Discrete Morse theory is a versatile tool in combinatorial algebraic topology for the investigation of cell complexes of many different flavors. Persistent topology is the study of filtered spaces using topological techniques, which are ubiquitous in topological data analysis.

In this talk, I will give a brief introduction to peristent topology and motivate the subject with topological data analysis. After that I will present results from my PhD thesis: an investigation of the inverse problem between discrete Morse functions on graphs and their induced (generalized) merge trees, as well as a novel model for the moduli space of discrete Morse functions and its relationship to smooth Morse theory, discrete Morse matchings, merge trees, and barcodes. In these projects, I mostly use concepts from graph theory, poset theory, combinatorial topology, combinatorial geometry, differential topology, and differential geometry.

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