We study the existence and uniqueness problem of compact minimal vertical graphs in $\mathbb{H}^{n}\times\mathbb{R}$, $n\geq2$, over bounded domains in the slice $\mathbb{H}^{n}\times\{0\}$, with non-connected boundary having a finite number of $C^{0}$ hypersufaces homeomorphic to the sphere $\mathbb{S}^{n-1}$, with prescribed bounded continuous boundary data, under hypotheses relating those data and the geometry of the boundary. We show the nonexistence of compact minimal vertical graphs in $\mathbb{H}^{n}\times\mathbb{R}$ having the boundary in two slices and the height greater than or equal to $\pi/(2n-2)$.

In this paper, we define a Rokhlin property for automorphisms of non-unital $C^{*}$-algebras and for endomorphisms. We show that the crossed product of a $C^{*}$-algebra by a Rokhlin automorphism preserves absorption of a strongly self-absorbing $C^{*}$-algebra, and use this result to deduce that the same result holds for crossed products by endomorphisms in the sense of Stacey. This generalizes earlier results of the second named author and W. Winter.

In this paper, we describe the surjective linear isometries on a vector valued little Bloch space with range space a smooth, strictly convex and reflexive complex Banach space. We also describe the hermitian operators and the generalized bi-circular projections supported by these spaces.

In this paper, we provide constructions to enumerate large numbers of CI-liaison classes. To this end, we introduce a liaison invariant and prove several results concerning it, notably that it commutes with hypersurface sections. This theory is applied to the CI-liaison classes of ruled joins of projective schemes, yielding strong obstructions for such joins to lie in the same liaison class. A second construction arises from the actions of automorphisms on liaison classes, allowing the enumeration of many liaison classes of perfect ideals of codimension at least three.

We identify all translation covers among triangular billiard surfaces. Our main tools are the holonomy field of Kenyon and Smillie and a geometric property of triangular billiard surfaces, which we call the fingerprint of a point, that is preserved under balanced translation covers.

Let $X$ be a compact connected Riemann surface of genus $g$, with $g\geq 2$, and let $\mathcal{O}_{X}$ denote the sheaf of holomorphic functions on $X$. Fix positive integers $r$ and $d$ and let $\mathcal{Q}(r,d)$ be the Quot scheme parametrizing all torsion coherent quotients of $\mathcal{O}^{\oplus r}_{X}$ of degree $d$. We prove that $\mathcal{Q}(r,d)$ does not admit a Kähler metric whose holomorphic bisectional curvatures are all nonnegative.

For a von Neumann algebra $\mathcal{M}$ with a faithful normal tracial state $\tau$ and a positive ergodic homomorpsism $\alpha :\mathcal{L}^{1}(\mathcal{M},\tau)\to\mathcal{L}^{1}(\mathcal{M},\tau)$ such that $\tau\circ\alpha =\tau$ and $\alpha $ does not increase the norm in $\mathcal{M}$, we establish a non-commutative counterpart of the classical Wiener–Wintner ergodic theorem.

Let $R=K[X_{1},\ldots,X_{n}]$ where $K$ is a field of characteristic zero. Let $I$ be an ideal in $R$ and let $M=H^{i}_{I}(R)$ be the $i$th-local cohomology module of $R$ with respect to $I$. Let $c=\operatorname{injdim} M$. We prove that if $P$ is a prime ideal in $R$ with Bass number $\mu_{c}(P,M)>0$ then $P$ is a maximal ideal in $R$.

The higher rank Lefschetz formula for $p$-adic groups is used to prove rationality of a several-variable zeta function attached to the action of a $p$-adic group on its Bruhat–Tits building. By specializing to certain lines one gets one-variable zeta functions, which then can be related to geometrically defined zeta functions.

A spherical $n$-gon is a bordered surface homeomorphic to a closed disk, with $n$ distinguished boundary points called corners, equipped with a Riemannian metric of constant curvature $1$, except at the corners, and such that the boundary arcs between the corners are geodesic. We discuss the problem of classification of these polygons and enumerate them in the case that two angles at the corners are not multiples of $\pi$. The problem is equivalent to classification of some second order linear differential equations with regular singularities, with real parameters and unitary monodromy.

Since the 1970’s, great interest has been taken in the study of pure $O$-sequences, which are in bijective correspondence to the Hilbert functions of Artinian level monomial algebras. Much progress has been made in classifying these by their shape. It has been shown that all monomial complete intersections, Artinian algebras in two variables and Artinian level monomial algebras with type two in both three and four variables have unimodal Hilbert functions. This paper proves that Artinian level monomial algebras of type three in three variables have unimodal Hilbert functions. We will also discuss the licciness of these algebras.

We construct a compact metric space that has any other compact metric space as a tangent at all points, with respect to the Gromov–Hausdorff distance. Furthermore, we give examples of compact sets in the Euclidean unit cube, that have almost any other compact set of the cube as a tangent at all points or just in a dense subset. Here the “almost all compact sets” means that the tangent collection contains a contracted image of any compact set of the cube and that the contraction ratios are uniformly bounded. In the Euclidean space, the distance of subsets is measured by the Hausdorff distance. Also the geometric properties and dimensions of such spaces and sets are studied.

We characterize and study Riemannian almost CR manifolds admitting characteristic connections, that is, metric connections with totally skew-symmetric torsion parallelizing the almost CR structure. Natural constructions are provided of new nontrivial examples. We study the influence of the curvature of the metric on the underlying almost CR structure. A global classification is obtained under flatness assumption of a characteristic connection, provided that the fundamental $2$-form of the structure is closed (quasi Sasakian condition).

We study some estimates for the sums of eigenvalues of the Dirichlet poly-harmonic operator $(-\Delta)^{l}$ restricted to a bounded domain $\Omega\subset\mathbb{R}^{d}$ with $d\ge2$, $l\ge1$. Our approach yields estimates sharper than the estimates recently obtained by Q.-M. Cheng, X. Qi and G. Wei (Pacific J. Math. 262 (2013) 35–47) and G. Wei and L. Zeng (Estimates for eigenvalues of poly-harmonic operators, preprint). Another central object of study in this paper is to establish some certain estimates for the sums of powers of the eigenvalues of the poly-harmonic operator $(-\Delta)^{l}|_{\Omega}$.