Explore the intricacies of sampling smooth manifolds using ellipsoids in this 46-minute lecture from the Applied Algebraic Topology Network. Delve into a common data science problem of determining space properties from samples, focusing on the homotopy equivalence between subspaces of Euclidean space and unions of balls around sample points. Examine the seminal work by Niyogi, Smale, and Weinberger on closed smooth submanifolds, and discover how using ellipsoids instead of balls can significantly reduce the required sample density for manifold approximation. Learn about new techniques developed to prove that unions of suitably sized ellipsoids can deformation retract to the manifold, overcoming challenges not present in ball-based approaches. Cover topics such as tangent space approximation, normal deformation retraction, and potential improvements to the method, concluding with a discussion on apparently optimal shapes for manifold sampling.