| Summary: | In this thesis we explore a variety of topics in the intersection of Topology and Statistics. This includes some object oriented data analysis of topological summary statistics, a method of performing shape statistics using topological summaries, and proving a reconstruction theorem for point cloud samples of compact sets.
A common theme in Applied Topology is the use of persistent homology and persistence diagrams as topological summaries to capture the "shape" of data. These topological summaries are inherently stochastic as they are constructed from data that is stochastic. We want to be able to analyze distributions of persistence diagrams. To this end, we prove results about the geometry of the space of persistence diagrams under a variety of metrics that are analogous to Lp metrics. We characterize the Frechet mean, and the analogously defined median, of finite populations of persistence diagrams.
We develop an application of Applied Topology too morphology and shape statistics more generally with the persistent homology transform. This uses a collection of persistence diagrams to describe simplicial complexes embedded in R3 or R2. This persistent homology transform of an object is stable under small perturbation of the shape and the space of persistent homology transforms inherits a metric from the space of persistence diagrams. We explore the analysis of simulated and real data sets using the persistent homology transform.
Another topic in this thesis considers the reconstruction of a compact subset K from a finite point cloud S that is Hausdorff close to it. We find sufficient condition in terms of the geometry of K for when the union of balls centered at the points in S deformation retracts to K..
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