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Rencontres du GDR Renormalisation

LMBP, Clermont-Ferrand, 12-16 novembre 2018

Programme (schedule)

Mini-cours

Ismaël Bailleul (Rennes) : A Swiss Knife for climbing Mount Rough Paths

Rough paths theory is a deterministic theory of differential equations driven by irregular controls. Introduced by T. Lyons in the mid-nineties for the purposes of stochastic analysis, it has now reached a level of maturity that makes its core ideas accessible to a wide audience at a very low cost. These three lectures will give you

• a photo of the mechanics of differential equations controlled by irregular paths,  

• a detailed study of the workhorse of the theory, the so-called sewing lemma,  

• the details on how to apply this machinery to the study of controlled differential equations.  

Loïc Foissy (Calais) : Bialgebras in interactions

The Hopf algebraic version of pairs of semidirect products of groups is the notion of pairs of bialgebras in interaction. In numerous cases, the two bialgebras are the same as algebras. Examples are known on graphs, posets, Feynman graphs, partitions... We shall explain in detail what bialgebras in interaction are, give structure theorems on them and on their monoids of characters, and explicit several applications.

Viviane Pons (Paris-Sud) : Algèbres de Hopf et treillis sur des objets combinatoires : des permutations aux relations binaires.

A travers des exemples clés telle que l'algèbre de Malvenuto-Reutenauer sur les permutations ou de Loday-Ronco sur les arbres binaires, nous verrons le lien étroit qui existe entre les algèbres de Hopf en combinatoire et les structures de treillis. Dans la même thématique, nous étudierons ensuite les récents résultats sur les algèbres de Hopf sur les relations binaires et posets.

Exposés

Olivier Bouillot (Marne la Vallée) : Mould and comould algebras with composition structure

Mould calculus is a powerful combinatorial tool which has been developed in the late 80's by J. Ecalle. It often provides some explicit formulae when there are no other available computational methods. Mould calculus has many elementary objects, as moulds (ie functions with a variable number of variables) or comoulds (ie dual objects of moulds) with specified symmetries.

Using a Hopf algebraic interpretation of (a part of) mould calculus, we will see in this talk how moulds can be multiplied or composed to obtain a mould algebra with composition structure, as well as numerous symmetry stability properties. Less known, we will present a similar comould algebra with composition structure.

Yvain Bruned (Imperial College, London) : Pre-Lie Structures in Geometric stochastic heat equations.

Pre-Lie structures have been an essential tool for establishing the renormalised equation of some singular SPDEs in the context of Regularity Structures. In this talk, we will present how we can use this formalism to derive geometric properties of solutions of Geometric stochastic heat equations such as equivariance under the action of diffeomorphisms and Itô's Isometry.

Ilya Chevyrev (Oxford) : Pre-Lie structures appearing in renormalised SPDEs

Recent work in regularity structures allows one to assign a renormalisation group to any subcritical system of semilinear SPDEs. Under mild additional assumptions, a large classes of driving noises can be lifted to a model following a procedure reminiscent of BPHZ renormalisation in perturbative QFT. I will report on work which further constructs an action of the renormalisation group on a class of SPDEs, which allows one to carry out the renormalisation directly on the level of the equation. The main algebraic tool in this construction is a certain free pre-Lie algebra which is in a suitable co-interaction with the renormalisation group.

Bérénice Delcroix-Oger (Paris 7) : Existence of rigidity theorems for operads

Cartier-Milnor-Moore theorem is a classical theorem in algebra stating that "Any commutative conilpotent cocommutative graded Hopf algebra is free and cofree over the vector space of its primitive elements" In 1997, Fox and Markl extended the notion of Hopf algebra to the notion of generalised bialgebra, by relaxing the relation between product and coproduct. In 2006, Loday and Ronco proved an analogue of Cartier-Milnor-Moore theorem for associative coassociative bialgebras, which was the first step for the formalisation of an extension of this theorem to generalised bialgebras, enounced by Loday in 2008 in terms of operads. In the work presented here, joint with Emily Burgunder, we answered to a question raised by Loday on the condition under which rigidity theorems exist.

