Lundi 11 Mai

09h30-10h00

Accueil des participants

10h00-11h00

Maxim Kontsevitch

Two word puzzles

The first puzzle concerns certain series (similar to the characteristic polynomial), associated with any element of the group ring of a group.

I've found few years ago that in the case of free groups this series is algebraic, using deep number-theoretic results (and also classical theory of noncommutative algebraic functions).

The challenge is to find a purely combinatorial proof.

The second puzzle is related with Bar-complexes for a non-unital algebra generated by a finite alphabet modulo the relations generated by arbitrary noncommutative monomials (words). The problem reduces to certain complexes associated with Dyck paths. Conjecturally, the cohomology groups are at most one-dimensional (joint work with Y.Vlassopoulos).

11h00-11h30

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11h30-12h30

David Jarossay

Multiple harmonic sums, p-adic multiple zeta values and finite multiple zeta values

We study multiple harmonic sums, in the framework of the fundamental group of the projective line minus three points. We are interested in particular in their p-adic aspects. This enables to compute explicitly the p-adic analogues of multiple zeta values, and to define a geometric notion of "finite multiple zeta values". (slides)

12h30-14h30

déjeuner

14h30-15h30

Gleb Koshevoy

Algebraic and geometric RSK correspondences

A geometric RSK is a pair of a birational isomporphism of torii  and a Laurent polynomial such that their composition is equal to the sum of coordinates.  An algebaic RSK is an injective homomorphism a  ring of polynomials to the ring of Laurent polynomials, such that there exists a basis of the former ring labeled by points of the valuation cone. A mirror is a case when  the valuation cone of an algebraic RSK coincides with the cone of defined by Laurent polynomial for a geometric RSK.  We give constructions of geometric and algebraic RSK, and mirrors.

15h30-16h00

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16h00-17h00

Vincent Rivasseau

Random tensors

We shall review recent progress on random tensors using the intermediate field representation. In particular we shall describe in some detail enhanced quartic tensor models of rank four whose single scaling limits can interpolate between continuous random trees and Brownian spheres. (slides)

Mardi 12 Mai

10h00-11h00

Karol Penson

Probabilistic and Free-Probabilistic Properties of Canstellation Numbers

After a short introduction about the properties of classical Catalan numbers and their rôle in Physics we shall dwell on a certain quite vast generalization of Catalan numbers called the constellation numbers. We solve exactly and explicitly the Hausdorff moment problem for these numbers and provide explicit expressions for appropriate positive weight functions. We have employed the technique of inverse Mellin transform and of Meijer-G functions. For certain values of parameters we analyse the properties of these solutions from the point of view of free probability (theorem of Nica and Speicher) and demonstrate their infinite divisibility with
respect to free additive convolution. We thus provide examples of probabilistic distributions which are infinitely divisible with respect to both conventional and free additive convolutions. Finally we introduce the idea of fractional constellation numbers and provide complete solutions to this particular case in terms of generalized hypergeometric functions. (slides)

11h00-11h30

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11h30-12h30

Dominique Manchon

The Hopf algebra of finite topologies and mould composition

In recent works by L. Foissy, Cl. Malvenuto and F. Patras, a Hopf algebra structure is constructed on the linear span of all topologies on the finite sets [n]={1,...,n}. We will describe a new "internal" coproduct which interacts with the abovementioned Hopf structure in a precise way. We exhibit a surjective Hopf algebra morphism L onto WQSym as well as a new "internal" coproduct on WQSym such that L respects both internal coproducts.

The picture is most conveniently outlined in the linear species formalism, which we will briefly explain. Characters of the Hopf algebra of finite topologies can be called "quasi-ormoulds" in J. Ecalle's terminology : convolution with respect to the external coproduct, resp. the internal coproduct, decribes quasi-ormould product, resp. quasi-ormould composition.

12h30-14h30

déjeuner

14h30-15h30

Vladimir Fock

Flag configurations, matroids and integrable systems

We study the configuration spaces of cyclically ordered flags in a vector space. On one hand this space has a class of simple rational (cluster) parameterisations. If we consider infinite collection of flags in a finite dimensional space and impose certain discrete symmetry, the the configuration space turn out to coincide with the space of local
systems on a Riemann surface. On the other hand, if the dimension is infinite, but the number of flags is finite we get phase spaces of a integrable system on affine SL(N) Poisson-Lie group.

15h30-16h00

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16h00-17h00

Hoang Ngoc Minh

Multiplicative Renormalization of Polyzetas and around

In order to renormalize divergent polyzetas, as well at positive indices as at non positive indices, we establish Abel like theorems for the noncommutative generating series of polylogarithms and of harmonic sums in the factorized form. These allow to draw a precise picture of their structure. (slides)

17h00-17h30

FIN

Web talk

Alin Bostan

Efficient Experimental Mathematics for Lattice Path Combinatorics

Classifying lattice walks in restricted lattices is an important problem in enumerative combinatorics. Recently, experimental mathematics has been used to explore and solve a number of difficult questions related to lattice walks. In this talk, we will give an overview of recent results on structural properties and explicit formulas for generating functions of walks in the quarter plane, with an emphasis on the algorithmic methodology. (slides)