Related papers: Quantitative Simplification of Filtered Simplicial…
We develop persistent homology in the setting of filtrations of (Cech) closure spaces. Examples of filtrations of closure spaces include metric spaces, weighted graphs, weighted directed graphs, and filtrations of topological spaces. We use…
We provide a simple characterization of simplicial complexes on few vertices that embed into the $d$-sphere. Namely, a simplicial complex on $d+3$ vertices embeds into the $d$-sphere if and only if its non-faces do not form an intersecting…
Cofaces -- simplices that contain a given simplex -- have multiple important uses in generating and using a Vietoris-Rips filtration: both in creating the coboundary matrix for computing persistent cohomology, and for generating the ordered…
Determining whether two graphs are isomorphic is a fundamental problem with practical applications in areas such as molecular chemistry or social network analysis, yet it remains a challenging task, with exact solutions often being…
We consider persistent homology obtained by applying homology to the open Rips filtration of a compact metric space $(X,d)$. We show that each decrease in zero-dimensional persistence and each increase in one-dimensional persistence is…
We combine Gromov's amenable localization technique with the Poincar\'{e} duality to study the traversally generic vector flows on smooth compact manifolds $X$ with boundary. Such flows generate well-understood stratifications of $X$ by the…
While sporadic examples of virtual resolutions with homology have been constructed, their occurrence is not well understood or controlled. Our results build a new set of tools for studying virtual resolutions of monomial ideals as arising…
We outline a novel clustering scheme for simplicial complexes that produces clusters of simplices in a way that is sensitive to the homology of the complex. The method is inspired by, and can be seen as a higher-dimensional version of,…
We present a parallelizable algorithm for computing the persistent homology of a filtered chain complex. Our approach differs from the commonly used reduction algorithm by first computing persistence pairs within local chunks, then…
Matrix reduction is the standard procedure for computing the persistent homology of a filtered simplicial complex with $m$ simplices. Its output is a particular decomposition of the total boundary matrix, from which the persistence diagrams…
We initiate the study of persistent homology of random geometric simplicial complexes. Our main interest is in maximally persistent cycles of degree-$k$ in persistent homology, for a either the \cech or the Vietoris--Rips filtration built…
We consider the topology of simplicial complexes with vertices the points of a random point process and faces determined by distance relationships between the vertices. In particular, we study the Betti numbers of these complexes as the…
While persistent homology has taken strides towards becoming a wide-spread tool for data analysis, multidimensional persistence has proven more difficult to apply. One reason is the serious drawback of no longer having a concise and…
A key point in the spin foam approach to quantum gravity is the implementation of simplicity constraints in the partition functions of the models. Here, we discuss the imposition of these constraints in a phase space setting corresponding…
We introduce a notion of discrete topological complexity in the setting of simplicial complexes, using only the combinatorial structure of the complex by means of the concept of contiguous simplicial maps. We study the links of this new…
In data clustering, it is often desirable to find not just a single partition into clusters but a sequence of partitions that describes the data at different scales (or levels of coarseness). A natural problem then is to analyse and compare…
Based on general and minimal properties of the {\it discrete} circuit complexity, we define the complexity in {\it continuous} systems in a geometrical way. We first show that the Finsler metric naturally emerges in the geometry of the…
A central problem in topological data analysis is that of computing the homology of a given simplicial complex. Said complexes can have arbitrary large number of simplices, as can happen, for example, if the space is the Rips-Vietoris or…
This paper proposes convolutional filtering for data whose structure can be modeled by a simplicial complex (SC). SCs are mathematical tools that not only capture pairwise relationships as graphs but account also for higher-order network…
There has been considerable recent interest, primarily motivated by problems in applied algebraic topology, in the homology of random simplicial complexes. We consider the scenario in which the vertices of the simplices are the points of a…