Complex Analysis, Dynamics and Geometry Seminar

Removability of sets for quasiconformal maps and Sobolev functions has applications in Complex Dynamics, in Conformal Welding, and in other problems that require "gluing" of functions to obtain a new function of the same class. We, therefore, seek geometric conditions on sets which guarantee their removability. In this talk, I will discuss some very recent results on the (non)-removability of the Sierpinski gasket. A first result is that the Sierpinski gasket is removable for continuous functions of the class W^{1,p} for p > 2. The method used applies to more general fractals that resemble the Sierpinski gasket, such as Apollonian gaskets and generalized Sierpinski gasket Julia sets. Then, I will sketch a proof that the Sierpinski gasket is non-removable for quasiconformal maps and thus for W^{1,p} functions, for 1 leq p leq 2. The argument involves the construction of a non-Euclidean sphere, and then the use of the Bonk-Kleiner theorem to embed it quasisymmetrically to the plane. Speaker(s): Dimitrios Ntalampekos (UCLA)