Hodge non-seminar, University of Edinburgh, 16 April 2020
Abstract Cayley's theorem counts the number of trees on a given vertex-set. A couple of things make it more than just another counting formula. First, André Joyal gave a really beautiful bijective proof of it (one of the proofs from the "Proofs from the Book" book), which I'll recount. Second, Cayley's theorem and Joyal's proof can be linearized, resulting in a simple new proof of an old theorem on the number of nilpotent operators on a finite vector space.
Slides In this pdf file.
Related resources Paper and various blog posts: 1, 2, 3.