離散数理モデリングセミナー

過去の記録 ~02/20次回の予定今後の予定 02/21~

担当者 時弘哲治, ウィロックス ラルフ

2016年12月17日(土)

10:00-18:00   数理科学研究科棟(駒場) 056号室
Anton Dzhamay 氏 (University of Northern Colorado) 10:00-10:50
Factorization of Rational Mappings and Geometric Deautonomization (ENGLISH)
[ 講演概要 ]
This talk is the first of two talks describing the joint project with Tomoyuki Takenawa and Stefan Carstea on geometric deautonomization.
The goal of this project is to develop a systematic approach for deautonomizing discrete integrable mappings, such as the QRT mappings, to non-automonous mappings in the discrete Painlevé family, based on the action of the mapping on the Picard lattice of the surface and a choice of an elliptic fiber. In this talk we will explain the main ideas behind this approach and describe the technique that allows us to recover explicit formulas defining the mapping from the known action on the divisor group (the factorization technique). We illustrate our approach by reconstructing the famous example of the q-PVI equation of Jimbo-Sakai from a simple QRT mapping.
Tomoyuki Takenawa 氏 (Tokyo University of Marine Science and Technology) 11:00-11:50
From the QRT maps to elliptic difference Painlevé equations (ENGLISH)
[ 講演概要 ]
This talk is the second part of the joint project with Anton Dzhamay and Stefan Carstea on geometric deautonomization and focuses on the elliptic case and the special symmetry groups. It is well known that two-dimensional mappings preserving a rational elliptic fibration, like the Quispel-Roberts-Thompson mappings, can be deautonomized to discrete Painlevé equations. However, the dependence of this procedure on the choice of a particular elliptic fiber has not been sufficiently investigated.
In this talk we establish a way of performing the deautonomization for a pair of an autonomous mapping and a fiber. Especially, in the case where the fiber is smooth elliptic, imposing certain restrictions on such non autonomous mappings, we obtain new and simple elliptic difference Painlevé equations, including examples whose symmetry groups do not appear explicitly in Sakai's classification.
Hiroshi Kawakami 氏 (Aoyama Gakuin University) 13:30-14:20
The complete degeneration scheme of four-dimensional Painlevé-type equations (ENGLISH)
[ 講演概要 ]
In the joint work with H. Sakai and A. Nakamura, we constructed the degeneration scheme of four-dimensional Painlevé-type equations associated with unramified linear equations. In this talk I present the "complete" degeneration scheme of the four-dimensional Painlevé-type equations, which is constructed by means of the degeneration of HTL forms of associated linear equations.
Akane Nakamura 氏 (Josai University) 14:30-15:20
Degeneration of the Painlevé divisors (ENGLISH)
[ 講演概要 ]
There are three types of curves associated with 4-dimensional algebraically completely integrable systems, namely the spectral curve, the Painlevé divisors, and the separation curve. I am going to explain these three curves of genus two taking examples derived from the isospectral limit of the 4-dimensional Painlevé-type equations and study the Namikawa-Ueno type degeneration.
Teruhisa Tsuda 氏 (Hitotsubashi University) 16:00-16:50
Rational approximation and Schlesinger transformation (ENGLISH)
[ 講演概要 ]
We show how rational approximation problems for functions are related to the construction of Schlesinger transformations. Also we discuss their applications to the theory of isomonodromic deformations or Painlevé equations. This talk is based on a joint work with Toshiyuki Mano.
Takafumi Mase 氏 (the University of Tokyo) 17:00-17:50
Spaces of initial conditions for nonautonomous mappings of the plane (ENGLISH)
[ 講演概要 ]
Spaces of initial conditions are one of the most important and powerful tools to analyze mappings of the plane. In this talk, we study the basic properties of general nonautonomous equations that have spaces of initial conditions. We will consider the minimization of spaces of initial conditions for nonautonomous systems and we shall discuss a classification of nonautonomous integrable mappings of the plane with a space of initial conditions.