Vendredi 4 février 2011

Le vendredi 4 février 2011, l'équipe Equations fonctionnelles de l'IRMA organise une journée Japon-France avec trois exposés de collègues japonais et trois exposés de collègues français.

  • Vendredi 4 février 2011

  • 09:00 - 09:45

    Foliations on the moduli space of rank two connections on the projective line minus four points

    — Frank Loray (Rennes)

    Joint work with Masa-Hiko Saito and Carlos TSimpson.
    We study two natural fibrations on the Painlevé VI moduli space
    of logarithmic connections of rank 2 on P¹ with four deleted points.
    See arXiv 1012.3612.
  • 10:00 - 10:45

    Degeneration of 4-dimensional Painlevé type equations

    — Hiroshi Kawakami (Tokyo)

    Fuchsian equations with 4 accessory parameters were classified by Oshima. According to the classification, there are three equations which admit one-dimensional isomonodromic deformation. Recently Sakai obtained the deformation equations of the three equations. The purpose of the present study is to make a scheme of 4-dimensional Painlevé type equations (like the degeneration scheme of the original Painlevé equations by considering the successive degeneration of the three deformation equations.
    This talk is based on joint work with H. Sakai and A. Nakamura.
  • 11:00 - 11:45

    The Plateau problem, Fuchsian systems and the Riemann-Hilbert problem

    — Laura Desideri (Tübingen)

    The Plateau problem is to prove than any given Jordan curve in R³ bounds at least one minimal surface of disk-type. In 1928 Garnier published a resolution for polygonal boundary curves which seems to have been forgotten. His proof relies on the fact that one can associate with each minimal disk with a polygonal boundary curve a real Fuchsian second-order equation defined on the Riemann sphere. The monodromy of the equation is determined by the oriented directions of the edges of the boundary. To solve the Plateau problem, we are thus led to solve a Riemann-Hilbert problem and to use isomonodromic deformations of Fuchsian equations. We will see that it is more convenient to use Fuchsian systems instead of equations. I will first give a sketch of Garnier's proof following the more achieved version I gave of it. I will then briefly explain how Garnier's result can be extended into Minkowski 3-space and also how we can deduce by polygonal approximations a resolution of the Plateau problem for piecewise C² boundary curves.

    (joint work with R. Jakob).
  • 13:45 - 14:30

    Special solutions of q-Painleve equations

    — Yousuke Ohyama (Osaka)

    Special solutions of differential Painleve equations were studied by many people. For example, algebraic solutions and solutions of
    hypergeometric type are completely classified. Moreover, we know some asymptotic expansions of special solutions, such as the Boutroux solution, the Ablowitz-Segur solutions or the Clarkson-McLeod solution.
    Special solutions of q-Painleve equations, however, are less familiar. In this lecture, we present some special solutions of q-Painleve equations. We exhibit algebraic solutions if q is a root of unity and meromorphic solutions around the origin.
  • 14:45 - 15:30

    Geometrical and dynamical aspects of mobile singularities in holomorphic foliations

    — Loïc Teyssier (IRMA)

  • 15:40 - 16:15


    Tea/Coffee break
  • 16:15 - 17:00

    Combinatorics on the Katz middle convolution

    — Kazuki Hiroe (Tokyo)

    The Katz middle convolution is defined as a generalization of the Euler transform (Riemann-Liouville transform) and has a lot of applications on the global analysis of Fuchsian ordinary differential equations. On the other hand, Crawley-Boevey found that Fuchsian equations and the middle convolution on them are realized as certain representations of quivers and their reflection functors respectively. In this lecture, I will explain the relationship between the middle convolution on differential equations with unramified irregular singular points and Weyl group action on Kac-Moody root systems.