the sense of Nakano and show that it is possible to regularise such metrics with a sequence of smooth, strictly Nakano negative metrics.