Painlevé's problem and analytic capacity (Q854501)

This is a survey of some recent results in connection with the Painlevé's problem, the semiadditivity of analytic capacity and other related questions. Painlevé's problem consists in characterizing removable singularities of bounded analytic functions in a geometric/metric way. Equivalently, the problem is to characterize compact sets of zero analytic capacity. Recently, the author proved the semiadditivity of analytic capacity. This property, combined with other recent results, led to important progress in Painlevé's problem. The present paper contains a detailed review of these recent results. It also includes the proof of the semiadditivity of analytic capacity.