Explore a novel topological data analysis tool for nonlinear dimensionality reduction in this 35-minute conference talk. Delve into the TALLEM (Topological Assembly of Locally Linear Euclidean Models) method, which combines local Euclidean coordinates with fiber bundle theory to create globally consistent maps that respect data topology. Learn about the algorithm's context within fiber bundles, theoretical impacts of local alignment quality, and comparisons with other popular dimensionality reduction techniques. Discover the benefits of fiber bundle-based decomposition and discuss potential applications and challenges of non-Euclidean local coordinates. Gain insights into topics such as principal bundles, correspondences, topological assembly, Euclidean assembly, and stability for TALLEM through practical examples like the Brey Faces dataset and topological assembly of circles.