The first step in topological data analysis is often the construction of a simplicial complex. This complex approximates the lost topology of a sampled point set. Current techniques often assume that the input is embedded in a metric – often Euclidean – space, and make significant use of the underlying geometry for efficient computation. Consequently, these techniques do not extend to non-Eucli...

Recently an alternate technique for numerical quantum gravity, dynamical triangulation, has been developed. In this method, the geometry is varied by adding and subtracting equilateral simplices from the simplicial complex. This method overcomes certain difficulties associated with the traditional approach in Regge calculus of varying geometry by varying edge lengths. However additional complic...

In this work we study the framework of mathematical morphology on simplicial complex spaces. Simplicial complexes are widely used to represent multidimensional data, such as meshes, that are two dimensional complexes, or graphs, that can be interpreted as one dimensional complexes. Mathematical morphology is one of the most powerful frameworks for image processing, including the processing of d...

The special properties of planar posets have been studied, particularly in the 1970's by I. Rival and others. More recently, the connection between posets, their corresponding polynomial rings and corresponding simplicial complexes has been studied by R. Stanley and others. This paper, using work of A. Bjorner, provides a connection between the two bodies of work, by characterizing when planar...

It is shown that every shifted simplicial complex ∆ is EKR of type (r, s), provided that the size of every facet of ∆ is at least (2s + 1)r − s. It is moreover proven that every i-near-cone simplicial complex is EKR of type (r, i) if depthK∆ > (2i + 1)r − i − 1, for some field K. Furthermore, we prove that if G is a graph having at least (2i + 1)r − i connected components, including i isolated ...

We consider a Hopf algebra of simplicial complexes and provide a cancellation-free formula for its antipode. We then obtain a family of combinatorial Hopf algebras by defining a family of characters on this Hopf algebra. The characters of these combinatorial Hopf algebras give rise to symmetric functions that encode information about colorings of simplicial complexes and their f -vectors. We al...

The usual definition of finite simplicial complex is a set of non-empty subsets of a finite set, closed under non-empty subset formation. For our purposes here, we will omit the non-emptiness and define a finite simplicial complex to be a down-closed subset of the set of subsets of a finite set. We can, and will suppose that the finite set is the integers 0,. . . ,N . We will denote by K the se...

In this paper we study a class of groups which may be described as groups of piecewise linear bijections of a circle or of compact intervals of the real line. We use the action of these groups on simplicial complexes to obtain homological and combinatorial information about them. We also identify large simple subgroups in all of them, providing examples of finitely presented infinite simple gro...