Abstract: Carpets are a natural class of metric spaces that has been the focus of much research recently. Here, I present recent results on quasisymmetrically embeddability of large classes of carpets; and other results showing that one can often not get bi-Lipschitz embeddings. In particular, this latter component, when specialized to the case of slit carpets, that were studied by Merenkov and others, leads to a resolution of an open question from the influential list of problems by Heinonen and Semmes. Some of this is joint work with Jeff Cheeger and Guy C. David, and some based on prior results by Bruce Kleiner.
摘要：地毯是一类近来常常研究的度量空间。在这个报告中，我讲展示一大类地毯的拟对称嵌入结果，以及其他一些不可双Lipschitz映射的嵌入结论。尤其是第二部分，主要讨论由Merenkov以及其他研究者研究的开缝地毯，这导致Heinonen与Semmes的一个问题的解决，这项工作是我与Jeff Cheeger, Guy David合作的成功，基于Bruce Kleiner之前的一些结果。