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H 3 Taming the coloured multizetas

Auteurs : Ecalle, Jean (Auteur de la Conférence)
CIRM (Editeur )

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flexions ARI-GARI symmetral-symmetril palindromy polar / bisymmetrals symmetry / exchanger singulator / singuland / singulate cannonical / irreducible extremal algebra satellites question of the audience

Résumé : 1. We shall briefly describe the ARI-GARI structure; recall its double origin in Analysis and mould theory; explain what makes it so well-suited to the study of multizetas; and review the most salient results it led to, beginning with the exchanger $adari(pal^\bullet)$ of double symmetries $(\underline{al}/\underline{il}) \leftrightarrow (\underline{al}/\underline{al})$, and culminating in the explicit decomposition of multizetas into a remarkable system of irreducibles, positioned exactly half-way between the two classical multizeta encodings, symmetral resp. symmetrel.

2. Although the coloured, esp. two-coloured, multizetas are in many ways more regular and better-behaved than the plain sort, their sheer numbers soon make them computationally intractable as the total weight $\sum s_i$ increases. But help is at hand: we shall show a conceptual way round this difficulty; make explicit its algebraic implementation; and sketch some of the consequences.

A few corrections and comments about this talk are available in the PDF file at the bottom of the page.

Keywords : uncoloured/bicoloured multizetas; flexions; bimoulds; irreducibles ; ari/gari biari/bigari; swap;
bialternal/bisymmetral; perinomal algebra; singulators/singulands/singulates; mould amplification satellites

Codes MSC :
11M32 - Multiple Dirichlet series and zeta functions and multizeta values

    Informations sur la Vidéo

    Réalisateur : Hennenfent, Guillaume
    Langue : Anglais
    Date de publication : 06/07/17
    Date de captation : 27/06/17
    Collection : Research talks
    Format : MP4 (.mp4) - HD
    Durée : 01:04:50
    Domaine : Number Theory ; Dynamical Systems & ODE
    Audience : Chercheurs ; Doctorants , Post - Doctorants
    Download : https://videos.cirm-math.fr/2017-06-27_Ecalle.mp4

Informations sur la rencontre

Nom du congrès : Algebraic combinatorics, resurgence, moulds and applications / Combinatoire algébrique, résurgence, moules et applications
Organisteurs Congrès : Chapoton, Frédéric ; Fauvet, Frédéric ; Malvenuto, Claudia ; Thibon, Jean-Yves
Dates : 26/06/17 - 30/06/17
Année de la rencontre : 2017
URL Congrès : http://conferences.cirm-math.fr/1599.html

Citation Data

DOI : 10.24350/CIRM.V.19190003
Cite this video as: Ecalle, Jean (2017). Taming the coloured multizetas. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19190003
URI : http://dx.doi.org/10.24350/CIRM.V.19190003

Voir aussi


  1. Broadhurst, D.J. (1996). Conjectured enumeration of irreducible multiple zeta values, from knots and Feynman diagrams. - https://arxiv.org/abs/hep-th/9612012

  2. Ecalle, J. (2016). Combinatorial tidbits from resurgence theory and mould calculus. Preprint - https://www.math.u-psud.fr/~ecalle/fichiersweb/WEB_combinat_0.pdf

  3. Ecalle, J. (2015). Eupolars and their bialternality grid. Acta Mathematica Vietnamica, 40(4), 545-636 - http://dx.doi.org/10.1007/s40306-015-0152-x

  4. Ecalle, J. (2014). Singulators vs Bisingulators. In Finitary Flexion Algebras. Preprint - https://www.math.u-psud.fr/~ecalle/fichiersweb/WEB_singulators.pdf

  5. Ecalle, J. (2011). The flexion structure and dimorphy: flexion units, singulators, generators, and the enumeration of multizeta irreducibles. In O. Costin, F. Fauvet, F. Menous, & D. Sauzin (Eds.), Asymptotics in dynamics, geometry and PDEs. Generalized Borel summation. Vol. II (pp. 27-211). Pisa: Edizioni della Normale - http://dx.doi.org/10.1007/978-88-7642-377-2_2

  6. Ecalle, J. (2003). ARI/GARI, la dimorphie et l'arithmétique des multizêtas: un premier bilan. Journal de Théorie des Nombres de Bordeaux, 15(2),411-478 - http://dx.doi.org/10.5802/jtnb.410

  7. Zagier, D. (1994). Values of Zeta Functions and their Applications. In A. Joseph, F. Mignot, F. Murat, B. Prum, & R. Rentschler (Eds.), First European congress of mathematics (ECM), Paris, France, July 6-10, 1992 (pp. 497-512). Basel: Birkhäuser - http://dx.doi.org/10.1007/978-3-0348-9112-7_23