What are the removable singularities of harmonic functions with bounded gradient?  This problem, that takes its origins in certain problems of complex analysis, which are 140 years old, was solved relatively recently. Its solution is based on extension to a new territory of classical theory of singular integrals.

Singular integrals are ubiquitous objects and play an important part in Geometric Measure Theory. The simplest ones are called Calderon–Zygmund operators. Their theory was completed in the 50′s by Zygmund and Calderon. Or it seemed like that. The last 25 years saw the need to consider CZ operators in very bad environment, so kernels are still very good, but the ambient set/measure has no regularity whatsoever.

Initially such situations appeared from the wish to solve some outstanding problems in complex analysis: such as problems of Painleve, Ahlfors, Denjoy and Vitushkin.

The analysis of CZ operators on very bad sets  is very fruitful in the part of Geometric Measure Theory that deals with removability mentioned above and rectifiability. It can be viewed as the study of very low regularity free boundary problems.  We will explain the genesis of ideas that led to several  long and difficult proves that culminated in the solutions of problems of Denjoy, Vitushkin, David-Semmes, and Bishop, and brought also the solution by Tolsa of Painleve’s problem.