Homology Inference for the Degree-Rips Bifiltration

Explore the density-sensitive degree-Rips bifiltration in this comprehensive lecture on applied algebraic topology. Delve into the motivation behind the "degree" aspect of the construction and examine a framework for studying homology inference questions. Analyze a practical example inspired by experiments from Blumberg and Lesnick's recent paper, focusing on point clouds concentrated on an annulus in the plane with and without outliers. Compare the homology of degree-Rips complexes for both point clouds and investigate why existing stability results for degree-Rips may not fully explain their relationship. Learn about the application of Adamaszek-Adams' computation of Vietoris-Rips complexes of the circle to determine the homotopy types of degree-Rips complexes for various metric probability spaces related to these experiments. Discover how a homology inference approach can provide powerful explanatory insights in this context, advancing your understanding of topological data analysis techniques.