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.
[after Assaf Naor]
De Alexandros Eskenazis
De Galina Perelman
De Julien Berestycki
De Denis Roze
De Ignacio Madrid Canales
De Juan Camilo Arosemena Serrato
De Amine Marrakchi
De Francesca Arici
De Shirly Geffen
![[1188] Average distortion embeddings, nonlinear spectral gaps, and a metric John theorem](/media/cache/video_light/uploads/video/video-fa2a499707d16a256b7cc98b5b17ad71.jpg)
![[1189] Local marked length spectrum rigidity](/media/cache/video_light/uploads/video/video-68d9e8d3d9280fc670e7fd49c67acdd8.jpg)
![[1191] Finite time blow up for the compressible fluids and for the energy supercritical defocusing nonlinear Schrödinger equation](/media/cache/video_light/uploads/video/video-34432d88d14a9311adf5b81a3f21c83c.jpg)
