Explore the concept of geodesic complexity in metric spaces and its relationship to the topological robot motion planning problem in this 55-minute lecture. Delve into the structure of cut loci in closed Riemannian manifolds and learn how it influences geodesic complexity. Discover the novel concept of fibered decomposition of the total cut locus and understand its role in establishing lower and upper bounds on geodesic complexity. Examine practical applications of these theories in projective spaces and lens spaces, based on joint research with Stephan Mescher.