Singularity theoretical research on Partial Differential Equations and Differential Geometry

在这个研究项目中，我们建立了关于偏微分方程弱解的奇点(或激波)传播的基本结果，以及作为奇点理论的应用的微分几何中新不变量的构造。这些结果不能用20世纪数学的主框架来研究。这些结果包括海洋声学理论中出现的Eikonal方程解的奇性的分类，构造了广义弱解的概念，它是熵和粘性解的推广，曲线的四个顶点定理的统一处理，多个平均曲率不变曲面的构造方法，多平面引力透镜的辛框架的构造，双曲高斯映射和光锥高斯映射的正弦性的研究。最后一年，我们给出了奇异平面曲线的辛微分同胚分类，并发现与普通微分同胚分类的差别是一个辛不变量。我们还发现了特殊空间曲线与直纹曲面之间的关系。此外，我们还研究了线同余，并利用拉格朗日同余的概念给出了正规线同余的一个刻画。

In this research project, we established the fundamental results on the propagation of singualnties (or shock waves) for weak solutions of partial differential equations and the construction on new invariants in Differential Geometry as an application of Singularity theory. Those results could not be studied by using the main frame of 20th century's Mathematics. Those results contain the classification of singularities for solutions of the Eikonal equation which appears in the theory of Ocean acoustics, construction of the generalized notion of weak solutions which is a generalization of both of the entropy and the viscosity solutions, the unified treatment on four vertices theorems of curves, the method to construct many mean curvature constant surfaces, construction of the symplectic framework for multiple-plane garvitation lensing and the study of sinuglarities of hyperbolic Gauss maps and lightcone Gauss maps. In the final year, we have given a classification of singular plane curves by symplectic diffeomorphisms and discovered that the difference from the classification by ordinary diffeomorphisms is a symplectic invariant. We have also found relations between special space curves and ruled surfaces. Moreover, we have studied line congruences and have given a characterization of normal line congruences by the notion of Lagrangian congruences.