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Explore the homotopy types of Vietoris-Rips complexes in metric gluings and wedge sums in this comprehensive lecture. Delve into the study of Vietoris-Rips complexes of metric wedge sums and metric gluings, discovering how the complex of a wedge sum is homotopy equivalent to the wedge sum of the individual complexes. Examine generalizations for gluing two metric spaces along a common isometric subset, with a focus on metric graphs joined along a sufficiently short path. Gain insights into describing persistent homology across all homological dimensions for a class of metric graphs. Conclude by discussing open research directions in metric gluings and metric graphs, providing a foundation for further exploration in applied algebraic topology.
