The development of Schramm-Loewner Evolutions, and other related objects, have lead to an explosion in the understanding of various conformally invariant scaling limits of discrete models from statistical physics. Of particular interest for this talk is the development of the conformal loop ensembles (CLE) gaskets which are designed to serve as the scaling limits of the outer cluster boundaries of various spin models. In parallel, there has been much development in the theory of quasi-conformal unifomization, most notably for this talk recent work by Mario Bonk on the quasisymmetric uniformization of carpets, provided that the carpets possess a certain degree of uniform geometric control. In this talk, I will discuss a current work-in-progress with Steffen Rohde investigating such uniformization type results for the CLE carpet. In a suitable sense, I will show that one can still gain some understanding of conformal uniformization of CLE, despite the lack of uniform control in the random setting.
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