Department of

Mathematics


Seminar Calendar
for events the day of Thursday, February 8, 2018.

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Thursday, February 8, 2018

2:00 pm in 243 Altgeld Hall,Thursday, February 8, 2018

(Non)-Removability of the Sierpiński Gasket

Dimitrios Ntalampekos (UCLA)

Abstract: 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 Sierpiński gasket. A first result is that the Sierpiński 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 Sierpiński gasket, such as Apollonian gaskets and generalized Sierpiński gasket Julia sets. Then, I will sketch a proof that the Sierpiński 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.