Related papers: Combinatorics of rank jumps in simplicial hypergeo…

We present a new notion of non-positively curved groups: the collection of discrete countable groups acting (AU-)acylindrically on finite products of $\delta$-hyperbolic spaces with general type factors. Inspired by the classical theory of…

We show that a finite regular cell complex with the intersection property is a Cohen-Macaulay space iff the top enriched cohomology module is the only nonvanishing one. We prove a comprehensive generalization of Balinski's theorem on convex…

The porpose of this article is to introduce and investigate properties of a tool (the a-hyperbolic rank) which enables us to obtain new examples of homogeneous spaces G/H which admit and do not admit almost compact Clifford-Klein forms. We…

It is easy to see that if $\cG$ is a non-abelian group of unitary matrices, then for no members $A$ and $B$ of $\cG$ can the rank of $AB-BA$ be one. We examine the consequences of the assumption that this rank is at most two for a general…

We prove that the $f$-vector of members in a certain class of meet semi-lattices satisfies Macaulay inequalities. We construct a large family of meet semi-lattices belonging to this class, which includes all posets of multicomplexes, as…

We show that the group of conformal homeomorphisms of the boundary of a rank one symmetric space (except the hyperbolic plane) of noncompact type acts as a maximal convergence group. Moreover, we show that any family of uniformly…

Let $M$ be complete nonpositively curved Riemannian manifold of finite volume whose fundamental group $\Gamma$ does not contain a finite index subgroup which is a product of infinite groups. We show that the universal cover $\tilde M$ is a…

Via the BGG-correspondence a simplicial complex D on [n] is transformed into a complex of coherent sheaves L(D) on the projective space n-1-space. In general we compute the support of each of its cohomology sheaves. When the Alexander dual…

In this short note we announce three formulas for the set of weights of various classes of highest weight modules $\V$ with highest weight \lambda, over a complex semisimple Lie algebra $\lie{g}$ with Cartan subalgebra $\lie{h}$. These…

For a finite set A of integral vectors, Gel'fand, Kapranov and Zelevinskii defined a system of differential equations with a parameter vector as a D-module, which system is called an A-hypergeometric (or a GKZ hypergeometric) system.…

Given a finite abelian $p$-group $F$, we prove an efficient recursive formula for $\sigma_a(F)=\sum_{\substack{H\leq F}}|H|^a$ where $H$ ranges over the subgroups of $F$. We infer from this formula that the $p$-component of the…

We address the questions (P1), (P2) asked in Kirchheim-M\"{u}ller-\v{S}ver\'{a}k (2003) concerning the structure of the Rank-$1$ convex hull of a submanifold $\mathcal{K}_1\subset M^{3\times 2}$ that is related to weak solutions of the two…

For even $k$, the matchings connectivity matrix $\mathbf{M}_k$ encodes which pairs of perfect matchings on $k$ vertices form a single cycle. Cygan et al. (STOC 2013) showed that the rank of $\mathbf{M}_k$ over $\mathbb{Z}_2$ is…

Let $V$ be a finite-dimensional vector space over the field with $p$ elements, where $p$ is a prime number. Given arbitrary $\alpha,\beta\in \mathrm{GL}(V)$, we consider the semidirect products $V\rtimes\langle \alpha\rangle$ and…

We call a real algebraic hypersurface in $(\mathbb{C}^*)^n$ simplicial if it is given by a real Laurent polynomial in $n$-variables that has exactly $n+1$ monomials with non-zero coefficients and such that the convex hull in $\mathbb{R}^n$…

The rank of a finite algebraic structure with a single binary operation is the minimum number of elements needed to express every other element under the closure of the operation. In the case of groups, the previous best algorithm for…

Let ${\cal O}$ be a quantizable coadjoint orbit of a semisimple Lie group $G$. Under certain hypotheses we prove that $#(\pi_1(\text{Ham}({\cal O})))\geq #(Z(G))$, where $\text{Ham}({\cal O})$ is the group of Hamiltonian symplectomorphisms…

In this note, given a pair $(\mathfrak{g}, \lambda)$, where $\mathfrak{g}$ is a complex semisimple Lie algebra and $\lambda \in \mathfrak{h}^*$ is a dominant integral weight of $\mathfrak{g}$, where $\mathfrak{h} \subset \mathfrak{g}$ is…

We introduce the monic rank of a vector relative to an affine-hyperplane section of an irreducible Zariski-closed affine cone $X$. We show that the monic rank is finite and greater than or equal to the usual $X$-rank. We describe an…

We develop a notion of rank one properly convex domains (or Hilbert geometries) in the real projective space. This is in the spirit of rank one non-positively curved Riemannian manifolds and CAT(0) spaces. We define rank one isometries for…