Senior Honors Thesis Presentation: Methods in Topological Data Analysis

Abstract: We introduce core methods in topological data analysis and show how topology can be used to understand high dimensional data. The main tool is persistent homology, which tracks when topological features appear and disappear across a filtration and summarizes this information using persistence diagrams and barcodes. We develop the theory behind persistent homology and introduce distances such as the bottleneck and Wasserstein metrics to compare datasets through their topological signatures. These ideas are applied to image analysis by studying the topology of painting styles. We then present the Mapper algorithm, which builds graph representations of data using a metric and a filter function. Finally, we explore an application to neuroscience, where Temporal Mapper is used to analyze time series data and recover transition networks that reflect underlying dynamical structure.

Faculty Advisor: Aliakbar Daemi