Abstract
In this talk we will begin by introducing the notion of analytic capacity as well as some of its essential properties. Using this concept, we identify the family of removable subsets of the complex plane, which are those such that, for any bounded holomorphic function defined on their complement, it can be analytically extended to the whole plane. From there, we discuss a possible geometric characterization for removable subsets, a question popularly known as the Painlevé problem. The previous study is carried out in terms of the Hausdorff dimension of these subsets, obtaining a complete classification for values different from 1. The remaining case, associated with the critical dimension of analytic capacity, must be treated separately. It is at this point that we invoke the theory of singular integrals in order to study a subfamily of these subsets: those contained in graphs of Lipschitz functions. We will conclude by giving a brief overview of the treatment of this case, introduced by Arnaud Denjoy at the beginning of the 20th century.