Mathematical Sciences: Problems in Complex Analysis

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This project is concerned with several problems in complex variables that are of interest in their own right and in connection with problems in the geometry of planar sets. The first set of problems deals with the iteration theory of meromorphic functions in the plane. The main question proposed for study here asks if the Julia set of any rational function and of a suitable entire or meromorphic function is uniformly perfect. This is tantamount to asking if the Julia set has a uniform lower bound for the local density of its logarithmic capacity. The second set of problems deals with the question of under what circumstances a convergence group in the plane is topologically conjugate to a Moebius group. The third set of problems involves the study of analytic inequalities that arise in the theory of Markov processes.