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

1.多面体调和与不变理论：岩崎解决了一个长期存在的关于多面体调和函数空间的有限维的猜想。他发现了有限反射群的新的基本不变量，以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，明确地确定了特殊正多面体的多面体调和函数。他计划写一本关于多面体调和和不变理论的书。递归关系和超几何函数的上同调：岩崎得到了某些差分方程解的一个精确的渐近公式。在这个公式的基础上，他计划发展一个递归关系的上同调理论。K.Iwaaki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。岩崎和松本…进一步得到了一个猜想，即与广义Aary函数相关的扭曲上同调群的交矩阵可以用Skew-Schur多项式来表示。证明这一点的尝试现在正在进行中。不变分支问题和奇异积分方程组：K.Iwa aki和Y.Kamimura建立了一类奇异积分方程解的可解性。他们利用这一结果证明了非线性Sturm-Liouville方程的逆分支问题解的存在性。Y.Kamimura正在写一本关于积分方程组的书，这本书详细介绍了他们的结果。Painlev方程和组合学：K.Iwaaki和H.K.Kawamuko发现了一个Leibniz型组合公式，它与第四个Painlev方程的多变量哈密顿结构有关。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方程的解的二次变换。他还确定了该方程的经典解。较少

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