Explore a comprehensive lecture on the application of topological data analysis, particularly persistent homology, in neuroscience. Delve into the topological representation of branching trees and its successful use in classifying and clustering neurons and microglia. Discover recent findings on the topological representation of brain cells, with a focus on neurons and microglia. Learn about the innovative solution for solving the inverse TDA problem on neurons, understanding the process of moving from persistence barcodes back to the space of neuronal trees. Gain insights into the collaborative work between the Blue Brain Project and various researchers in this cutting-edge field of neuroscience and applied topology.