09h30-10h30

Accueil des participants

10h30-11h30

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 half-axis, 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 Marchenko-Pastur 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.

11h45-12h45

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.

We show that the cluster complex for cluster algebra of the base affine space for GL(n) can be implemented as a simplicial fan in the space span by interval one-column semistandard Young tableaux filled in the alphabet {1,...,n}. For n>6, such a fan contains infinitely many cones and its support covers semistandard Young tableaux corresponding to real elements of the dual canonical basis. For types ADE, cluster complexes of the corresponding finite cluster algebras are subfans of our cluster complex with appropriate n's.

12h45-14h00

Lunch

14h00-15h00

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 L-values of modular forms, and study the associated Galois group.

1) Arithmetic. After reviewing the classical theory, I will explain how to regularise iterated integrals of modular forms, first considered by Manin, with respect to a tangential base point at infinity. This leads to explicit and rapidly convergent formulae  which are suitable for computations.

2) Combinatorics. The numbers defined in 1) conjecturally admit an action by a motivic Galois group. I will describe the action of this Galois group, and the corresponding dual coaction, in elementary terms. In the case of SL_2(Z), there is a natural quotient of this group whose Lie algebra is the free Lie algebra generated by one element in every odd degree >1.

I will aim to make the talk as concrete and explicit as possible.

15h15-16h15

Adrian Tanasa

Some Combinatorics of Random Tensor Models

Tensor models, seen as quantum field theoretical models, represent a natural generalization of the celebrated 2-dimensional 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 2-dimensional sphere S^2). In this talk I will present such a 1/N asymptotic expansion for some particular class of 3-dimensional tensor models (called multi-orientable models). The leading order, the next-to-leading order and finally some considerations on the combinatorics of the general term of this asymptotic expansion will be given.

16h15-16h45

Tea/cofee Break

16h45-17h45

Hoang Ngoc Minh

From Zoology of Shuffle Products to Their $\phi$-Deformations

Calculus with integro-differential operators is often a calculation in an associative algebra with unit and it is essentially a non-commutative computation. But, by adjonction a co-commutative co-product, it operates in a bi-algebra 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.

In this talk, we give the most general co-commutative deformations of the shuffle co-product and an effective construction of pairs of bases in duality.

18h00-19h00

Maxim Kontsevitch

Geometry of cluster mutations