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Stability of Universal Constructions for Persistent Homology
thesisposted on 01.08.2019 by Janis Lazovskis
In order to distinguish essays and pre-prints from academic theses, we have a separate category. These are often much longer text based documents than a paper.
In this thesis I describe poset stratifications of the product of the Ran space and the nonnegative real numbers, as a universal space for the Cech construction of simplicial complexes. This leads to a cosheaf valued in diagrams of simplicial complexes for which every restriction to a finite collection recovers the persistent homology of the collection. For the stratification, I describe a partial order on isomorphism classes of abstract simplicial complexes, which allows spaces stratified by them to have entrance paths uniquely interpreted as simplicial maps. Decomposing entrance paths gives a sheaf structure, which has higher information that the cosheaf does not capture.