ARI, GARI, Zig and Zag: An introduction to Ecalle's theory of multiple zeta values

Abstract

This text has two goals. The first is to give an introduction to Ecalle's work on mould theory, multiple zeta values and double shuffle theory and relate this work explicitly to the classical theory of multiple zeta values and double shuffle expressed in the usual terms of non-commutative variables. The second is to provide complete proofs of those of his main results and identities which are strictly useful in the context of (non-colored) multiple zeta values. Many of these proofs are difficult, laborious and not enlightening and have been relegated to appendices. The emphasis in the text is to provide an easily approachable introduction to Ecalle's language while placing it almost from the start in the context of multiple zeta value theory. Disclaimer: This text is not final and is not submitted for publication. The intention is to continue to add to and complete it over time.