We introduce the tidy set, a minimal simplicial set that captures the topology of a simplicial complex. The tidy set is particularly effective for computing the homology of clique complexes. This family of complexes include the Vietoris-Rips complex and the weak witness complex, methods that are popular in topological data analysis. The key feature of our approach is that it skips constructing the clique complex. We give algorithms for constructing tidy sets, implement them, and present experiments. Our preliminary results show that tidy sets are orders of magnitude smaller than clique complexes, giving us a homology engine with small memory requirements.

Persistent homology captures the topology of a filtration - a one-parameter family of increasing spaces - in terms of a complete discrete invariant. This invariant is a multiset of intervals that denote the lifetimes of the topological entities within the filtration. In many applications of topology, we need to study a multifiltration: a family of spaces parameterized along multiple geometric dimensions. In this paper, we show that no similar complete discrete invariant exists for multidimensional persistence. Instead, we propose the rank invariant, a discrete invariant for the robust estimation of Betti numbers in a multifiltration, and prove its completeness in one dimension.

In this paper, we provide the theoretical foundation and an effective algorithm for localizing topological attributes such as tunnels and voids. Unlike previous work that focused on 2-manifolds with restricted geometry, our theory is general and localizes arbitrary-dimensional attributes in arbitrary spaces. We implement our algorithm and present experiments to validate our approach in practice.

MOTIVATION: Membrane fusion constitutes a key stage in cellular processes such as synaptic neurotransmission and infection by enveloped viruses. Current experimental assays for fusion have thus far been unable to resolve early fusion events in fine structural detail. We have previously used molecular dynamics simulations to develop mechanistic models of fusion by small lipid vesicles. Here, we introduce a novel structural measurement of vesicle topology and fusion geometry: persistent voids. RESULTS: Persistent voids calculations enable systematic measurement of structural changes in vesicle fusion by assessing fusion stalk widths. They also constitute a generally applicable technique for assessing lipid topological change. We use persistent voids to compute dynamic relationships between hemifusion neck widening and formation of a full fusion pore in our simulation data. We predict that a tightly coordinated process of hemifusion neck expansion and pore formation is responsible for the rapid vesicle fusion mechanism, while isolated enlargement of the hemifusion diaphragm leads to the formation of a metastable hemifused intermediate. These findings suggest that rapid fusion between small vesicles proceeds via a small hemifusion diaphragm rather than a fully expanded one. AVAILABILITY: Software available upon request pending public release. SUPPLEMENTARY INFORMATION: Supplementary data are available on Bioinformatics online.

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 give an algorithm for computing individual persistent homology groups over an arbitrary PIDs in any dimension.

We present algorithms for constructing a hierarchy of increasingly coa rse Morse complexes that decompose a piecewise linear 2-manifold. While Morse complexes are defined only in the smooth category, we extend the construction to the piecewise linear category by ensuring structural integrity and simulating differentiability. We then simplify Morse complexes by cancelling pairs of critical points in order of increasing persistence.

We present a fast implementation of a hybrid algorithm for reporting box and cube intersections.Our algorithm initially takes a divide-and-conquer approach and switches to simpler algorithms for low numbers of boxes.We use our implementations as engines to solve problems about geometric primitives.We look at two such problems in the category of quality analysis of surface triangulations.

We formalize a notion of topological simplification within the framework of a filtration, which is the history of a growing complex. We classify a topological change that happens during growth as either a feature or noise, depending on its life-time or persistence within the filtration. We give fast algorithms for completing persistence and experimental evidence for their speed and utility.