Alexandros Eskenazis — Average distortion embeddings, nonlinear spectral gaps, and a metric John theorem after Assaf Naor

In this lecture we shall discuss some geometric applications of the theory of nonlinear spectral gaps. Most notably, we will present a proof of a deep theorem of Naor asserting that for any norm \(\|\cdot\|\) on \(\mathbf{R}^d\), the metric space \((\mathbf{R}^d, \sqrt{\|x-y\|})\) embeds into Hilbert space with quadratic average distortion \(O(\sqrt{\log d})\). As a consequence, we will deduce that any n-vertex expander graph does not admit a \(O(1)\)-average distortion embedding into any \(n^{o(1)}\)-dimensional normed space.

NB: A youtube link is available on bourbaki.fr