Average distortion embeddings, nonlinear spectral gaps, and a metric John theorem (after Assaf Naor)

 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.