Frédéric Fauvet

Renormalization and Galois Theories

This article is an introduction to some aspects of Ecalles mould calculus, a powerful combinatorial tool which yields surprisingly explicit formulas for the normalising series attached to an analytic germ of singular vector field or of map. This is illustrated on the case of the saddle-node, a two-dimensional vector field which is formally conjugate to Eulers vector field

A comodule-bialgebra structure for word-series substitution and mould composition

x^2\frac{\pa}{\pa x}+(x+y)\frac{\pa}{\pa y}

Asymptotics in dynamics, geometry and PDEs : generalized Borel summation. Vol. I

, and for which the formal normalisation is shown to be resurgent in~

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

1/x

Renormalization and Galois theories. Selected papers of the CIRM workshop, Luminy, France, March 2006

. Resurgence monomials adapted to alien calculus are also described as another application of mould calculus.We describe various combinatorial aspects of the Birkhoff-Connes-Kreimer factorization in perturbative renormalisation. The analog of Bogoliubovs preparation map on the Lie algebra of Feynman graphs is identified with the pre-Lie Magnus expansion. Our results apply to any connected filtered Hopf algebra, based on the pro-nilpotency of the Lie algebra of infinitesimal characters.

Stokes phenomenon for the prolate spheroidal wave equation

Abstract An internal coproduct is described, which is compatible with Hoffmans quasi-shuffle product. Hoffmans quasi-shuffle Hopf algebra, with deconcatenation coproduct, is a comodule-Hopf algebra over the bialgebra thus defined. The relation with Ecalles mould calculus, i.e., mould composition and contracting arborification is precised.

The Hopf algebra of finite topologies and mould composition

These are the proceedings of a one-week international conference centered on asymptotic analysis and its applications. They contain major contributions dealing with: mathematical physics: PT symmetry, perturbative quantum field theory, WKB analysis, local dynamics: parabolic systems, small denominator questions, new aspects in mould calculus, with related combinatorial Hopf algebras and application to multizeta values, and a new family of resurgent functions related to knot theory.

Global Behavior of Solutions of Nonlinear ODEs: First Order Equations

Numerical analysis of time-integration algorithms has been applying advanced algebraic techniques for more than fourty years. An explicit description of the group of characters in the Butcher-Connes-Kreimer Hopf algebra first appeared in Butchers work on composition of integration methods in 1972. In more recent years, the analysis of structure preserving algorithms, geometric integration techniques and integration algorithms on manifolds have motivated the incorporation of other algebraic structures in numerical analysis. In this paper we will survey structures that have found applications within these areas. This includes pre-Lie structures for the geometry of flat and torsion free connections appearing in the analysis of numerical flows on vector spaces. The much more recent post-Lie and D-algebras appear in the analysis of flows on manifolds with flat connections with constant torsion. Dynkin and Eulerian idempotents appear in the analysis of non-autonomous flows and in backward error analysis. Non-commutative Bell polynomials and a non-commutative Fa\`a di Bruno Hopf algebra are other examples of structures appearing naturally in the numerical analysis of integration on manifolds.These lecture notes contain a review of the results of [15, 16, 17, 19] about combinatorial Dyson-Schwinger equations and systems. Such an equation or system generates a subalgebra of a Connes-Kreimer Hopf algebra of decorated trees, and we shall say that the equation or the system is Hopf if the associated subalgebra is Hopf. We first give a classication of the Hopf combinatorial Dyson-Schwinger equations. The proof of the existence of the Hopf subalgebra uses pre-Lie structures and is different from the proof of [15, 17]. We consider afterwards systems of Dyson-Schwinger equations. We give a description of Hopf systems, with the help of two families of special systems (quasicyclic and fundamental) and four operations on systems (change of variables, dilatation, extension, concatenation). We also give a few result on the dual Lie algebras. Again, the proof of the existence of these Hopf subalgebras uses pre-Lie structures and is different from the proof of [16].

Operads of Finite Posets

This article is an introduction to some aspects of Ecalles mould calculus, a powerful combinatorial tool which yields surprisingly explicit formulas for the normalising series attached to an analytic germ of singular vector field or of map. This is illustrated on the case of the saddle-node, a two-dimensional vector field which is formally conjugate to Eulers vector field

New Characterizations for the Eigenvalues of the Prolate Spheroidal Wave Equation

x^2\frac{\pa}{\pa x}+(x+y)\frac{\pa}{\pa y}