One of the main open questions in liaison theory is whether every homogeneous Cohen-Macaulay ideal in a polynomial ring is glicci, i.e. if it is in the G-liaison class of a complete intersection. We give an affirmative answer to this question for StanleyReisner ideals defined by simplicial complexes that are weakly vertex-decomposable. This class of complexes includes matroid, shifted and Goren...

Definition 20 (simplex). A k-simplex σ is the convex hull of a set P of k + 1 affinely independent points. In particular, a 0-simplex is a vertex, a 1-simplex is an edge, a 2-simplex is a triangle, and a 3-simplex is a tetrahedron. A k-simplex is said to have dimension k. A face of σ is a simplex that is the convex hull of a nonempty subset of P. Faces of σ come in all dimensions from zero (σ’s...

We study the homology of a filtered d-dimensional simplicial complex K as a single algebraic entity and establish a correspondence that provides a simple description over fields. Our analysis enables us to derive a natural algorithm for computing persistent homology over an arbitrary field in any dimension. Our study also implies the lack of a simple classification over non-fields. Instead, we ...

As a generalization of the use of graphs to describe pairwise interactions, simplicial complexes can be used to model higher-order interactions between three or more objects in complex systems. There has been a recent surge in activity for the development of data analysis methods applicable to simplicial complexes, including techniques based on computational topology, higher-order random proces...

Currently, the de facto representational choice for networks is graphs which capture pairwise relationships between entities. This dyadic approach fails to adequate capture the array of group relationships that are more than the sum of their parts and prevalent in real-world situations. For example, collaborative teams, wireless broadcast, and political coalitions all contain unique group dynam...

We show that monomial ideals generated in degree two satisfy a conjecture by Eisenbud, Green and Harris. In particular we give a partial answer to a conjecture of Kalai by proving that h-vectors of flag Cohen-Macaulay simplicial complexes are h-vectors of Cohen-Macaulay balanced simplicial complexes.

Complex networks can be used to represent complex systems which originate in the real world. Here we study a transformation of these complex networks into simplicial complexes, where cliques represent the simplices of the complex. We extend the concept of node centrality to that of simplicial centrality and study several mathematical properties of degree, closeness, betweenness, eigenvector, Ka...

In the spirit of topological entropy we introduce new complexity functions for general dynamical systems (namely groups and semigroups acting on closed manifolds) but with an emphasis on the dynamics induced on underlying simplicial complexes. For expansive systems remarkable properties are observed. Known examples are revisited and new examples are presented.

Given a graph G, the number of nowhere-zero Zq-flows φG(q) is known to be a polynomial in q. We extend the definition of nowhere-zero Zq-flows to simplicial complexes ∆ of dimension greater than one, and prove the polynomiality of the corresponding function φ∆(q) for certain q and certain subclasses of simplicial complexes. Résumé. Et́ant donné une graphe G, on est connu que le nombre de Zq-flot...

We consider a q-analogue of abstract simplicial complexes, called q-complexes, and discuss the notion shellability for such complexes. It is shown that q-complexes formed by independent subspaces q-matroid are shellable. Further, we explicitly determine homology corresponding to uniform q-matroids. also outline some partial results concerning determination arbitrary shellable q-complexes.