"We propose to study different questions in the area of the so called geometric analysis. Most of the topics we are interested in deal with the connection between the behavior of singular integrals and the geometry of sets and measures. The study of this connection has been shown to be extremely helpful in the solution of certain long standing problems in the last years, such as the solution of the Painlev\'e problem or the obtaining of the optimal distortion bounds for quasiconformal mappings by Astala.More specifically, we would like to study the relationship between the L^2 boundedness of singular integrals associated with Riesz and other related kernels, and rectifiability and other geometric notions. The so called David-Semmes problem is probably the main open problem in this area. Up to now, the techniques used to deal with this problem come from multiscale analysis and involve ideas from Littlewood-Paley theory and quantitative techniques of rectifiability. We propose to apply new ideas that combine variational arguments with other techniques which have connections with mass transportation. Further, we think that it is worth to explore in more detail the connection among mass transportation, singular integrals, and uniform rectifiability.We are also interested in the field of quasiconformal mappings. We plan to study a problem regarding the quasiconformal distortion of quasicircles. This problem consists in proving that the bounds obtained recently by S. Smirnov on the dimension of K-quasicircles are optimal. We want to apply techniques from quantitative geometric measure theory to deal with this question.Another question that we intend to explore lies in the interplay of harmonic analysis, geometric measure theory and partial differential equations. This concerns an old problem on the unique continuation of harmonic functions at the boundary open C^1 or Lipschitz domain. All the results known by now deal with smoother Dini domains."