Speaker: Kai Rajala, University of Jyväskylä

Title: Rigidity of circle domains and conformal removability

Abstract: The long-standing Koebe conjecture asserts that every subdomain of the complex plane admits a conformal map onto a circle domain—a domain with boundary components that are circles and points. A circle domain is called rigid if every conformal map onto another circle domain is a Möbius transformation. While there exist non-rigid circle domains, meaning the map in Koebe's conjecture is not always unique, He and Schramm proved that both Koebe's conjecture and rigidity hold for countably connected domains. They further conjectured that rigidity is equivalent to the conformal removability of the boundary. We discuss the basic properties of conformally removable sets and present a negative answer to the rigidity conjecture.