Let $(R,\mathfrak{m})$ be a d-dimensional commutative complete Noetherian local ring and Λ be a Noetherian R-algebra. Motivated by the notion of Cohen–Macaulay Artin algebras of Auslander and Reiten, we say that Λ is Cohen–Macaulay if there is a finitely generated Λ-bimodule ω that is maximal Cohen–Macaulay over R such that the adjoint pair of functors $(\mathit{\omega }{\otimes }_{{\mathrm{\Lambda }}^{\prime }}-,{\mathsf{Hom}}_{{\mathrm{\Lambda }}^{\prime }}(\mathit{\omega },-))$ induces quasi-inverse equivalences between the full subcategories of finitely generated ${\mathrm{\Lambda }}^{\prime }$-modules consisting of modules of finite projective dimension, ${\mathcal{P}}^{\mathrm{\infty }}({\mathrm{\Lambda }}^{\prime })$, and the modules of finite injective dimension, ${\mathcal{I}}^{\mathrm{\infty }}({\mathrm{\Lambda }}^{\prime })$, whenever ${\mathrm{\Lambda }}^{\prime }=\mathrm{\Lambda },{\mathrm{\Lambda }}^{\mathrm{op}}$. It is proved that such a module ω is unique, up to isomorphism, as a ${\mathrm{\Lambda }}^{\prime }$-module. It is also shown that Λ is a Cohen–Macaulay algebra if and only if there is a semidualizing Λ-bimodule ω of finite injective dimension and ${\mathcal{P}}^{\mathrm{\infty }}({\mathrm{\Lambda }}^{\prime })$ and ${\mathcal{I}}^{\mathrm{\infty }}({\mathrm{\Lambda }}^{\prime })$ are contained in the Auslander and Bass classes, respectively. We prove that Cohen–Macaulayness behaves well under reduction modulo system of parameters of R. Indeed, it will be observed that if Λ is a Cohen–Macaulay algebra, then for any system of parameters $x=x_1,\dots ,x_d$ of R, the Artin algebra $\mathrm{\Lambda }∕x\mathrm{\Lambda }$ is Cohen–Macaulay as well. Assume that ω is a semidualizing Λ-bimodule of finite injective dimension that is maximal Cohen–Macaulay as an R-module. It will turn out that Λ being a Cohen–Macaulay algebra is equivalent to saying that the pair $(\mathsf{CM}({\mathrm{\Lambda }}^{\prime }),{\mathcal{I}}^{\mathrm{\infty }}({\mathrm{\Lambda }}^{\prime }))$ forms a hereditary complete cotorsion theory and the pair $(\mathsf{CM}({\mathrm{\Lambda }}^{\prime \mathrm{op}}),{\mathcal{P}}^{\mathrm{\infty }}({\mathrm{\Lambda }}^{\prime }))$ forms a $\mathsf{Tor}$-torsion theory, where $\mathsf{CM}({\mathrm{\Lambda }}^{\prime })$ is the class of all finitely generated ${\mathrm{\Lambda }}^{\prime }$-modules admitting a right resolution by modules in $\mathsf{add}\mathit{\omega }$. Finally, it is shown that Cohen–Macaulayness ascends from R to $R\mathrm{\Gamma }$ and $R\mathcal{Q}$, where Γ is a finite group and $\mathcal{Q}$ is a finite acyclic quiver.

We consider the R-matrix of the quantum toroidal algebra of type ${\mathfrak{gl}}_1$, both abstractly and in Fock space representations. We provide a survey of a certain point of view on this object which involves the elliptic Hall and shuffle algebras, and we show how to obtain certain explicit formulas.

Let $\mathrm{\Gamma }\setminus G∕K$ be a compact Hermitian locally symmetric space, where G is simple. We study the components of a de Rham cohomology class of $\mathrm{\Gamma }\setminus G∕K$ with respect to the Matsushima decomposition, where the class is obtained by taking the Poincaré dual of a totally geodesic complex analytic submanifold. Using an improved version of the vanishing result of Kobayashi and Oda, we specify the existence of certain components of such cohomology classes when $G=\mathrm{SU}(p,q)$.

On établit des relations entre la fonction de Hilbert d’un groupe de points et ses propriétés géométriques. On utilise ce résultat pour aborder une conjecture de Gruson–Lazarsfeld–Peskine sur la postulation d’une courbe de l’espace projectif et l’existence d’une multisécante.

We study the Gruson–Lazarsfeld–Peskine conjecture about equations defining space curves and multisecant lines.

For the irreducible holomorphic symplectic eightfold Z associated to a cubic fourfold Y not containing a plane, we show that a natural Abel–Jacobi map from $H_{\mathrm{prim}}^4(Y)$ to $H_{\mathrm{prim}}^2(Z)$ is a Hodge isometry. We describe the full $H^2(Z)$ in terms of the Mukai lattice of the K3 category $\mathcal{A}$ of Y. We give numerical conditions for Z to be birational to a moduli space of sheaves on a K3 surface or to ${\mathrm{Hilb}}^4$(K3). We propose a conjecture on how to use Z to produce equivalences from $\mathcal{A}$ to the derived category of a K3 surface.

We discuss natural operations on loops in a quasisurface and show that these operations define a structure of a quasi-Lie bialgebra in the module generated by the set of free homotopy classes of noncontractible loops.

For any 0-cell B in a 2-category $\mathcal{B}$, we introduce the notion of adjoint algebra ${\mathcal{A}d}_B$. This is an algebra in the center of $\mathcal{B}$. We prove that if $\mathcal{C}$ is a finite tensor category, this notion applied to the 2-category ${}_{\mathcal{C}}\mathrm{Mod}$ of $\mathcal{C}$-module categories coincides with the one introduced by Shimizu. As a consequence of this general approach, we obtain new results on the adjoint algebra for tensor categories.

In this paper, we confirm a conjecture of Orlik and Randell from 1977 on the Seifert form of chain-type invertible singularities. We use Lefschetz bifibration techniques as developed by Seidel (inspired by Arnold and Donaldson) and take advantage of the symmetries at hand. We believe that our method will be useful in understanding the homological/categorical version of Berglund–Hübsch mirror conjecture for invertible singularities.

In this paper we introduce a class of Hamilton delay equations which arise as critical points of an action functional motivated by orbit interactions. We show that the kernel of the Hessian at each critical point of the action functional satisfies a uniform bound on its dimension.

In this article we explore the relationship between the systole and the diameter of closed hyperbolic orientable surfaces. We show that they satisfy a certain inequality, which can be used to deduce that their ratio has a (genus-dependent) upper bound.

We characterize the subexponential densities on $(0,\mathrm{\infty })$ for compound Poisson distributions on $[0,\mathrm{\infty })$ with absolutely continuous Lévy measures. In particular, we show that the class of all subexponential probability density functions on $[0,\mathrm{\infty })$ is closed under generalized convolution roots for compound Poisson sums. Moreover, we give an application to the subexponential density on $(0,\mathrm{\infty })$ for the distribution of the supremum of a random walk.