Suppose $G$ is a connected complex Lie group and $H$ is a closed complex subgroup. Then there exists a closed complex subgroup $J$ of $G$ containing $H$ such that the fibration $\pi :G/H\to $$G/J$ is the holomorphic reduction of $G/H$i.e., $G/J$ is holomorphically separable and $\mathcal{O}(G/H)\cong $${{\pi }^{*}}\mathcal{O}(G/J)$. In this paper we prove that if $G/H$ is pseudoconvex, i.e., if $G/H$ admits a continuous plurisubharmonic exhaustion function, then $G/J$ is Stein and $J/H$ has no non-constant holomorphic functions.

It follows from earlier work of Silver and Williams and the authors that twisted Alexander polynomials detect the unknot and the Hopf link. We now show that twisted Alexander polynomials also detect the trefoil and the figure-8 knot, that twisted Alexander polynomials detect whether a link is split and that twisted Alexander modules detect trivial links. We use this result to provide algorithms for detecting whether a link is the unlink, whether it is split, and whether it is totally split.

Let $D$ be an integral domain, ${{X}^{1}}\left( D \right)$ be the set of height-one prime ideals of $D$, $\left\{ {{X}_{\beta }} \right\}$ and $\left\{ {{X}_{\alpha }} \right\}$ be two disjoint nonempty sets of indeterminates over $D$, $D\left[ \left\{ {{X}_{\beta }} \right\} \right]$ be the polynomial ring over $D$, and $D\left[ \left\{ {{X}_{\beta }} \right\} \right]{{\left[\!\left[ \left\{ {{X}_{\alpha }} \right\} \right]\!\right]}_{1}}$ be the first type power series ring over $D\left[ \left\{ {{X}_{\beta }} \right\} \right]$. Assume that $D$ is a Prüfer $v$-multiplication domain $\left( \text{P}v\text{MD} \right)$ in which each proper integral $t$-ideal has only finitely many minimal prime ideals (e.g., $t$-$\text{SFT}$$\text{P}v\text{MDs}$, valuation domains, rings of Krull type). Among other things, we show that if ${{X}^{1}}\left( D \right)\,=\,\phi$ or ${{D}_{p}}$ is a $\text{DVR}$ for all $P\,\in \,{{X}^{1}}\left( D \right)$, then $D\left[ \left\{ {{X}_{\beta }} \right\} \right]{{\left[\!\left[ \left\{ {{X}_{\alpha }} \right\} \right]\!\right]}_{1D-\left\{ 0 \right\}}}$ is a Krull domain. We also prove that if $D$ is a $t$-$\text{SFT}\text{P}v\text{MD}$, then the complete integral closure of $D$ is a Krull domain and $\text{ht}\left( M\left[ \left\{ {{X}_{\beta }} \right\} \right]{{\left[\!\left[ \left\{ {{X}_{\alpha }} \right\} \right]\!\right]}_{1}} \right)\,=\,1$ for every height-one maximal $t$-ideal $M$ of $D$.

We prove an analogue of the Riemann–Hurwitz theorem for computing Euler characteristics of pullbacks of coherent sheaves through finite maps of smooth projective varieties in arbitrary dimensions, subject only to the condition that the irreducible components of the branch and ramification locus have simple normal crossings.

In 2012, Gubeladze (Adv. Math. 2012) introduced the notion of $k$-convex-normal polytopes to show that integral polytopes all of whose edges are longer than $4d(d+1)$ have the integer decomposition property. In the first part of this paper we show that for lattice polytopes there is no diòerence between $k$- and $(k+1)$-convex-normality (for $k\ge 3$) and improve the bound to $2d(d+1)$. In the second part we extend the definition to pairs of polytopes. Given two rational polytopes $P$ and $\text{Q}$, where the normal fan of $P$ is a refinement of the normal fan of $\text{Q}$, if every edge ${{e}_{P}}$ of $P$ is at least $d$ times as long as the corresponding face (edge or vertex) ${{e}_{\text{Q}}}$ of $\text{Q}$, then $(P+\text{Q})\cap {{\mathbb{Z}}^{d}}=(P\cap {{\mathbb{Z}}^{d}})+(\text{Q}\cap {{\mathbb{Z}}^{d}})$.

