Malik Younsi (Hawaii)

Title: Holomorphic motions, conformal welding and capacity
Abstract: The notion of a holomorphic motion was introduced by Mane, Sad and Sullivan in the 1980's, motivated by the observation that Julia sets of rational maps often move holomorphically with holomorphic variations of the parameters. Even though the original motivation for their study came from complex dynamics, holomorphic motions have found over the years to be of fundamental importance in other related areas of Complex Analysis, such as the theory of Kleinian groups and Teichmuller theory for instance. Holomorphic motions also played a central role in the seminal work of Astala on distortion of dimension and area under quasiconformal mappings. In this talk, I will first review the basic notions and results related to holomorphic motions, including quasiconformal mappings and the (extended) lambda lemma. I will then present some recent results on the behavior of logarithmic capacity and analytic capacity under holomorphic motions. As we will see, conformal welding (of quasicircle Julia sets) plays a fundamental role. This is joint work with Tom Ransford and Wen-Hui Ai.

For zoom seminar log info please contact dmitry.jakobson [at] mcgill.ca