9/11/17

09h00-09h30

Accueil des participants

09h30-10h30

Maxim Kontsevitch

Multiplication kernels

About 10 years I proposed an approach to theory of special functions based on a notion of associative commutative integral kernels, associated with integrable systems. It has origins in Langlands correspondence for curves over finite fields, but formulas make sense in any characteristic. All this was done analyzing one specific example related to Heun equation. I'll talk about recent progress in this direction based on interesting manipulations with formal power series.

10h30-11h00

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11h00-11h45

Nicolas Behr

Combinatorics of chemical reaction systems

Reporting on recent work with GHE Duchamp and KA Penson, I will present a treatment of the theory of chemical reactions from the viewpoint of so-called stochastic mechanics in the spirit of Doi's formalism. This approach allows to uncover some deep relationships between the combinatorial techniques of boson normal-ordering and the dynamics of chemical reaction networks: each semi-linear reaction type induces an evolution within a space of probability distributions that can be computed explicitly via our techniques. For the interesting remaining types of reactions, some results involving systems of orthogonal polynomials will be presented. (paper)

11h45-12h30

Gérard H.E. Duchamp

Noncommutative evolution equation

In this talk, I will show tools and sketch proofs about Noncommutative Evolution Equations (in particular, preparing Hoang Ngoc Minh's talk about associators). Starting from the very simple setting of Noncommutative Formal Power Series with variable coefficients, we can explore in a compact and effective (in the sense of computability) way the Hausdorff group of Lie exponentials in particular the one linked to Hyperlogarithms (and then Polylogarithms) and show some multiplicative renormalisations (as those of Drinfeld). Some parts of this work are connected with Dyson series and take place within the project "Evolution Equations in Combinatorics and Physics". (slides)

12h30-14h00

déjeuner

14h00-15h00

Gleb Koshevoy

Combinatorics of canonical bases and superpotentials

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15h00-15h30

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15h30-16h30

Vladimir Fock

TBA

(slides)

16h30-17h00

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17h00-18h00

Dimity Grigoryev

TBA

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10/11/17

09h30-10h30

Marek Bozejko

Anyonic Fock spaces, q-CCR relation for complex q in the unit disc and relations with Hecke-Yang-Baxter operators with applications to non-commutative functional analysis

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10h30-11h00

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11h00-11h45

Karol Penson

Aerated Poisson Distributions and their Continuous Approximants

We analyze the properties of combinatorial numbers appearing in the normal ordering of powers of certain differential operators. They are natural generalizations of the conventional Bell numbers. We explicitly construct the solutions of the Stieltjes moment problems with these combinatorial sequences as moments. It turns out that in certain cases one encounters as solutions discrete probability distributions located on lacunary subsets of positive integers. They generalize the standard Poisson laws and are called aerated Poisson distributions. We furnish explicit approximants of the aerated Poisson distributions through continuous positive functions via reparametrization of auxiliary solutions for other generalized Bell numbers. These approximants can be expressed via Meijer's G-functions (joined work with Wlotkowski). (slides)

h45-12h30

Pierre Lairez

Calcul des périodes : applications aux sommes binomiales et au volume d'ensembles semi-algébriques

L'étude des intégrales de fonctions algébriques est très fructueuse en géométrie algébrique, comme l'illustre, par exemple, les travaux d'Euler sur les intégrales elliptiques. Les applications aux calculs effectifs sont moins connues. J'en présenterai quelques unes. Plus précisément, je vais m'intéresser aux « périodes », c'est le résultat de l'intégration sur un cycle d'une fractions rationnelle en plusieurs variables. Je montrerai comment les calculer, sous la forme d'équations différentielles, et j'expliquerai deux applications à des problèmes bien différents : la démonstrations automatiques d'identités entre sommes binomiales et le calcul du volume de certains ensembles semi-algébriques (i.e., définis par un ensemble d'inégalités polynomiales). (slides)

h30-14h00

déjeuner

14h00-15h00

V. Hoang Ngoc Minh

On a group of « associators »

Starting from the equation $KZ_3$ and its differential Galois group, we describe a group of “associators”, containing the unique $\Phi_ {KZ}$ (determined by asymptotic conditions). We also exhibit non trivial examples of “ associator” with rational coefficients. (slides)

15h00-15h30

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15h30-16h30

Leila Schneps

TBA

(slides)

16h30-17h00

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17h00-18h00

Pierre Cartier

TBA

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