We compute: * the cohomology with group ring coefficients of Artin groups (or actually, of their associated Salvetti complexes), Bestvina-Brady groups, and graph products of groups, * the L^2-Betti numbers of Bestvina-Brady groups and of graph...
We show that any action of a finite group on a finitely presentable group arises as the action of the group of self-homotopy equivalences of a space on its fundamental group. In doing so, we prove that any finite connected (abstract) simplicial compl...
We give a new proof of the fact that the complement of the complexification of a real hyperplane arrangement is homotopy equivalent to the Salvetti complex of the associated oriented matroid. Our proof involves no choices, is relatively easy to visua...
We establish some connections between nonresonant $A$-hypergeometric systems and de Rham-type complexes. This allows us to determine which of these $A$-hypergeometric systems "come from geometry."...
We characterize folding patterns, the combinatorial options of folding minimal alcove-to-alcove galleries in affine Coxeter complexes positively with respect to Weyl chamber orientations of the Coxeter complex, by drawing a connection to the Bruhat m...
We prove that for $n\geq 2$, a non-uniform lattice in $\text{PU}(n,1)$ does not admit a relatively geometric action on a $\mathrm{CAT}(0)$ cube complex, in the sense of Einstein and Groves. As a consequence, if $Γ$ is a non-uniform lattice in a non-...
"Concurrence topology" (Ellis and Klein \emph{Homology, Homotopy, and Applications,} \textbf{16}) is a TDA method for binary data. The idea is to construct a filtration consisting of Dowker complexes then compute persistent homology. Persistent class...
We give a characterization for asymptotic dimension growth. We apply it to CAT(0) cube complexes of finite dimension, giving an alternative proof of N. Wright's result on their finite asymptotic dimension. We also apply our new characterization to ge...
We present complete simplicial fan realizations of any spherical subword complex of type $A_n$ for $n\leq 3$. This provides complete simplicial fan realizations of simplicial multi-associahedra $Δ_{2k+4,k}$, whose facets are in correspondence with $...
We prove that a hyperplane in a CAT(0) cubical complex X has no self-intersections and separates X into two convex complementary components. These facts were originally proved by Sageev. Our argument shows that his theorem is a corollary of Gromov'...
We introduce a new homological machine for the study of secondary geometric invariants. The objects, called spark complexes, occur in many areas of mathematics. The theory is applied here to establish the equivalence of a large family of spark comp...
In that paper, we prove that the composition of two roofs is another roof by using mapping cone of a morphism of cochain complexes....
In his seminal 1983 paper, Jim Lawrence introduced lopsided sets and featured them as asymmetric counterparts of oriented matroids, both sharing the key property of strong elimination. Moreover, symmetry of faces holds in both structures as well as i...
Higher-order network analysis uses the ideas of hypergraphs, simplicial complexes, multilinear and tensor algebra, and more, to study complex systems. These are by now well established mathematical abstractions. What's new is that the ideas can be te...
For the cyclic group $C_2$ we give a complete description of the derived category of perfect complexes of modules over the constant Mackey ring $\underline{\mathbb{Z}/\ell}$, for $\ell$ a prime. This is fairly simple for $\ell$ odd, but for $\ell=2$...
In this article, we present a dynamical homotopical cancellation theory for Gutierrez-Sotomayor singular flows $\varphi$, GS-flows, on singular surfaces $M$. This theory generalizes the classical theory of Morse complexes of smooth dynamical systems...
Giza). It includes the pyramid complexes of Giza, Abusir, Saqqara and Dahshur and is recognized by UNESCO as a World Heritage Site. In addition to many
Recently, Tao and Mo (TM) derived a meta-generalized gradient approximation functional based on a model exchange-correlation hole. In this work, the performance of this functional is assessed on standard test sets, using the 6-311++G(3df,3pd) basis s...
The stability of (C20)N chains with N = 3 - 7 is analyzed by numerical simulation using a tight-binding potential and molecular dynamics. Various channels of losing the cluster-chain structure of the (C20)N complexes are observed, including the dec...
Good atlases are defined for effective orbifolds, and a spark complex is constructed on each good atlas. It is proved that this process is 2-functorial with compatible systems playing as morphisms between good atlases, and that the spark character 2-...