Wolff Memorial Lectures

There are many situations in which understanding distortion growth is deeply related to a range of isoperimetric questions, from continuous settings to discrete and algorithmic issues. We will see how the Sparsest Cut problem, which is a central open question in approximation algorithms, relates to distortion growth. We will describe how this link led over 3 decades of intensive research to a resolution of a major question about the performance of a well-studied algorithm for Sparsest Cut, with the final step occurring a few months ago. During those decades of work, this endeavor featured multiple twists and turns that benefited both computer science and pure mathematics, exhibiting deep interactions with areas such as geometric measure theory, harmonic analysis, probability, combinatorics, group theory, functional analysis, complexity theory and algorithm design. We will explain these developments and their ramifications, including the recently demonstrated extremal property of the observable diameter of the Euclidean sphere.