Topological analysis of data on 911 calls in the Boston area

Among the newer approaches to data analysis are topological methods (TDA), which proved to be effective in analyzing data. In this thesis we analyze data on 911 calls that include a large number of calls. Firstly, we prepare data by grouping calls together using the Vietoris-Rips complex. We do this because it enables us to also analyze smaller areas and connect them. We analyze this complex in two ways: by using Morse theory and persistent homology. Morse theory is used to acquire critical simplices from the complex. They give us new information about the data. Using persistent homology, we produce persistent diagrams that illustrate how homology of a complex changes depending on a parameter. The initiative to use the TDA on such data came from the Department of Sociology at Harvard, where they had already tried to analyze this data by using various mechanical and mathematical models

Spatial analysis, decision support systems (DSS) and land use design: the case-study of antique viability system in San Martino valley (Lombardy, Italy)

This paper concerns the development of a Decision Support System (DSS), which is a system able to support temporal and spatial choices about land use design, in order to project and manage the antique viability system in San Martino valley (located in Lombardy, Italy) The main purpose is providing to a project manager necessary information to help him to understand problems (in particular concerning the spatial system of viability), therefore assists him to analyze the question from different points of view. This process needs a particular informative architecture, based on a complex and relational structured system (DSS) able to produce response for the whole decision process. The DSS is interfaced with a GIS in order to manage cartography and alphanumeric files with geo-referenced data. It works on information which are supposed to be indispensable for the planners of the San Martino valley.

Computing Topological Persistence for Simplicial Maps

Algorithms for persistent homology and zigzag persistent homology are well-studied for persistence modules where homomorphisms are induced by inclusion maps. In this paper, we propose a practical algorithm for computing persistence under $\mathbb{Z}_2$ coefficients for a sequence of general simplicial maps and show how these maps arise naturally in some applications of topological data analysis. First, we observe that it is not hard to simulate simplicial maps by inclusion maps but not necessarily in a monotone direction. This, combined with the known algorithms for zigzag persistence, provides an algorithm for computing the persistence induced by simplicial maps. Our main result is that the above simple minded approach can be improved for a sequence of simplicial maps given in a monotone direction. A simplicial map can be decomposed into a set of elementary inclusions and vertex collapses--two atomic operations that can be supported efficiently with the notion of simplex annotations for computing persistent homology. A consistent annotation through these atomic operations implies the maintenance of a consistent cohomology basis, hence a homology basis by duality. While the idea of maintaining a cohomology basis through an inclusion is not new, maintaining them through a vertex collapse is new, which constitutes an important atomic operation for simulating simplicial maps. Annotations support the vertex collapse in addition to the usual inclusion quite naturally. Finally, we exhibit an application of this new tool in which we approximate the persistence diagram of a filtration of Rips complexes where vertex collapses are used to tame the blow-up in size.Comment: This is the revised and full version of the paper that is going to appear in the Proceedings of 30th Annual Symposium on Computational Geometr

Identifiers for structural warnings of malfunction in power grid networks

Although its uninterrupted supply is essential for everyday life, the electricity occasionally experiences disruptions and outages. The work presented in the current paper aims to initiate the research to design a strategy based on advanced approaches of algebraic topology to prevent such malfunctions in a power grid network. Simplicial complexes are constructed to identify higher-order structures embedded in a network and, alongside a new algorithm for identifying delegates of the simplicial complex, are intended to pinpoint each element of the power grid network to its natural layer. Results of this methodology for analysis of a power grid network can single out its elements that are at risk to cause cascade problems which can result in unintentional islanding and blackouts. Further development of the outcomes of research can find implementation in the algorithms of the energy informatics research applications

Computing The Cubical Cohomology Ring (Extended Abstract)

The goal of this work is to establish a new algorithm for computing the cohomology ring of cubical complexes. The cubical structure enables an explicit recurrence formula for the cup product. We derive this formula and, next, show how to extend the Mrozek and Batko [7] homology coreduction algorithm to the cohomology ring structure. The implementation of the algorithm is a work in progress. This research is aimed at applications in electromagnetism and in image processing, among other fields