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H 1 Gamma functions, monodromy and Apéry constants

Auteurs : Vlasenko, Masha (Auteur de la Conférence)
CIRM (Editeur )

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    Résumé : In 1978 Roger Apéry proved irrationality of zeta(3) approximating it by ratios of terms of two sequences of rational numbers both satisfying the same recurrence relation. His study of the growth of denominators in these sequences involved complicated explicit formulas for both via sums of binomial coefficients. Subsequently, Frits Beukers gave a more enlightening proof of their properties, in which zeta(3) can be seen as an entry in a monodromy matrix for a differential equation arising from a one-parametric family of K3 surfaces. In the talk I will define Apéry constants for Fuchsian differential operators and explain the generalized Frobenius method due to Golyshev and Zagier which produces an infinite sequence of Apéry constants starting from a single differential equation. I will then show a surprising property of their generating function and conclude that the Apéry constants for a geometric differential operator are periods.
    This is work in progress with Spencer Bloch and Francis Brown.

    Codes MSC :
    11F23 - Relations of number theory with algebraic geometry and topology
    14G10 - Zeta-functions and related questions
    34M35 - Singularities, monodromy, local behavior of solutions, normal forms

      Informations sur la Vidéo

      Langue : Anglais
      Date de publication : 31/05/2018
      Date de captation : 30/05/2018
      Collection : Research talks ; Dynamical Systems and Ordinary Differential Equations ; Algebraic and Complex Geometry ; Number Theory
      Format : MP4
      Durée : 00:53:46
      Domaine : Number Theory ; Algebraic & Complex Geometry ; Dynamical Systems & ODE
      Audience : Chercheurs ; Doctorants , Post - Doctorants
      Download : https://videos.cirm-math.fr/2018-05-30_Vlasenko.mp4

    Informations sur la rencontre

    Nom de la rencontre : Algebra, arithmetic and combinatorics of differential and difference equations / Algèbre, arithmétique et combinatoire des équations différentielles et aux différences
    Organisateurs de la rencontre : Adamczewski, Boris ; Delaygue, E. ; Raschel, Kilian ; Roques, Julien
    Dates : 28/05/2018 - 01/06/2018
    Année de la rencontre : 2018
    URL Congrès : https://conferences.cirm-math.fr/1761.html

    Citation Data

    DOI : 10.24350/CIRM.V.19409303
    Cite this video as: Vlasenko, Masha (2018). Gamma functions, monodromy and Apéry constants. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19409303
    URI : http://dx.doi.org/10.24350/CIRM.V.19409303

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