Kurusch Ebrahimi-Fard (NTNU, Trondheim) : Planar rooted trees and iterated integrals

Pre-Lie algebras and the Butcher-Connes-Kreimer Hopf algebra of non-planar rooted trees form the foundations of Butcher’s B-series. The theory of Lie-Butcher series combines Lie series with B-series and is based on post-Lie algebras and the MKW Hopf algebra on planar rooted forests, introduced by Munthe-Kaas and Wright. We will discuss the notion of rough paths as characters of the MWK Hopf algebra. This talk is based on joint work with C. Curry, D. Manchon and H. Munthe-Kaas.

Frédéric Fauvet (IRMA, Strasbourg) : Comoulds, bialgebras and moduli

Moulds and comoulds have been introduced by Jean Ecalle in the context of the classification of complex analytic dynamical systems at singularities, as a combinatorial apparatus that enables effective computations of moduli. Reporting in particular on a joint work with O. Bouillot (U. Marne la Vallée), I will describe some elements of mould calculus, highlighting its power to obtain, beyond the algebraic calculations, estimates which are crucial for the analyst -- and possibly also for the probabilist.

Joachim Kock (Univ. Autonoma de Barcelona) : The incidence comodule bialgebra of the Baez-Dolan construction

Starting from an operad P, one can consider on one hand the free operad on P, and on the other hand the Baez--Dolan construction on P, which I will explain in detail. These two operads have the same space of operations, but with very different notions of arity and substitution. Together the incidence bialgebras of the two operads constitute a comodule bialgebra. If P is the terminal operad, then the result is the Calaque--Ebrahimi-Fard--Manchon comodule bialgebra (except that it is with operadic trees instead of combinatorial trees).  Another example is to take as P any monoid, considered as a one-coloured operad with only unary operations.  In this case the resulting comodule bialgebra is the dual of the near-semiring of moulds under product and composition.

Ulf Kühn (Hamburg) : Multiple q-zeta values and bimoulds

We report on our approach to study the algebra of multiple q-zeta values by means of Ecalle’s theory of bimoulds.

Cécile Mammez (Univ. Nice) : Combinatorics of dissection diagrams

In his thesis, Cl. Dupont introduces a family of combinatorial Hopf algebras of dissection diagrams where the product is given by the disjoint union and the coproduct is given by a selection-quotient process with a parameter. Then, he associates each dissection diagram to an absolutely convergent integral called period. In this work, considering a scalar x, we denote by H the graded connected Hopf algebra of dissection diagrams of parameter x. We are interested in the following question: the coalgebra associated to H is it cofree or not cofree. If x=-1, we can conclude it is not cofree. But, if x is different from -1, the problem is still open. To tackle this problem, we consider the dual Hopf algebra H* because it can be equipped with a pre-Lie structure. So, we can build a pre-Lie morphism between dissection diagrams and the free pre-Lie algebra of rooted trees with one generator and then study the pre-Lie algebra generated by the dissection diagram of degree 1. Then we can prove that this sub-object of the pre-Lie algebra of dissection diagrams is not trivial and not free either. So it doesn't permit to answer the cofreedom problem. 

Nikolas Tapia (NTNU, Trondheim) : The geometry of the space of branched rough paths

We construct an explicit transitive free action of a Banach space of Hölder functions on the space of branched rough paths, which yields in particular a bijection between theses two spaces. This allows us to describe the space of branched rough paths with a fixed regularity as a principal homogeneous space (or torsor) for this group. The construction is based on an explicit form of the Baker-Campbell-Hausdorff formula due to Reutenauer and on the Hairer-Kelly map, which allows to describe branched rough paths in terms of what we call anisotropic geometric rough paths.

Yannic Vargas (IVIC, Caracas) : An operad structure over the coalgebra of permutations of Malvenuto-Reutenauer

An operad is defined over the coalgebra structure of the Hopf algebra of permutations of Malvenuto-Reutenauer FQSym, from the notion of composition of coalgebras developed by S. Forcey, A. Lauve and F. Sottile. In particular, this constructions induces the classical shuffle product over permutations, just starting from the coalgebra structure of FQSym and the operad structure.This operad allows also to define a Hopf algebra over painted binary trees with increasing labels. We will define a partially ordered set between these objects, using weak Bruhat order and the operad structure, and compare it with the construction of the Stellohedron, a polytope constructed by L. Berry and S. Forcey form some kind of painted binary trees.