Frédéric Menous

From dynamical systems to renormalization

In this paper we study logarithmic derivatives associated to derivations on completed graded Lie algebra, as well as the existence of inverses. These logarithmic derivatives, when invertible, generalize the exp-log correspondence between a Lie algebra and its Lie group. Such correspondences occur naturally in the study of dynamical systems when dealing with the linearization of vector fields and the non linearizability of a resonant vector fields corresponds to the non invertibility of a logarithmic derivative and to the existence of normal forms. These concepts, stemming from the theory of dynamical systems, can be rephrased in the abstract setting of Lie algebra and the same difficulties as in perturbative quantum field theory (pQFT) arise here. Surprisingly, one can adopt the same ideas as in pQFT with fruitful results such as new constructions of normal forms with the help of the Birkhoff decomposition. The analogy goes even further (locality of counter terms, choice of a renormalization scheme) and shall lead to more interactions between dynamical systems and quantum field theory.

Formal differential equations and renormalization

The study of solutions of differential equations (analytic or formal) can often be reduced to a conjugacy problem, namely the conjugation of a given equation to a much simpler one, using identity-tangent diffeomorphisms. On one hand, following Ecalle’s work (with a different terminology), such diffeomorphisms are given by characters on a given Hopf algebra (here a shuffle Hopf algebra). On the other hand, for some equations, the obstacles in the formal conjugacy are reflected in the fact that the associated characters appear to be ill-defined. The analogy with the need for a renormalization scheme (dimensional regularization, Birkhoff decomposition) in quantum field theory becomes obvious for such equations and deliver a wide range of toy models. We discuss here the case of a simple class of differential equations where a renormalization scheme yields meaningful results.

Combinatorial Hopf algebras from renormalization

In this paper we describe the right-sided combinatorial Hopf structure of three Hopf algebras appearing in the context of renormalization in quantum field theory: the non-commutative version of the Faà di Bruno Hopf algebra, the non-commutative version of the charge renormalization Hopf algebra on planar binary trees for quantum electrodynamics, and the non-commutative version of the Pinter renormalization Hopf algebra on any bosonic field.We also describe two general ways to define the associative product in such Hopf algebras, the first one by recursion, and the second one by grafting and shuffling some decorated rooted trees.

Free cumulants, Schröder trees, and operads

Abstract The functional equation defining the free cumulants in free probability is lifted successively to the noncommutative Faa di Bruno algebra, and then to the group of a free operad over Schroder trees. This leads to new combinatorial expressions, which remain valid for operator-valued free probability. Specializations of these expressions give back Speichers formula in terms of noncrossing partitions, and its interpretation in terms of characters due to Ebrahimi-Fard and Patras.

The well-behaved Catalan and Brownian averages and their applications to real resummation

The aim of this expository paper is to introduce the well-behaved uniformizing averages, which are useful in resummation theory. These averages associate three essential, but often antithetic, properties: respecting convolution; preserving realness; reproducing lateral growth. These new objects are serviceable in real resummation and we sketch two typical applications: the unitary iteration of unitary diffeomorphisms and the real normalization of real, local, analytic, vector fields.

Combinatorics of Poincaré’s and Schröder’s equations

We investigate the combinatorial properties of the functional equation ϕ[h(z)] = h(qz) for the conjugation of a formal diffeomorphism ϕ of ℂ to its linear part z ↦ qz. This is done by interpreting the functional equation in terms of symmetric functions, and then lifting it to noncommutative symmetric functions. We describe explicitly the expansion of the solution in terms of plane trees and prove that its expression on the ribbon basis has coefficients in ℕ[q] after clearing the denominators (q) n . We show that the conjugacy equation can be lifted to a quadratic fixed point equation in the free triduplicial algebra on one generator. This can be regarded as a q-deformation of the duplicial interpretation of the noncommutative Lagrange inversion formula. Finally these calculations are interpreted in terms of the group of the operad of Stasheff polytopes, and are related to Ecalle’s arborified expansion by means of morphisms between various Hopf algebras of trees.

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

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.