Composition of Permutons

Permutons are objects from probabilistic combinatorics that can be thought of as "continuum limits of permutations." Just as permutations are equipped with a natural notion of composition, so too are permutons. I'll discuss some properties of this composition operation, some limitations, and what it can tell us about compositions of random permutations. We will find that taking a functional-analytic perspective allows us to prove some surprising results in a rather "soft" way.