### List of papers/presentations

Slots :
• Tuesday June 24, 17:00-19:00
• Wednesday June 25 11:00-13:00
1. Cyril Banderier
Dirichlet series, Mellin transform, number theory and physics.
Scheduled : Wednesday June 25, 11:00-11:30
Numerous problems in combinatorics and number theory are related to some integer sequences having some "multiplicative properties". For such sequences, we will illustrate how Dirichlet series and Mellin transforms are a powerful tool to get formulas and their asymptotics. We will also give recent applications to the number of residues in Z/nZ, integer factorization, and to problems in physics (in quantum tomography).
2. Bui van Chiên
Algebra on words with $q$-deformed stuffle product and expressing polyzetas
Scheduled : Tuesday June 24, 17:30-18:00
From an infinite alphabet $\{y_s: s\in \mathbb{N} ^* \}$, we construct a completed Hopf algebraic structure with bases in duality thanks to concatenation product and $q$-deformed stuffle product. We also establish structures of polyzetas by expressing on these bases.
3. Gérard H. E. Duchamp and Hoang Ngoc Minh
An interface between physics and number theory via an analytic version of Cartier-Quillen-Milnor-Moore theorem
Scheduled : Tuesday June 24, 17:00-17:30
The functional expansions were common in physics as well as in engineering and have been developped to represent the evolution systems in QED. The main difficulty is the divergence of these expansions at $0$ or at $+\infty$ and leads to the problems of {\it regularization} and {\it renormalization} which can be solved by combinatorial technics. Recently, the combinatorial aspects of noncommutative formal power series (monoidal factorization, transcendence bases, PBW bases, CQMM, \ldots) were intensively amplified for the asymptotic expansions, the computation of the monodromy and of the Galois differential groups of the KZ equation, \ldots facilitating mainly the renormalization and the computation of the associators via polylogarithms and their special values. In this work, we show how a fine study of convolution allows to understand the meaning of the nilpotence of this operator and we focus on this approach to study the renormalization at the singularities in $\{0,1,+\infty\}$ of the solutions of nonlinear differential equations involved in QED
4. Hoan Quoc Ngo
A scheme of noncommutative Combinatorial Number Theory and Physics
Scheduled : Tuesday June 24, 18:00-18:30
In this talk, we say about a result of the polylogarithm and the harmonic sum. We wil talk about an analytic presentation of the harmonic sums and the polylogarithms. By this way, we define the polylogarithm on the negative points. And so we will study about the values of zetafunction at negative points. On the other hand, we also give some applications of Bernoulli’s numbers in the study. Finally, we consider the polylogarithm and harmonic sums in a relation with words on an alphabet.
Combinatoire et algorithmique des factorisations tangentes à l'identité.
Scheduled : Wednesday June 25, 12:00-12:30
Dans cet exposé, nous présenterons un résumé des résultats les plus importants et de nos apports sur l'étude des factorisations tangentes à l'identité grâce à l'ulisation d'outils combinatoires et algorithmiques. L'écriture des factorisations tangentes à l'identité passe par la construction effective d'une paire de bases en dualité et permet l'écriture d'un produit infini. Ce produit ne donne exactement l'identité que sous des conditions très restrictives que nous préciserons. Dans bien des cas, la construction d'une paire de bases en dualité passe par celle d'une base duale à partir d'une base dont on connaît certaines propriétés. Nous nous proposons donc de déterminer les conditions requises que doivent satisfaire la base dont nous partons de sorte que la base duale permette l'écriture des factorisations.
6. Christian Lavault
On Miki-Gessel Bernoulli identities
Scheduled : Wednesday June 25, 11:30-12:00
The Miki-Gessel (MG) and the Faber-Pandharipande-Zagier (FPZ) Bernoulli numbers identities can be proved and expressed under several forms (e.g. Matiyasevich's). Generalizations may be investigated in plenty directions, such as Bernoulli, Euler and Gennochi polynomials, "mixed" MG and FPZ identities extended Bernoulli convolution identities, etc. They also provide interesting new proofs using various tools, such as digamma and polygamma function, and methods, e.g. Dunne and Schubert's approach from quantum field theory and topological string theory.
7. Nguyên Hoàng Nghia
A combinatorial non-commutative Hopf algebra of graphs
Scheduled : Tuesday June 24, 2014 18:30-19:00
A non-commutative, planar, Hopf algebra of planar rooted trees was defined independently by one of the authors in Foissy (2002) and by R. Holtkamp in Holtkamp (2003). In this paper we propose such a non-commutative Hopf algebra for graphs. In order to define a non-commutative product we use a quantum field theoretical (QFT) idea, namely the one of introducing discrete scales on each edge of the graph (which, within the QFT framework, corresponds to energy scales of the associated propagators).
8. Benoit Cagnard
Sarkovski et les Automates
Scheduled : Wednesday June 25, 12:30-13:00
Le théorème de Sarkovski porte sur la cardinalité des orbites périodiques des fonctions continues de la variable réelle. Il existe un ordre, $\triangleright$, dit ordre de Sarkovski sur les entiers non nuls tel que si une fonction $f$ continue a une orbite périodique de cardinalité $n$ et $n \triangleright m$ alors $f$ possède une orbite périodique de cardinalité $m$. Nous remarquons tout d'abord que l'ordre de Sarkovski est réalisable par automate si l'on écrit les entiers en base $2$. Nous faisons ensuite le lien avec les fonctions dont le graphe est définissable par automate de mots infinis synchrone . Lorsque la numération est en base Pisot, on sait que pour de telles fonctions on peut décider si elles définissent des fonctions continues sur les réels. Si tel est le cas, on peut aussi décider si elles admettent des orbites périodiques de cardinalité $n$ pour tout $n$. En particulier, on peut décider si elles ont des orbites de toutes cardinalités. De plus, ces fonctions permettent de construire des exemples de fonctions ayant une orbite périodique de cardinalité $m$ fixé et aucune orbite périodique de cardinalité $n \triangleright m$ dans l'ordre de Sarkovski.