Ebrahimi-Fard K., Fauvet F. (eds.) Faà di Bruno Hopf algebras, Dyson-Schwinger equations, and Lie-Butcher series

Strasbourg: European Mathematical Society, 2015. — 468 p.Since the seminal works of Gian-Carlo Rota and his school in the 1960s, combinatorics has become a respected branch of modern mathematics. Today it appears at the intersection of a wide variety of mathematical sciences. In fact, techniques and methods from combinatorics have become indispensable in such diverse areas as, for instance, statistical physics, number theory, and probability theory. The work of Rota and his collaborators had a particularly profound impact on the development of algebraic combinatorics. As a result research on ltered bialgebras and Hopf algebras has seen a formidable expansion over the last two decades. Notably, some speci c algebras, such as Faà di Bruno Hopf algebra, Rota– Baxter algebra, Grossman–Larson algebra, Malvenuto–Reutenauer Hopf algebra, and some pre-Lie algebras, among others, have appeared in different guises and in various and often seemingly unrelated domains. Such include renormalization in perturbative quantum eld theory, Terry Lyons’ rough path theory, dynamical systems, control theory and the analysis of numerical ows on manifolds.Forewordy
Pre-Lie algebras and systems of Dyson–Schwinger equations
Five interpretations of Faà di Bruno’s formula
A Faà di Bruno Hopf algebra for analytic nonlinear feedback control systems
On algebraic structures of numerical integration on vector spaces and manifolds
Simple and contracting arbori cation
Strong QCD and Dyson–Schwinger equations