Explore a comprehensive lecture on topological data analysis, focusing on the inverse problem of recovering input data from barcodes to merge trees. Delve into the connection between barcodes and symmetric groups, and discover how this relationship can be applied to study neuron distributions modeled as trees. Learn about a novel approach to coordinatize the barcode space, opening avenues for statistical and probabilistic analysis through geometric group theory. Examine the characterization of the persistence map between merge trees and barcodes using monotone maps between subset and partition lattices. Gain insights into ongoing research and potential future applications in this field of applied algebraic topology.