Wolff Memorial Lectures

This self-contained talk will focus on a concept that has a central role in the proofs of results that were described in the previous two talks, including the most recent progress, though it could be understood and appreciated on its own without knowledge of the content of the previous two talks. A random zero set is a distribution over random subsets of a given metric space that has the following paradoxical-sounding property: with a fixed definite positive probability, for every two points in the metric space, one of those points belongs to the random subset and the other one is far from the random subset. We will explain how such objects occur, how they can be used and what they are good for, and discuss how they can be constructed, thus leading to fine structural information about a large and useful class of metric spaces.