Contact Geometry of Second Order

研究结果摘要如下：首席研究员研究了有限类型线性微分方程组(完整系统)与其基本解的射影嵌入像(射影解)的等价问题之间的对应关系。作为前一个问题的Seashi理论的应用，我们证明了除了E(3，6)外，超几何系统E(n，k)的射影解的像不在Grassman流形Gr(k-1，n-1)(Plucker嵌入像)中。Izumiya对拟线性一阶偏微分方程解曲面的一般奇性进行了分类，并对空间维Hamilton-Jacobi方程粘性解的一般分支进行了分类。Kiyohara定义了测地流可积的两类黎曼流形Liouville流形和Kahler-Liouville流形的概念，并详细研究了它们的结构。Ishikawa证明了Corank 1到辛流形的迷向映射空间的Thom-Mather型横截定理，给出了关于迷向映照辛稳定性和Lagrange稳定性的Mather型或Arnold型刻画。川泉发展了计算超椭圆映射类群有限域上的上同调群的新工具，并综述了复解析Gelfond-Fuchs上同调群的研究。

The summary of Research Results is as follows. The head investigator studied the correspondence between the equivalence problem of the systems of linear differential equations of finite type (Holonomic system) and that of the projective embedding images of their fundamental solutions (projective solution). As the application of Seashi's theory for the former problem, we showed that the image of the projective solution of the Hyper geometric system E(n, k) does not lie in the Grassmannian manifold Gr(k-1 , n-1) (the image of Plucker embedding) except for E(3,6). Furthermore we discussed the generalization of E.Cartan's paper on 5 variables.Izumiya classified the generic singularities of solution surfaces forquasi-linear first order partial differential equations and also classified the generic bifurcation of viscosity solutions for Hamilton-Jacobi equations of space dimension 1.Kiyohara defined the notion of Liouville manifolds and Kahler-Liouville manifolds, which are two classes of riemannian manifolds whose geodesic flows are integrable and studied their structures in detail. He carried out a part of classification and found out a new family of so-called "Cl-metrics" on the Sphere.Ishikawa showed the transversality theorem of Thom-Mather type for the space of isotropic mappings of corank 1 into symplectic manifolds and gave the characterization of Mather type or Arnold type for the Symplectic stability and Lagrange stability of the isotropic mappings.Kawazumi developed new tools to calculate the cohomology groups over finite fields of the hyperelliptic mapping class groups and gave an overview on the study of the cohomology groups of the moduli spaces of Riemann surfaces by the complex analytic Gelfond-Fuchs cohomology.