Let $\mathbb{L}$ be a length function on a group $G$, and let ${{M}_{\mathbb{L}}}$ denote the operator of pointwise multiplication by $\mathbb{L}$ on ${{\ell }^{2}}\left( G \right)$. Following Connes, ${{M}_{\mathbb{L}}}$ can be used as a “Dirac” operator for the reduced group ${{C}^{*}}$-algebra $C_{r}^{*}\left( G \right)$. It defines a Lipschitz seminorm on $C_{r}^{*}\left( G \right)$, which defines a metric on the state space of $C_{r}^{*}\left( G \right)$. We show that for any length function satisfying a strong form of polynomial growth on a discrete group, the topology from this metric coincides with the weak-$*$ topology (a key property for the definition of a “compact quantum metric space”). In particular, this holds for all word-length functions on finitely generated nilpotent-by-finite groups.

We study linear projections on Plücker space whose restriction to the Grassmannian is a non-trivial branched cover. When an automorphism of the Grassmannian preserves the fibers, we show that the Grassmannian is necessarily of $m$-dimensional linear subspaces in a symplectic vector space of dimension $2m$, and the linear map is the Lagrangian involution. The Wronski map for a self-adjoint linear diòerential operator and the pole placement map for symmetric linear systems are natural examples.

It is known that if $E$ is a closed subset of an open Riemann surface $R$ and $f$ is a holomorphic function on a neighbourhood of $E$, then it is “usually” not possible to approximate $f$ uniformly by functions holomorphic on all of $R$. We show, however, that for every open Riemann surface $R$ and every closed subset $E\subset R$, there is closed subset $F\subset E$ that approximates $E$ extremely well, such that every function holomorphic on $F$ can be approximated much better than uniformly by functions holomorphic on $R$.

In the Simpson moduli space $M$ of semi-stable sheaves with Hilbert polynomial $dm\,\text{-}\,\text{1}$ on a projective plane we study the closed subvariety ${{M}^{'}}$ of sheaves that are not locally free on their support. We show that for $d\ge 4$, it is a singular subvariety of codimension 2 in $M$. The blow up of $M$ along ${{M}^{'}}$ is interpreted as a (partial) modification of $M\backslash {{M}^{'}}$ by line bundles (on support).

We previously obtained a congruence modulo four for the number of real solutions to many Schubert problems on a square Grassmannian given by osculating flags. Here we consider Schubert problems given by more general isotropic flags, and prove this congruence modulo four for the largest class of Schubert problems that could be expected to exhibit this congruence.

Let $X$ be a real normed space, $Y$ a Banach space, and $f\,:\,X\,\to \,Y$. We prove theUlam–Hyers stability theorem for the cubic functional equation

$$f\left( 2x\,+\,y \right)\,+\,f\left( 2x\,-\,y \right)\,-\,2f\left( x\,+\,y \right)\,-\,2f\left( x\,-\,y \right)\,-\,12f\left( x \right)\,=\,0$$

in restricted domains. As an application we consider a measure zero stability problem of the inequality

$$\left\| f\left( 2x\,+\,y \right)\,+\,f\left( 2x\,-\,y \right)\,-\,2f\left( x\,+\,y \right)\,-\,2f\left( x\,-\,y \right)\,-\,12f\left( x \right) \right\|\,\le \,\varepsilon$$

for all $\left( x,\,y \right)$ in $\Gamma \,\subset \,{{\mathbb{R}}^{2}}$ of Lebesgue measure 0.

Fix an irreducible (finite) root system $R$ and a choice of positive roots. For any algebraically closed field $k$ consider the almost simple, simply connected algebraic group ${{G}_{k}}$ over $k$ with root system $k$. One associates with any dominant weight $\lambda $ for $R$ two ${{G}_{k}}$-modules with highest weight $\lambda $, the Weyl module $V{{(\lambda )}_{k}}$ and its simple quotient $V{{(\lambda )}_{k}}$. Let $\lambda $ and $\mu $ be dominant weights with $\mu <\lambda $ such that $\mu $ is maximal with this property. Garibaldi, Guralnick, and Nakano have asked under which condition there exists $k$ such that $L{{(\mu )}_{k}}$ is a composition factor of $V{{(\lambda )}_{k}}$, and they exhibit an example in type ${{E}_{8}}$ where this is not the case. The purpose of this paper is to to show that their example is the only one. It contains two proofs for this fact: one that uses a classiffication of the possible pairs $(\lambda ,\mu )$, and another that relies only on the classiûcation of root systems.

