# Metric differentiation and embeddings of the Heisenberg group

##
Robert Young
(Courant Institute, New York)

created by pinamonti on 04 Mar 2021

modified on 09 Mar 2021

16 mar 2021
-- 16:00
[open in google calendar]

Zoom seminar

Please write to Andrea.pinamonti@unitn.it or to Andrea.marchese@unitn.it if you want to attend the seminar.

**Abstract.**

Pansu and Semmes used a version of Rademacher's differentiation theorem to show that there is no bilipschitz embedding from the Heisenberg groups into Euclidean space. More generally, the non-commutativity of the Heisenberg group makes it impossible to embed into any $L_p$ space for $p\in (1,\infty)$. Recently, with Assaf Naor, we proved sharp quantitative bounds on embeddings of the Heisenberg groups into $L_1$ and constructed a metric space based on the Heisenberg group which embeds into $L_1$ and $L_4$ but not in $L_2$; our construction is based on constructing a surface in $\mathbb{H}$ which is as bumpy as possible. In this talk, we will describe what are the best ways to embed the Heisenberg group into Banach spaces, why good embeddings of the Heisenberg group must be "bumpy" at many scales, and how to study embeddings into $L_1$ by studying surfaces in $\mathbb{H}$.