Explore a 58-minute lecture on self-similar solutions to extension and approximation problems presented by Robert Young from NYU at the Institut des Hautes Etudes Scientifiques (IHES). Delve into higher-dimensional generalizations of Kaufman's 1979 construction of a surjective Lipschitz map from a cube to a square with rank 1 derivative almost everywhere. Examine Lipschitz and Hölder maps with intriguing properties, including topologically nontrivial maps from S^m to S^n with derivative of rank n-1, 2/3-ε-Hölder approximations of surfaces in the Heisenberg group, and Hölder maps from disc to disc preserving signed area while approximating arbitrary continuous maps. Cover topics such as self-similar maps, Kaufman's construction, topological obstructions, map construction techniques, homotopic map construction, maps with signed area 0, limit maps, uniform continuity, Oberman's Theorem, and density results. Conclude with a Q&A session to further explore these complex mathematical concepts. Read more