Conformal removability of Schramm-Loewner evolutions

Abstract:

 

A subset K of the complex plane is said to be conformally removable if each homeomorphism of the complex plane which is conformal on the complement of K is also conformal on K. The question of conformal removability of Schramm-Loewner evolutions (SLE) has been of considerable interest as it concerns uniqueness of weldings quantum surfaces. The conformal removability of SLE(kappa) for kappa < 4 was proved over 20 years ago, but the case of kappa in [4,8) has proved to be very elusive. In this talk, we will review the topic of conformal removability, its connections to SLE and present new results: that SLE(kappa) is indeed conformally removable for kappa = 4 and whenever its adjacency graph of complementary connected components is connected.