Applied singularity theory to exterior differential systems

得到以下结果：一类蒙日-安培方程几何解的一般奇异性。澄清Goursat-Legendre通信。奇异Legendre子流形的基本研究，特别是奇异Legendre曲线。可展物奇点的分岔。投影对偶下双退化子流形的估计。引入奇异共同性映射的概念。辛模空间上局部化定理的证明。映射空间商的基础研究。奇异Legendre节的分类理论。三维蒙日-安培方程解的新奇点的发现。单模曲线奇异点的分类及其辛模空间的确定。发现平面曲线奇点的微分同胚分类与相关勒让德曲线奇点的接触同胚分类之间有显著的相似性。

The following results are obtained : Classification of generic singularities in geometric solutions to Monge-Ampere equations. Clarification of Goursat-Legendre correspondence. Basic investigations on singular Legendre submanifolds, in particular singular Legendre curves. Bifurcation of singularities of developables. Estimate on doubly degenerate submanifolds under projective duality. Introduction of the notion of singular coisotropic mappings. Proof of the localization theorem on the symplectic moduli spaces. Basic study on mapping space quotients. Classification theory of singular Legendre knots. Discovery of new singularities of solutions to Monge-Ampere equations in dimension three. Classification of uni-modal curve singularities and determination of their symplectic moduli spaces. Discovery of remarkable similarity between diffeomorphism classification of plane curve singularities and contactomorphism classification of associated Legendre curve singularities.