Topological Data Analysis: Functional Summaries and Locating Cosmic Voids and Filament Loops

Jessi Cisewski Kehe

Data exhibiting complicated spatial structures are common in many areas of science (e.g. cosmology, biology), but can be difficult to analyze. Persistent homology is a popular approach within the area of Topological Data Analysis (TDA) that offers a way to represent, visualize, and interpret complex data by extracting topological features, which can be used to infer properties of the underlying structures. For example, TDA may be useful for analyzing the large-scale structure (LSS) of the Universe, which is an intricate and spatially complex web of matter. The output from persistent homology, called persistence diagrams, summarizes the different ordered holes in the data (e.g. connected components, loops, voids). I will introduce persistent homology, present functional transformations of persistence diagrams useful for inference, and discuss how it can be used to locate cosmological voids and filament loops in the LSS of the Universe.

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