Study of Classical and Quantum Integrable Systems and their discretizations

K.Hasegawa研究了K.Kajiwara，M.Noumi和Y. Yamada提出的离散Painleve方程的量子化及其对称性（Backlund变换）。特别是他成功地构造了Jimbo和Sakai的离散Painleve第六方程的量子化。他成功地重建长谷川的量子化的Bcklund变换的观点的着装链和几何晶体。此外，他试图制定量化理论的变形共形场理论下的对称性W-代数。Y.山田研究了热带化的结构有关的二维可解晶格统计模型，产生了许多组合correspondances。还研究了离散Painleve系统。得到了椭圆和/或退化离散Painleve方程的超几何解。本文将此方法应用于微分情形的哈密顿结构，T.Ikeda研究了孤子方程的约化。他利用费米福克空间获得了舒尔Q-函数的组合公式。此外，他成功地确定舒尔的Q-职能的一些类等变上同调，这表明研究特殊职能的设置环面行动及其不动点将是富有成效的。

K.Hasegawa studied the quantization of the discretized Painleve equation and its symmetries (Backlund transformations) proposed by K.Kajiwara, M.Noumi and Y.Yamada. Especially he succeeded to construct the quatization of the discrete Painleve VIth equation of Jimbo and Sakai.G.Kuroki studied the quantization and discretization of the monodromy preserving deformation in general. He succeeded to reconstruct Hasegawa's quantization of Bcklund transformations from the viewpoint of dressing chains and geometric crystals. Also he tried to formulate the quantized theory as the deformation of the conformal field theory under the symmetry of W-algebras.Y.Yamada studied the tropicalization of the structures relevant in the two dimensional solvable lattice statistical models, yielding many combinatorial corresspondances. Also studied are the discrete Painleve systems. Hypergeometric solutions for the elliptic and/or degenerate discrete Painleve equations are obtained. The method was applied to the Hamiltonian sturucture for the differential case.T.Ikeda studied the reductions of solitonic equations. Heused the Fermionic Fock space to obtain a combinatorial formula for Schur's Q- functions. Also he succeded to identify Schur's Q-functions as some classes in equivariant cohomologies, suggesting that the study of special functions in the setting of torus action and its fixed points will be fruitful.