Singularity theoretical research on Partial Differential Equations and Differential Geometry

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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.