Caltech/UCLA Joint Analysis Seminar

Carpets are metric spaces that are homeomorphic to the standard Sierpinski carpet. By a theorem of Whyburn, they have a natural topological characterization. Consequently, they arise in many contexts involving dynamics, self similarity or geometric group theory. In these contexts, the Loewner property would have many structural and geometric implications for the space. However, unfortunately, we do not know if the Loewner property would be satisfied, or even could be satisfied, in many of the applications of interest. I will discuss recent results on finding large families of Loewner carpets, and some properties they must enjoy, such as explicit and inexplicit uniformizations in the plane.