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Abstract

One approach to combining geometry, topology and statistics in the analysis of data consists of the following steps: (1) use the data to construct a geometric object; (2) apply topology to obtain a summary; and (3) apply statistics to the resulting summaries. From a statistical viewpoint, it is fruitful to replace the standard topological summary, the persistence diagram, with a vector (or better yet, a point in a Hilbert space). One such construction with particularly nice properties (e.g. reversability) is the persistence landscape. I will give an overview of this pipeline and apply it to analyze protein data and brain imaging data.