Institut für Algebra, Geometrie, Topologie und deren Anwendungen (ALTA)
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Item-typ:Veröffentlichung, New invariants in topological data analysis and their applications in material and biological sciences(2025-10-29); ; Topological data analysis has proven to be a versatile and powerful enrichment for data analysis of diverse selections of data in all types of quantitative analyses. Strategies include incorporating descriptors and characterizing topological properties of datasets ranging from scalar fields, point clouds, graphs, or any other scientific dataset. Furthermore, more data-oriented combinatorial constructions can be used from which the topological properties are taken. In the following thesis, the focus is on three applications: A new combinatorial structure for describing phylogenetic networks as well as general filtered spaces, surface classification via roughness computations and its comparison to topological invariants on its scalar field, and an application for the spatial analysis of microscopic images. More specifically, the cliquegram and facegram models are established for phylogenetic models and their theoretical properties investigated, as well as their computational complexity, and algorithms for their efficient computation proposed. In addition, an approach to classify surfaces based on their roughness is presented and persistent homology is employed to extract multiscale topological features from surface data and integrate these features into machine learning models. Finally, embryonic and neural stem cells are distinguished after protein staining results in distinct spatial structures of their 3D microscopic images which are captured using cubical persistent homology and analyzed. These different approaches demonstrate the benefits of combining new topological and combinatorial methods with existing ones, providing a more comprehensive toolkit for complex network analysis. Code for reproducing and reusing the techniques is provided.Dissertation41 47 - Some of the metrics are blocked by yourconsent settings
Item-typ:Veröffentlichung, On different witness complex filtrations and their landmark choices(2025-06-16); ; ; The main aim of this thesis is to study different Witness Complex filtrations. By the notion of Witness Complex we mean filtrations of simplicial complexes characterized by the fact that the data set they are based on is not automatically included in the complex but rather landmark points are chosen and the rest of the points is used for additional information on the higher dimensional simplices between them. We consider two constructions that are commonly referred to in literature simply by the term Witness Complex, two constructions based on the Cech and Vietoris-Rips Complexes as well as a possible construction for a Witness Complex bifiltration. First, we compare them to one another and to other known filtrations, look at methods of choosing the landmark points and search for approximations to the Vietoris-Rips Complex before introducing multiparameter filtrations. One of the main parts of this work is after that to construct a different method of acquiring landmark points that is robust with respect to outliers in the data set. This method is derived from the DBSCAN algorithm and with it we can compare one of the Witness Complex constructions to the Degree-Rips filtration which is a density-sensitive bifiltration of the classical Vietoris-Rips Complex. Another goal of this work is then to expand on some existing interleaving results for Witness Complexes to all of the different introduced constructions and especially to expand them before finally stating some additional stability results for all of the constructions which allow both the landmark points as well as the data points to shift to a certain extent.Dissertation67 331