David--Semmes conjecture relates Singular Integrals with Geometric Measure Theory. We are in R^d.
If classical singular integrals (of singularity m) are becoming bounded operators after restriction to an m-dimensional set, does this imply that the set is necessarily ``smooth" (for example, is a subset of m-dimensional Lipschitz manifold)? Everybody believed that the answer is positive. It has been proved for only one case: d=2, m=1. This has been done in the combination of papers by Peter Jones, Pertti Mattila, Mark Melnikov, Joan Verdera, Guy David. However, if d>2 the method explored in these papers did not work, and this was a big roadblock in this part of Harmonic Analysis and Geometric Measure Theory. It still is for d>2, m< d-1. But for any dimension d, and m=d-1, Fedja Nazarov, Xavier Tolsa, and myself, we recently answered positively to this question of Guy David and Steven Semmes.