We give a differential-geometric construction of compact manifolds with holonomy $\operatorname{Spin}(7)$ which is based on Joyce's second construction of compact $\operatorname{Spin}(7)$-manifolds and Kovalev's gluing construction of compact $G_2$-manifolds. We provide several examples of compact $\operatorname{Spin}(7)$-manifolds, at least one of which is new. Here in this paper we need orbifold admissible pairs $(\overline{X}, D)$ consisting of a compact Kähler orbifold $\overline{X}$ with isolated singular points modelled on $\mathbb{C}^4/\mathbb{Z}_4$, and a smooth anticanonical divisor $D$ on $\overline{X}$. Also, we need a compatible antiholomorphic involution $\sigma$ on $\overline{X}$ which fixes the singular points on $\overline{X}$ and acts freely on the anticanoncial divisor $D$. If two orbifold admissible pairs $(\overline{X}_1, D_1)$, $(\overline{X}_2, D_2)$ and compatible antiholomorphic involutions $\sigma_i$ on $\overline{X}_i$ for $i=1,2$ satisfy the gluing condition, we can glue $(\overline{X}_1 \setminus D_1)/\langle\sigma_1\rangle$ and $(\overline{X}_2 \setminus D_2)/\langle\sigma_2\rangle$ together to obtain a compact Riemannian 8-manifold $(M, g)$ whose holonomy group $\operatorname{Hol}(g)$ is contained in $\operatorname{Spin}(7)$. Furthermore, if the $\widehat{A}$-genus of $M$ equals 1, then $M$ is a compact $\operatorname{Spin}(7)$-manifold, i.e. a compact Riemannian manifold with holonomy $\operatorname{Spin}(7)$.

We prove a scale-invariant boundary Harnack principle for inner uniform domains over a large family of Dirichlet spaces. A novel feature of our work is that we do not assume volume doubling property for the symmetric measure.

In this article, we establish the notion of strong $(r,k,a,b)$-stability related to closed hypersurfaces immersed in the hyperbolic space $\mathbb{H}^{n+1}$, where $r$ and $k$ are nonnegative integers satisfying the inequality $0 \leq k<r \leq n-2$ and $a$ and $b$ are real numbers (at least one nonzero). In this setting, considering some appropriate restrictions on the constants $a$ and $b$, we show that geodesic spheres are strongly $(r,k,a,b)$-stable. Afterwards, under a suitable restriction on the higher order mean curvatures $H_{r+1}$ and $H_{k+1}$, we prove that if a closed hypersurface into the hyperbolic space $\mathbb{H}^{n+1}$ is strongly $(r,k,a,b)$-stable, then it must be a geodesic sphere, provided that the image of its Gauss mapping is contained in the chronological future (or past) of an equator of the de Sitter space.

Let $\varphi$ be a linear map between operator spaces. To measure the intensity of $\varphi$ being isometric we associate with it a number, called the isometric degree of $\varphi$ and written $\mathrm{id}(\varphi)$, as follows. Call $\varphi$ a strict $m$-isometry with $m$ a positive integer if it is an $m$-isometry, but is not an $(m+1)$-isometry. Define $\mathrm{id}(\varphi)$ to be 0, $m$, and $\infty$, respectively if $\varphi$ is not an isometry, a strict $m$-isometry, and a complete isometry, respectively. We show that if $\varphi:M_n\to M_p$ is a unital completely positive map between matrix algebras, then $\mathrm{id}(\varphi) \in \{0,\,1,\,2,\,\dots,\,[({n-1})/{2}],\,\infty\}$ and that when $n\ge 3$ is fixed and $p$ is sufficiently large, the values $1,\,2,\,\dots,\,[({n-1})/{2}]$ are attained as $\mathrm{id}(\varphi)$ for some $\varphi$. The ranges of such maps $\varphi$ with $1 \le \mathrm{id}(\varphi)<\infty$ provide natural examples of operator systems that are isometric, but not completely isometric, to $M_n$. We introduce and classify, up to unital complete isometry, a certain family of such operator systems.

The logarithmic derivative of a point process plays a key rôle in the general approach, due to the third author, to constructing diffusions preserving a given point process. In this paper we explicitly compute the logarithmic derivative for determinantal processes on $\mathbb{R}$ with integrable kernels, a large class that includes all the classical processes of random matrix theory as well as processes associated with de Branges spaces. The argument uses the quasi-invariance of our processes established by the first author.

Categories of W*-bimodules are shown in an explicit and algebraic way to constitute an involutive W*-bicategory.

