Rectifiability, Menger Curvature, Singular Integrals

Rectifiability is one of the basic concepts of geometric measure theory and singular integrals of harmonic analysis. One of the themes of the course is that natural singular integral operators behave well on m-dimensional rectifiable subsets of n-dimensional space, and conversely such a good behaviour leads to rectifiability. A particular emphasis will be on the Cauchy kernel 1/z in the complex plane. In that case Menger curvature of triples of points provides a powerful tool via a remarkable identity of Melnikov. Consequences of this to analytic capacity and removable sets of bounded analytic functions will also be discussed.