Alfred Schild has established conditions that Lorentz transformationsmap world-vectors $(ct,x,y,z)$ with integer coordinates onto vectors of the same kind. These transformations are called integral Lorentz transformations.

This paper contains supplements to our earlier work with a new focus on group theory. To relate the results to the familiar matrix group nomenclature, we associate Lorentz transformations with matrices in $\text{SL}(2,\mathbb{C})$. We consider the lattice of subgroups of the group originated in Schild's paper and obtain generating sets for the full group and its subgroups.

In this paper we determine the ${{L}^{1}}\to {{L}^{1}}$ and ${{L}^{\infty }}\to {{L}^{\infty }}$ norms of an integral operator $\mathcal{N}$ related to the gradient of the solution of Poisson equation in the unit ball with vanishing boundary data in sense of distributions.

Let $R$ be a commutative ring with non-zero identity. In this paper, we introduce the weakly nilpotent graph of a commutative ring. The weakly nilpotent graph of $R$ denoted by ${{\Gamma }_{w}}(R)$ is a graph with the vertex set ${{R}^{\star }}$ and two vertices $x$ and $y$ are adjacent if and only if $x\,y\in N{{(R)}^{\star }}$, where ${{R}^{\star }}=R\backslash \{0\}$ and $N{{(R)}^{\star }}$ is the set of all non-zero nilpotent elements of $R$. In this article, we determine the diameter of weakly nilpotent graph of an Artinian ring. We prove that if ${{\Gamma }_{w}}(R)$ is a forest, then ${{\Gamma }_{w}}(R)$ is a union of a star and some isolated vertices. We study the clique number, the chromatic number, and the independence number of ${{\Gamma }_{w}}(R)$. Among other results, we show that for an Artinian ring $R$, ${{\Gamma }_{w}}(R)$ is not a disjoint union of cycles or a unicyclic graph. For Artinan rings, we determine diam $\overline{({{\Gamma }_{w}}(R))}$. Finally, we characterize all commutative rings $R$ for which $\overline{({{\Gamma }_{w}}(R))}$ is a cycle, where $\overline{({{\Gamma }_{w}}(R))}$ is the complement of the weakly nilpotent graph of $R$.

A Nikishin–Maurey characterization is given for bounded subsets of weak-type Lebesgue spaces. New factorizations for linear and multilinear operators are shown to follow.

This paper introduces a unified operator theory approach to the abstract Plancherel (trace) formulas over homogeneous spaces of compact groups. Let $G$ be a compact group and let $H$ be a closed subgroup of $G$. Let ${G}/{H}\;$ be the left coset space of $H$ in $G$ and let $\mu$ be the normalized $G$-invariant measure on ${G}/{H}\;$ associated with Weil’s formula. Then we present a generalized abstract notion of Plancherel (trace) formula for the Hilbert space ${{L}^{2}}\left( {G}/{H,\,\mu }\; \right)$.

We prove that the moduli space of smooth primitively polarized $\text{K3}$ surfaces of genus 33 is unirational.

We prove that for $d\in \left\{ 2,3,5,7,13 \right\}$ and $K$ a quadratic (or rational) field of discriminant $D$ and Dirichlet character $\chi $, if a prime $p$ is large enough compared to $D$, there is a newform $f\in {{S}_{2}}({{\Gamma }_{0}}(d{{p}^{2}}))$ with sign $(+1)$ with respect to the Atkin–Lehner involution ${{w}_{{{p}^{2}}}}$ such that $L(f\otimes \chi ,1)\ne 0$. This result is obtained through an estimate of a weighted sum of twists of $L$-functions that generalises a result of Ellenberg. It relies on the approximate functional equation for the $L$-functions $L(f\otimes \chi ,\cdot )$ and a Petersson trace formula restricted to Atkin–Lehner eigenspaces. An application of this nonvanishing theorem will be given in terms of existence of rank zero quotients of some twisted jacobians, which generalises a result of Darmon and Merel.

A ${{C}^{*}}$-algebra Ahas the ideal property if any ideal $I$ of $A$ is generated as a closed two-sided ideal by the projections inside the ideal. Suppose that the limit ${{C}^{*}}$-algebra $A$ of inductive limit of direct sums of matrix algebras over spaces with uniformly bounded dimension has the ideal property. In this paper we will prove that $A$ can be written as an inductive limit of certain very special subhomogeneous algebras, namely, direct sum of dimension-drop interval algebras and matrix algebras over 2-dimensional spaces with torsion ${{H}^{2}}$ groups.