It is commonplace today for scientists to study networks where interactions among the participants happen at higher orders, where any number of participants can be related at once. Graphs, the typical mathematical model for networks, can only display pairwise relationships, and are sometimes inadequate in these situations. The combinatorial structures that have been used to model higher order networks and generalize graphs, and which will play a major role in this thesis, are hypergraphs and simplicial complexes. Hypergraphs have grown in popularity over the past decades, and are becoming standard combinatorial objects used in network representations. They differ from graphs only in that an edge in a hypergraph can have any number of vertices, not just two.Simplicial complexes, another generalization of graphs, have become ubiquitous in algebraic topology. In part, this is because the homology of simplicial complexes is standard. Homology is a tool for studying properties of the shape of topological spaces, and has proven to be useful in topology for classifying spaces as it is a topological invariant. In data science, researchers study the homology of simplicial complexes that change with the data. Keeping track of the simplicial complexes over time allows them to ascertain when changes in the homology are caused by meaningful changes in the data.Hypergraphs are also generalizations of simplicial complexes, however, the homology of hypergraphs is currently a problematic gap in the theory. No universal theory of homology for hypergraphs has been established, and the definition of homology for simplicial complexes does not extend obviously. Nonetheless, there has been research done into the homology of hypergraphs. The two homology theories for hypergraphs studied in this thesis are called the restricted barycentric homology and the relative barycentric homology. We present novel combinatorial definitions, topological and classification results, and methods for computation. This thesis aids in the development of theory, interpretability, and data science of topological hypergraph analytics.
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