Marcelo Aguiar (Texas A&M University), « Hopf algebras from cooperads »
Résumé. The notion of surspecies is a variant of the notion of species; both notions are due to Joyal. The category of surspecies is not a braided monoidal category; rather, it carries two monoidal structures which commute with each other in an appropriate sense. One can then consider bimonoids in the category of surspecies. Graded bialgebras can be constructed from bimonoids in surspecies by means of appropriate bilax monoidal functors. It turns out that this includes as a special case the construction of a Hopf algebra from an arbitrary cooperad. This talk will discuss the above notions and constructions and is based on joint work with Swapneel Mahajan. Related constructions of Hopf algebras from operads have been introduced independently by Mendez and van der Laan. There is additional related work by Brouder and Frabetti, Chapoton and Livernet, Schmitt, and Vallette.

Michael Batanin (Macquarie University), « Locally constant n-operads as higher braided operads »
Résumé It is known that contractible nonsymmetric operads detect 1-fold loop spaces, contractible braided operads detect 2-fold loop spaces and that contractible symmetric operads detect ∞-fold loop spaces. It is natural to ask: is there a sequence of groups G⁽ⁿ⁾={G⁽ⁿ⁾_k}_k≥0 together with a notion of G⁽ⁿ⁾-operad such that the algebras of a contractible such operad are n-fold loop spaces up to group completion? One can prove that the answer on the above question is negative. I will show, however, that there is a category of operads based on planar trees with n-levels which we can think of as a correct replacement for the nonexistent category of G⁽ⁿ⁾-operads in all dimensions. I call them locally constant n-operads. For n=1,2,∞ the homotopy category of locally constant n-operads is equivalent to the homotopy category of classical nonsymmetric, braided and symmetric operads correspondingly.

Christian Brouder (Université Paris 6), « Une opérade en théorie des champs »
Résumé. Stefan Hollands a proposé récemment une approche nouvelle pour un des fondements de la théorie quantique des champs : le développement en produits d'opérateurs (operator product expansion). Sa construction utilise des outils très algébriques, tels que la cohomologie de Hochschild et la cohomologie cyclique. Il définit aussi un objet qui ressemble fort à une opérade. L'exposé présentera l'origine physique du problème et les points essentiels de l'approche de Hollands.

Emily Burgunder (Université de Montpellier 2), « Une version non-symétrique du complexe de graphes de Kontsevich »
Résumé. Kontsevich a prouvé que l'algèbre de Lie des champs de vecteurs peut être reconstruite à partir d'un graphe-complexe. Nous prouvons une version non-symétrique de ce théorème : l'homologie de Leibniz de cette algèbre peut être reconstruite grâce à un nouveau complexe de graphes. On démontre que l'isomorphisme construit est un isomorphisme de bigèbres Zinbiel associatives.

Gérard H. E. Duchamp (Université Paris XIII), « Algèbres de diagrammes et ordre normal »
Résumé. In this talk, we consider two aspects of the product formula for formal power series applied to combinatorial field theories. Firstly, we remark that the case when the functions involved in the product formula have a constant term is of special interest as often these functions give rise to substitutional groups which are strongly connected to classical notions like Riordan groups, classical Sheffer conditions and the « exponential formula ». Secondly, we discuss deformations of the algebra of Feynman-like diagrams arising from the product formula of two free exponentials (counting natural graph parameters as crossings and superpositions). This leads to a true Hopf deformation of this algebra. We end with some concluding remarks on the algebraic and combinatorial nature of the deformation.

Kurusch Ebrahimi-Fard (Université de Mulhouse), « The pre-Lie relation, binary rooted trees and the Magnus expansion: there is still life in ordinary differential equations »

Bertfried Fauser (Université de Konstanz) , « On the ribbon Hopf algebra structure of plethystically generated subcharacter rings of GL(∞) »
Résumé. Symmetric functions inevitably play a central role in many areas of mathematics and physics. Moreover, they form in a natural way the universal positive self adjoint bi-commutative Hopf algebra over one generator. This structure resembles the (formal) character ring of Char_GL(∞)≃U_(∞). A much more interesting structure arises from studying character rings of certain (plethystically generated) subgroups. We show, that these character rings still carry a Hopf algebra structure, but that these Hopf algebras are not necessarily positive, not self adjoint and exhibit a nontrivial ribbon structure. Using the categorical definition of trace and dimension one can proceed to compute invariants, which can be interpreted as knot and link invariants. Unfortunately, one gets trivial (dimension, trace) invariants for the GL↓ H_π case, but we will show that other branchings H_μ↓ H_ν could be nontrivial. Anyhow, a q- or q,t- deformation using Hall-Littlewood or Macdonald symmetric functions will give nontrivial invariants, those which belong to non-semi-simple Lie groups should be new since we work in a larger category as the framework leading to the universal Kontsevich invariant.

