Title: Almost additivity of analytic capacity and boundedness of Cauchy operators
Abstract: The notion of analytic capacity was introduced in 1947 by Ahlfors in connection with the celebrated Painlev\'e problem: describe in metric/geometric terms the sets of removable singularities for bounded analytic functions. We give a shot survey of results and methods of this theory. They allow us to prove that under certain assumptions the analytic capacity of a union of sets is comparable with the sum of their analytic capacities. As an application we consider the problem of boundedness of the Cauchy operator generated by a union of measures, if the Cauchy operators of these measures are uniformly bounded. The results obtained jointly with Alexander Reznikov and Alexander Volberg.