Robin Belton

The purpose of this article is to study directed collapsibility of directed Euclidean cubical complexes. One application of this is in the nontrivial task of verifying the execution of concurrent programs. The classical definition of collapsibility involves certain conditions on a pair of cubes of the complex. The direction of the space can be taken into account by requiring that the past links of vertices remain homotopy equivalent after collapsing. We call this type of collapse a link-preserving directed collapse. In this paper, we give combinatorially equivalent conditions for preserving the topology of the links, allowing for the implementation of an algorithm for collapsing a directed Euclidean cubical complex. Furthermore, we give conditions for when link-preserving directed collapses preserve the contractability and connectedness of directed path spaces, as well as examples when link-preserving directed collapses do not preserve the number of connected components of the path space between the minimum and a given vertex.

In the directed setting, the spaces of directed paths between fixed initial and terminal points are the defining feature for distinguishing different directed spaces. The simplest case is when the space of directed paths is homotopy equivalent to that of a single path; we call this the trivial space of directed paths. Directed spaces that are topologically trivial may have non-trivial spaces of directed paths, which means that information is lost when the direction of these topological spaces is ignored. We define a notion of directed collapsibility in the setting of a directed Euclidean cubical complex using the spaces of directed paths of the underlying directed topological space relative to an initial or a final vertex. In addition, we give sufficient conditions for a directed Euclidean cubical complex to have a contractible or a connected space of directed paths from a fixed initial vertex. We also give sufficient conditions for the path space between two vertices in a Euclidean cubical complex to be disconnected. Our results have applications to speeding up the verification process of concurrent programming and to understanding partial executions in concurrent programs.

Topological Data Analysis (TDA) studies the shape of data. A common topological descriptor is the persistence diagram, which encodes topological features in a topological space at different scales. Turner, Mukeherjee, and Boyer showed that one can reconstruct a simplicial complex embedded in R^3 using persistence diagrams generated from all possible height filtrations (an uncountably infinite number of directions). In this paper, we present an algorithm for reconstructing plane graphs K=(V,E) in R^2 , i.e., a planar graph with vertices in general position and a straight-line embedding, from a quadratic number height filtrations and their respective persistence diagrams.

Placental chorionic surface vascular networks (PCSVNs) are essential high-capacitance, low-resistance distribution and drainage networks, and are hence important to placental function and to fetal and newborn health. It was hypothesized that variations in the PCSVN structure may reflect both the overall effects of genetic and environmentally regulated variations in branching morphogenesis within the conceptus and the fetus’s vital organs. A critical step in PCSVN analysis is the extraction of blood vessel structure, which has only been done manually through a laborious process, making studies in large cohorts and applications in clinical settings nearly impossible. The large variation in the shape, color, and texture of the placenta presents significant challenges to both machine and human to accurately extract PCSVNs. To increase the visibility of the vessels, colored paint can be injected into the vascular networks of placentas, allowing PCSVNs to be manually traced with a high level of accuracy. This paper provides a proof-of-concept study to explain the geometric differences between manual tracings of paint-injected and un-manipulated PCSVNs under the framework of a shape-context model. Under this framework, paint-injected and un-manipulated tracings of PCSVNs can be matched with nearly 100% accuracy. The implication of our results is that the manual tracing protocol yields faithful PCSVN representations modulo a set of affine transformations, making manual tracing a reliable method for studying PCSVNs. Our work provides assurance to a new pre-processing approach for studying vascular networks by ways of dye-injection in medical imaging problems.