Projective contact geometry and singularity theory

我们组织了联合工作：首席研究员和研究员Toru Morimoto（见参考文献）;首席研究员和研究员Reiko Miyabka和Makoto Kimura（已提交）;首席研究员和Ilya Bogaevski（见参考文献）;首席研究员和S. Janeczko（已提交）;首席调查员和V. Zakalyukin;以及与调查员合作的其他几个项目。例如，我们得到了如下结果：在约化空间中，迷向分支问题被约化为簇的分类问题。给出了Bruce-Gaffhey平面曲线奇异点的完全辛分类，并应用于自然地得到拉格朗日开度。此外，我们还从接触几何的角度研究了Monge-Ampere方程，研究了Monge-Ampere方程的整体结构和奇性。我们也有结果的奇异性高斯映射和对偶品种，出现在共形几何和格拉斯曼几何和应用微分方程。利用极小子流形、调和映射和标定构造可展子流形。

We have organized the joint works: The head investigator and the investigator Toru Morimoto (see the references); the head investigator and the investigators Reiko Miyabka and Makoto Kimura (submitted); the head Investigator and Ilya Bogaevski(see the references); the head investigator and S. Janeczko (submitted); the head investigator and V. Zakalyukin; and several other projects with investigators. For instance, we have the following result: The isotropic bifurcation problem is reduced to the classification of varieties by symplectomorphisms in the reduced space. The complete symplectic classification of Bruce-Gaffhey's plane curve singularites is provided and is applied to obtain naturally the Lagrangian openings. Moreover we study the Monge-Ampere equations from the view points of contact geometry and investigate the global structure and singularities of Monge-Ampere equations. Also we have the results on the singularities of Gauss mappings and dual varieties, appearing in the com formal geometry and Grassmannian geometry and application to differential equations. We construct developable submanifolds by using minimal submanifolds, harmonic mappings and calibrations.