Volterra series are universal asymptotic expansions of solutions of well-posed evolution equations. These series have rich algebraic and combinatorial structures which we will focus on. The mentioned structures are all related to the combinatorial analysis of words, trees and permutations.

We will study free Lie algebras, Hall and Lindon words, shuffles, chronological algebras, descent algebras and related things, and we apply all these tools to the high-order variations of evolution equations. I am going to start from the very beginning and do not require any particular prerequisites.

This is a new course and I do not plan to follow any book. A useful source for classical algebraic aspects of the course is the book “Free Lie algebras”, by Christophe Reutenauer. Almost all other aspects can be found in the published papers, which I’ll list during the course.