Lusin and Whitney theorems in Carnot groups: when geometry meets real analysis

Whitney extension results characterize when one can extend a mapping from a compact subset to a smooth mapping on a larger space. Lusin approximation results give conditions under which one can approximate a rough map by a smoother map after discarding a set of small measure. We first recall relevant results in the Euclidean setting, then describe recent work extending them to horizontal curves in the Heisenberg group.