The aim of this paper is to give an upper bound for the dimension of a torus $T$ which acts on a GKM manifold $M$ effectively. In order to do that, we introduce a free abelian group of finite rank, denoted by $\mathcal{A}(\Gamma,\alpha,\nabla)$, from an (abstract) $(m,n)$-type GKM graph $(\Gamma,\alpha,\nabla)$. Here, an $(m,n)$-type GKM graph is the GKM graph induced from a $2m$-dimensional GKM manifold $M^{2m}$ with an effective $n$-dimensional torus $T^{n}$-action which preserves the almost complex structure, say $(M^{2m},T^{n})$. Then it is shown that $\mathcal{A}(\Gamma,\alpha,\nabla)$ has rank $\ell(> n)$ if and only if there exists an $(m,\ell)$-type GKM graph $(\Gamma,\widetilde{\alpha},\nabla)$ which is an extension of $(\Gamma,\alpha,\nabla)$. Using this combinatorial necessary and sufficient condition, we prove that the rank of $\mathcal{A}(\Gamma_{M},\alpha_{M},\nabla_{M})$ for the GKM graph $(\Gamma_{M},\alpha_{M},\nabla_{M})$ induced from $(M^{2m},T^{n})$ gives an upper bound for the dimension of a torus which can act on $M^{2m}$ effectively. As one of the applications of this result, we compute the rank associated to $\mathcal{A}(\Gamma,\alpha,\nabla)$ of the complex Grassmannian of 2-planes $G_{2}(\mathbb{C}^{n+2})$ with the natural effective $T^{n+1}$-action, and prove that this action on $G_{2}(\mathbb{C}^{n+2})$ is the maximal effective torus action which preserves the standard complex structure.

Homological tilting modules of finite projective dimension are investigated. They generalize both classical and good tilting modules of projective dimension at most one, and produce recollements of derived module categories of rings in which generalized localizations of rings are involved. To decide whether a good tilting module is homological, a sufficient and necessary condition is presented in terms of the internal properties of the given tilting module. Consequently, a class of homological, non-trivial, infinitely generated tilting modules of higher projective dimension is constructed, and the first example of an infinitely generated $n$-tilting module which is not homological for each $n \ge 2$ is exhibited. To deal with both tilting and cotilting modules consistently, the notion of weak tilting modules is introduced. Thus similar results for infinitely generated cotilting modules of finite injective dimension are obtained, though dual technique does not work for infinite-dimensional modules.

Let $F_t$, where $t \in \mathbb{R}$, be an analytic family of plane-to-plane mappings with $F_0$ having a critical point at the origin. The paper presents effective algebraic methods of computing the number of those cusp points of $F_t$, where $0 < |t|\ll 1$, emanating from the origin at which $F_t$ has a positive/negative local topological degree.

Motivated by the study of kernels of bilinear pseudodifferential operators with symbols in a Hörmander class of critical order, we investigate boundedness properties of strongly singular Calderón–Zygmund operators in the bilinear setting. For such operators, whose kernels satisfy integral-type conditions, we establish boundedness properties in the setting of Lebesgue spaces as well as endpoint mappings involving the space of functions of bounded mean oscillations and the Hardy space. Assuming pointwise-type conditions on the kernels, we show that strongly singular bilinear Calderón–Zygmund operators satisfy pointwise estimates in terms of maximal operators, which imply their boundedness in weighted Lebesgue spaces.

Let $R$ be a Cohen–Macaulay local ring with a canonical module. We consider Auslander's (higher) delta invariants of powers of certain ideals of $R$. Firstly, we shall provide some conditions for an ideal to be a parameter ideal in terms of delta invariants. As an application of this result, we give upper bounds for orders of Ulrich ideals of $R$ when $R$ has Gorenstein punctured spectrum. Secondly, we extend the definition of indices to the ideal case, and generalize the result of Avramov–Buchweitz–Iyengar–Miller on the relationship between the index and regularity.

We define an “ample canonical height” for an endomorphism on a projective variety, which is essentially a generalization of the canonical heights for polarized endomorphisms introduced by Call–Silverman. We formulate a dynamical analogue of the Northcott finiteness theorem for ample canonical heights as a conjecture, and prove it for endomorphisms on varieties of small Picard numbers, abelian varieties, and surfaces. As applications, for the endomorphisms which satisfy the conjecture, we show the non-density of the set of preperiodic points over a fixed number field, and obtain a dynamical Mordell–Lang type result on the intersection of two Zariski dense orbits of two endomorphisms on a common variety.

For a metric measure space, we consider the set of distributions of 1-Lipschitz functions, which is called the 1-measurement. On the 1-measurement, we have the Lipschitz order relation introduced by M. Gromov. The aim of this paper is to study the maximum and maximal elements of the 1-measurement of a metric measure space with respect to the Lipschitz order. We present a necessary condition of a metric measure space for the existence of the maximum of the 1-measurement. We also consider a metric measure space that has the maximum of its 1-measurement.

This article presents an analysis of the normalized Yamabe flow starting at and preserving a class of compact Riemannian manifolds with incomplete edge singularities and negative Yamabe invariant. Our main results include uniqueness, long-time existence and convergence of the edge Yamabe flow starting at a metric with everywhere negative scalar curvature. Our methods include novel maximum principle results on the singular edge space without using barrier functions. Moreover, our uniform bounds on solutions are established by a new ansatz without in any way using or redeveloping Krylov–Safonov estimates in the singular setting. As an application we obtain a solution to the Yamabe problem for incomplete edge metrics with negative Yamabe invariant using flow techniques. Our methods lay groundwork for studying other flows like the mean curvature flow as well as the porous medium equation in the singular setting.