Studies on Differential and Difference Equations by Means of Geometric and Algebraic Methods

1. 多面体谐波与不变量理论：岩崎解决了一个长期存在的关于多面体谐波函数空间有限维数的猜想。他发现了有限反射群的新的基本不变量，以便明确地确定具有充分对称性的多面体的多面体调和函数。他还与A.Kenma和K.Matsumoto合作，明确确定了特殊规则多面体的多面体调和函数。他计划写一本关于多面体谐波和不变量理论的书。递归关系与超几何函数的上同调：K.Iwasaki得到了一类差分方程解的尖锐渐近公式。基于这个公式，他计划发展递归关系的上同调理论。iwasaki和M.Kita在与超几何函数相关的扭曲de Rham上同调群上发现了一种外部幂结构。K.Iwasaki和K.Matsumoto…More得到了广义Airy函数相关的扭曲上同群的交矩阵可以用斜舒尔多项式表示的猜想。目前正在努力证明这一点。Invese分岔问题与奇异积分方程：K.Iwasaki和y.k imura建立了一类奇异积分方程的可解性。他们应用这一结果证明了非线性Sturm-Liouville方程逆分岔问题解的存在性。神村正写一本关于积分方程的书，详细说明了积分方程的结果。Painlev_方程和组合：K.Iwasaki和h.t awamuko发现了一个莱布尼茨型的组合公式，该公式与若干变量的第四个Painlev_方程的哈密顿结构有关。作为应用，他们得到了Gegenbauer正交多项式之间的一种新的二次关系。H.Watanabe发现了第六个Painlev_方程解的双变换。他还确定了该方程的经典解。少

1. Polyhedral harmonics and invariant theory : K.Iwasaki settled a longstanding conjecture concerning the finite dimen- sionality of the space of polyhedral harmonic functions. He discovered new basic invariants of finite reflection groups in order to explicitly determine the polyhedral harmonic functions for polytopes with ample symmetry. In collaboration with A.Kenma and K.Matsumoto, he also made an explicit determination of polyhedral harmonic functions for the exceptional regular polytopes. He is planning to write a book on polyhedral harmonics and invariant theory.2. Cohomology for reccurence relations and hypergeometric functions : K.Iwasaki obtained a sharp asymptotic formula for solutions to certain difference equations. Based on this formula, he is planning to develop a cohomology theory for recurrence relations. K.Iwasaki and M.Kita discovered an exterior power structure on the twisted de Rham cohomology groups associated to hypergeometric functions. K.Iwasaki and K.Matsumoto … More obtained a conjecture that the intersection matrix of the twisted cohomology groups associated to generalized Airy functions can be expressed in terms of skew-Schur polynomials. Attempts to prove this are now in progress.3. Invese bifurcation problem and singular integral equations : K.Iwasaki and Y.Kamimura established the solvability of a class of singular integral equations. They applied this result to prove the existence of solutions to the inverse bifurcation problem for nonlinear Sturm-Liouville equations. Y.Kamimura is writing a book on integral equations which gives a detailed account of their results.4. Painlev_ equations and combinatorics : K.Iwasaki and H.Kawamuko discovered a combinatorial formula of Leibniz type associated to the Hamiltonian structure of the fourth Painlev_ equation in several variables. As an application, they obtained a new quadratic relation among Gegenbauer's orthogonal polynomials. H.Watanabe found birational transformations of solutions to the sixth Painlev_ equation. He also determined classical solutions to that equation. Less