Philémon Bordereau: Uniformization of CLE

Abstract: Koebe’s conjecture asks whether every domain of the complex plane, with possibly infinitely many holes, is conformally equivalent to a domain whose boundary components are disks. We study a variant of this question in the context of the \(CLE_\kappa\) carpet (\(8/3 < \kappa \leq 4\)), a random Sierpiński carpet corresponding to the complement of a countable collection of disjoint Jordan loops in the unit disk.