Monson, NathanielIn this dissertation, we present a novel stability result for the persistent homology of the Rips complex associated to a point cloud. Our theorem is narrower than the classic result of Cohen-Steiner, Edelsbrunner, and Harer in that it does not apply to Cech complexes, nor to functions which are not measuring distance to a point cloud. It is broader than the classic result in that it is “local”; if a function approximately preserves distances in some range, but is contractionary below or expansionary above that range, our result still applies. The novel stability result is paired with the Johnson-Lindenstrauss Lemma to show that, with high probability, random projection approximately preserves persistent homology. An experimental analysis is given of the computational speedup granted by this dimension reduction. This is followed by some observations suggesting that even when the theoretical bound is loose enough that we have no guarantee of homology preservation, thereis still a high chance that significant features of the dataset are preserved.enDissertationMathematicsDimension ReductionPersistent HomologyTopology