Title : Removability in Conformal Welding and Koebe’s Uniformization Conjecture
Ever since the seminal work of Ahlfors and Beurling in the middle of the 20th century, the study of removable plane sets with respect to various classes of analytic functions has proven over the years to be of fundamental importance to a wide variety of problems in complex analysis and geometric function theory. Questions revolving around necessary and sufficient geometric conditions for removability have held a prominent role in the development of valuable techniques, leading to deep results in various fields of mathematical analysis.
In recent years, attention has been drawn to the more modern notion of conformal removability, which continues to reveal connections with an ever-growing variety of central problems in complex analysis and related fields. Striking examples include injectivity of conformal welding, as well as the observation by He and Schramm in the 1990′s of the close relationship between conformal removability and Koebe’s uniformization conjecture.
The first part of the talk will consist of a brief introduction to conformal welding. I will discuss how removability appears naturally in the study of the injectivity of the welding correspondence.
In the second part of the talk, I will present new results on the conformal rigidity of circle domains and uniqueness in Koebe’s conjecture, following the work of He and Schramm.