Explore computational topology challenges and solutions in this lecture on parameterized Vietoris-Rips filtrations using covers. Delve into methods for handling large filtered geometric complexes built from point cloud data, including parallel computation and compression techniques. Learn about the extension of acyclic carriers to persistent homology, providing interleavings between restricted and full Vietoris-Rips filtration constructions. Discover how these filtrations can be applied to study data over a base space and guide cover selection. Cover key topics such as cover complexes, persistent homology, interleavings, the Cyclic Carrier Theorem, cover homology, and nested landmarking. Gain insights into practical applications and engage with a Q&A session addressing the concept of R in this context.