Metric differentiation and embeddings of the Heisenberg group

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 $\mathbb{H}$

differential geometrygeneral mathematicsmetric geometryoptimization and controlspectral theory

Audience: researchers in the topic

( slides | video )

Comments: VIRTUAL SESSION

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Sub-Riemannian Seminars

Series comments: The "Sub-Riemannian seminars" are the union of the "Séminaire de géométrie et analyse sous-riemannienne" (held in Paris since 2011) and the "International Sub-Riemannian Seminars", which were born in spring 2020 as a reaction to the COVID-19 pandemic.

The new format will gather every 3 weeks on average, alternating between these types of sessions:

- physical session in Paris (Laboratoire Jacques-Louis Lions), also transmitted online on Zoom.

- fully online session on Zoom.

- special session hosted physically somewhere else, and transmitted online.

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