Mathematical Conversations

In 1979, Kaufman constructed a remarkable surjective Lipschitz map from a cube to a square whose derivative has rank $1$ almost everywhere. In this talk, we will present some higher-dimensional generalizations of Kaufman's construction that lead to maps with low-rank derivatives and strange properties, including topologically nontrivial maps from $S^m$ to $S^n$ with derivative of rank $n-1$ and maps from the disc to the disc that send every closed curve to a curve with signed area zero. This is joint work with Stefan Wenger and Larry Guth.