Wolff Memorial Lectures

It is typically impossible to embed with finite distortion an infinite metric space X of interest, such as most finitely generated groups equipped with their word metric, into a "nice" metric space Y, such as a Hilbert space. In those common situations, one of the most pertinent and useful questions becomes determining the asymptotic growth rate of the smallest D=D(n) such that every n-point subset of X embeds into Y with distortion D. This talk will explain the geometric and analytic challenges that this question leads to, its rich history, including both major past achievements and very recent progress, as well as longstanding mysteries that remain elusive despite investigations over many decades. No background will be assumed beyond undergraduate mathematics.