Equipe
CALIN
GDR
CNRS Renormalisation
Bures
sur Yvette, le 13 Mars
2014.
IHES, 35 Route de Chartres 91440 BuressurYvette, France .
Rencontre organisée par Gérard H.E. DUCHAMP, Maxim KONTSEVITCH, Gleb KOSHEVOY et HOANG NGOC MINH
Combinatorics and Arithmetic for Physics : a special day
Les algèbres de Hopf combinatoires et diagrammatiques sont des outils efficaces pour la Renormalisation. Les calculs dans ces structures mènent souvent à des problèmes arithmétiques comme des valeurs spéciales des fonctions multiformes, formes modulaires, des identités entre les périodes. Par exemple aussi, la théorie de motifs offre un cadre suffisamment souple pour permettre des interprétations galoisiennes.
In order to facilitate discussions a free lunch will be taken on the spot.
Pour toute réservation (train, hôtel, …), veuillez contacter Monsieur Aimé Bayonga (responsable financier du LIPN  7030 UMR CNRS).
L'inscription est gratuite dans la mesure des places disponibles et il suffit d'envoyer à Aimé Bayonga (avec copies à HNM & GHED) les informations suivantes
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Liste des participants/List of paricipants
Programme ()
09h3010h30 


Accueil des participants 
10h3011h30 
Karol Penson 
Combinatorial Sequences in Classical and in Free Probability 
We consider several families of combinatorial sequences, which are related to binomial coefficients. Some of them are certain generalizations of the binomial numbers and Catalan numbers, in the form $\binom{np+r}{n}$, $\binom{np+r}{n}\frac{r}{np+r}$ or $\binom{3n}{n}\frac{2}{n+1}$. We consider these sequences as moment sequences of probability measures defined on the positive halfaxis, i.e. we find the range of parameters $p,r$, for which they are positive definite. We employ the method of inverse Mellin transform to construct exact and explicit expressions for these densities in terms of the Meijer $G$functions and generalized hypergeometric functions. These densities represent generalizations of the MarchenkoPastur and of the Wigner distributions. We prove that certain of these distributions are infinite divisible with respect to the free additive convolution. Work done in collaboration with W.~Mlotkowski. 
11h4512h45 
Gleb Koshevoy 
Cluster Fans 
One
of the main motivation of S.Fomin and A.Zelevinsky for
introducing cluster algebras was the desire to provide a
combinatorial framework to understand the structure of ''dual
canonical bases'' in coordinate rings of various algebraic
varieties related to semisimple groups. For a finite cluster
algebra, they show that the cluster complex can be implemented as
a simplical fan in the vector space span by the simple roots of
the corresponding Lie algebra. 
12h4514h00 


Lunch 
14h0015h00 
Françis Brown 
Multiple Modular Values and Their Galois Action 
There
are increasingly many situations in high energy physics where
processes are described by numbers which go outside the realm of
polylogarithms, zeta values and their usual generalisations, due
to the presence of modular forms. In this talk I will construct a
large new class of numbers which generalises both multiple
zeta values and Lvalues of modular forms, and study the
associated Galois group.

15h1516h15 
Adrian Tanasa 
Some Combinatorics of Random Tensor Models 
Tensor models, seen as quantum field theoretical models, represent a natural generalization of the celebrated 2dimensional matrix models. One of the main results of the study of these matrix models is that their perturbative series can be reorganized in powers of 1/N (N being the matrix size). The leading order in this expansion is given by planar graphs (which pave the 2dimensional sphere S^2). In this talk I will present such a 1/N asymptotic expansion for some particular class of 3dimensional tensor models (called multiorientable models). The leading order, the nexttoleading order and finally some considerations on the combinatorics of the general term of this asymptotic expansion will be given. 
16h1516h45 


Tea/cofee Break 
16h4517h45 
Hoang Ngoc Minh 
From Zoology of Shuffle Products to Their $\phi$Deformations 
Calculus with
integrodifferential operators is often a calculation in an
associative algebra with unit and it is essentially a
noncommutative computation. But, by adjonction a cocommutative
coproduct, it operates in a bialgebra isomorphic to an
enveloping algebra. We then obtain an adequate framework for an
implementation on computer algebra via monoidal factorization,
transcendence bases and PBW bases. 
18h0019h00 
Maxim Kontsevitch 
Geometry of cluster mutations 

Cyril Banderier (CNRSParis 13), Philippe Biane (CNRSMarneslaVallée), Olivier Bouillot (MarneslaVallée), Samir Bouslamti (Paris 6), Françis Brown (CNRSIHES), Van Chiên Bui (Paris 13), Stéphane Dartois (Paris 13), Gérard H.E. Duchamp, (Paris 13), Bertrand Duplantier (IPhT Saclay), MJeanYves Enjalbert (Paris 13), Sylvia Goodenough (Paris 13), Dmitry Grigoryev (CNRSLille 1), Hoàng Ngoc Minh (Lille 2/Paris 13), Martin Hyland (Cambridge/IHES), axim Kontsevitch (IHES), Gleb Koshevoy (Poncelet Lab, Moscow/IHES), Chrsitian Lavault (Paris 13), PaulAndré Mellies (CNRSParis 7), Wojciech Mlotkowski (Wroclaw Univerity/Paris 13), Quôc Hoàn Ngô (Paris 13), Nikolay Nikolov (IHES), Karol Penson (Paris 6), Pierre Simonnet (CNRSMarneslaVallée), Andrea Sportiello (CNRSParis 13), Adrian Tanasa (Paris 13), Christophe Tollu (Paris 13), Tony Yue Yu (Paris 7)