Loïc Foissy (Université de Reims), « The infinitesimal Hopf algebra of planar trees and the poset of planar forests. »
Résumé. We introduce an infinitesimal Hopf algebra of planar trees, generalising the construction of the non-commutative Connes-Kreimer Hopf algebra. A non-degenerate pairing and a dual basis are defined, and a combinatorial interpretation of the pairing in terms of orders on the vertices of planar forests is given. Moreover, the coproduct and the pairing can also be described with the help of a partial order on the set of planar forests, making it isomorphic to the Tamari poset. As a corollary, the dual basis can be computed with a Möbius inversion.

Ralf Holtkamp (Université de Bochum), « Free commutative operations, n-labeled trees, and power series »
Résumé. We consider abstract rooted trees with labeled leaves, such that the arity of each vertex lies in a given subset of the set of natural numbers. We introduce a shuffle product on the vector space of such trees. The dual coproduct together with abelian-magmatic grafting products leads to a cocommutative generalized bialgebra type. We obtain a structure theorem, discuss projectors on the spaces of primitives, and obtain tree power series with interesting properties.

Herbert Gangl (Université de Durham), « Hopf algebras on polygons, trees and algebraic cycles »
Résumé. A basic Hopf algebra constructed from algebraic cycles is found to have an interesting sub-Hopf algebra whose combinatorics can be captured in terms of (a bar construction using decorated) trees, and the latter in turn leads to an even smaller Hopf algebra on polygons. The polygons themselves can be thought of as pictorial versions of multiple polylogarithms (joint work with A.B. Goncharov and A. Levin).

Thomas Krajewski (Centre de Physique Théorique), « Power series of non linear operators, effective actions and some combinatorial illustrations »
Résumé. In this talk, we present some common algebraic framework for the resolution of non linear equations and effective actions computations, based on the Hopf algebras of rooted trees and Feynman diagrams. To solve non linear equations, we interpret trees as indices for power series of non linear operators and introduce the associated geometric series. The latter is used to solve fixed point equations, as we illustrate on the derivation of Postnikov's hook length formula and on the counting of plane trees with depth less than a given integer in terms of Catalan numbers. In the very same vein, we also interpret Feynman diagrams as indices for the powers of the background field operator and present an composition law analogous to the B-series of numerical analysis. Finally, we give a new proof of the universality of Tutte polynomial and its relation to the q-state Potts model partition function, based on diagrammatic differential equations and compositions laws of effective actions.

Jean-Louis Loday (CNRS, Strasbourg), « Braces, arbres et algèbres de Hopf »
Résumé. On construit plusieurs algèbres de Hopf combinatoires à partir d'algèbres munies d'opérations braces et multi-braces. On en donne une description explicite en termes d'arbres et une interprétation opéradique (dendriforme, diptère). On montre en quoi elles sont des variations de l'algèbre de Hopf introduite par Connes et Kreimer. Travail en commun avec M. Ronco.

Frédéric Menous (Université Paris-Sud) « Mould Calculus, Combinatorial Hopf Algebras and the Jacobian Conjecture »
Résumé. In the study of analytic Vector fields, Diffeomorphisms, Jean Ecalle's mould calculus provide a way to compute and handle formal identity-tangent diffeomorphisms (in any dimension) as characters on combinatorial Hopf algebras (Shuffle, Quasishuffle, Connes-Kreimer). I will illustrate this in the framework of the Jacobian conjecture and derive some conjectures in such Hopf algebras that imply the Jacobian conjecture.

Dominique Manchon (Université Blaise Pascal) « Deux algèbres de Hopf d'arbres en interaction » i.e. « Two Hopf algebras of trees interacting »

Frédéric Patras (Université de Nice), « The Hopf algebraic structure of Hopf operads »
Résumé. Joint work with Muriel Livernet. A Hopf triple is such that the tensor product of two algebras over the triple carry naturally the structure of an algebra over the triple. We investigate the properties of Hopf operads and show that many properties of the classical Lie theory carry over to this setting.

Bruno Vallette (Université de Nice), « Théorie de déformation de structures algébriques »
Résumé : Pour tout morphisme de props, nous définirons, à la Quillen, un complexe de chaines qui mesure les déformations de ce morphisme. Lorsque le prop but est le prop des endomorphismes d'un module A, ceci définit la théorie homologique des déformations de A comme (bi)gèbre sur le prop source. Nous retrouvons de cette manière les différents complexes de chaines de la littérature : (co)homologie de Hochschild des algèbres associatives, de Chevalley-Eilenberg des algèbres de Lie, de Harrison des algèbres commutatives, de Lecomte-Roger des bigèbres de Lie, de Gerstenhaber-Schack des bigèbres associatives. Grâce à ce point de vue, nous monterons que ce complexe de chaines est toujours une algèbre de Lie à homotopie près (stricte dans le cas Koszul). Les solutions de Maurer-Cartan généralisées correspondent alors aux structures déformées de (bi)gèbre sur A. De plus, nous construirons des opérations supérieures (non binaires) agissant sur ce complexe qui généralisent les opérations braces du complexe de Hochschild. Ceci permet, entre autre, de montrer une version généralisée de la conjecture de Deligne.