Explore geometric measure theory and its fundamental concept of n-rectifiable sets in this mathematics lecture that delves into the parametrization of sets using countably many Lipschitz images of n-dimensional Euclidean space. Learn how characterizations of rectifiable subsets in Euclidean space impact partial differential equations, harmonic analysis, and fractal geometry. Discover recent developments in non-Euclidean metric spaces, examining characterizations of rectifiable subsets through non-linear projections and tangent spaces. Gain essential background knowledge and understanding of geometric properties in non-smooth sets during this hour-long presentation from the Hausdorff Center for Mathematics.