Explore the collapse of Riemannian manifolds and metric spaces to lower-dimensional spaces in this 50-minute lecture by Mikhail Katz. Delve into the specific case of surfaces collapsing to circles or segments, utilizing Grove–Shiohama critical points theory, basic intersection theory, and the Toponogov comparison theorem. Discover how these concepts give meaning to the technique of 'discrete integration against the Euler characteristic' by examining the well-defined homotopy type of 'fibers' of M over X. Cover topics such as critical points, three scales, Borsa coulomb theorem, degree, taurus, Jung's Theorem, Gauss-Mannet Theorem, and open questions in the field of applied algebraic topology.