# Faà di Bruno Hopf Algebras, Dyson–Schwinger Equations, and Lie–Butcher Series

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*Kurusch Ebrahimi-Fard; Frédéric Fauvet*

A publication of the European Mathematical Society

Since the early works of G.-C. Rota and his school, Hopf algebras
have been instrumental in algebraic combinatorics. In a seminal 1998 paper,
A. Connes and D. Kreimer presented a Hopf algebraic approach to
renormalization in perturbative Quantum Field Theory (QFT). This work
triggered an abundance of new research on applications of Hopf algebraic
techniques in QFT as well as other areas of theoretical physics.

Furthermore, these new developments were complemented by progress made
in other domains of applications, such as control theory, dynamical
systems, and numerical integration methods. Especially in the latter
context, it became clear that J. Butcher's work from the early 1970s was
well ahead of its time.

This volume emanated from a conference hosted in June 2011 by
IRMA at Strasbourg University in France. Researchers from different
scientific communities who share similar techniques and objectives
gathered at this meeting to discuss new ideas and results on
Faà di
Bruno algebras, Dyson–Schwinger equations, and Butcher series.
The purpose of this book is to present a coherent set of lectures
reflecting the state of the art of research on combinatorial Hopf
algebras relevant to high energy physics, control theory, dynamical
systems, and numerical integration methods. More specifically,
connections between Dyson–Schwinger equations, Faà Bruno algebras,
and Butcher series are examined in great detail.

This volume is aimed at researchers and graduate students interested in
combinatorial and algebraic aspects of QFT, control theory, dynamical
systems and numerical analysis of integration methods. It contains
introductory lectures on the various constructions that are emerging and
developing in these domains.

A publication of the European Mathematical Society. Distributed within the Americas by the American Mathematical Society.

#### Readership

Graduate students and researchers interested in combinatorial and algebraic aspects of QFT, control theory, dynamical systems and numerical analysis of integration methods.