Geometry and Topology at Many Scales

Explore a lecture by Robert Young from NYU on multiscale geometry and topology, examining Kaufman's 1979 construction of a surjective Lipschitz map from a cube to a square with almost everywhere rank-1 derivative. Learn about the fascinating multiscale structure of this map, which builds complexity at progressively smaller scales. Discover key concepts in multiscale and nonsmooth geometry, including the limits of constructions that perturb maps or surfaces across multiple scales. The talk covers applications such as bounding the complexity of maps and surfaces, Heisenberg group geometry, and topologically nontrivial maps from S^m to S^n with derivative of rank n−1.