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.