New invariants in topological data analysis and their applications in material and biological sciences

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.