#22: Robert Young - Metric differentiation and Lipschitz embeddings in Lp paces

Robert Young (NYU)

Abstract. Kadec and Pełczyński showed that if $1\le p\lt 2\lt q\lt \infty$ and $X$ is a Banach space that embeds into both $L_p$ and $L_q$, then $X$ is isomorphic to a Hilbert space. The search for metric analogues of such a result is intertwined with the Ribe program and metric theories of type and cotype. Recently, with Assaf Naor, we have constructed a metric space based on the Heisenberg group which embeds into $L_1$ and $L_4$ but not in $L_2$. In this talk, we will describe this example, explain why embeddings of the Heisenberg group into Banach spaces must be "bumpy" at many scales, and discuss how to bound the bumpiness of Lipschitz maps to Banach spaces.