Projective contact geometry and singularity theory

我们组织了联合工作：首席研究员和研究员森本彻（见参考文献）；首席调查员和调查员宫卡玲子和木村诚（提交）；首席调查员和伊利亚·博加耶夫斯基（见参考资料）；首席调查员和S. Janeczko（提交）；首席调查员和v·扎卡柳金；还有一些其他的研究项目。例如，我们得到以下结果：将各向同性分岔问题简化为在约简空间中通过辛形态对品种进行分类。给出了Bruce-Gaffhey平面曲线奇点的完全辛分类，并将其应用于自然地得到拉格朗日开。从接触几何的角度研究了蒙奇-安培方程，研究了蒙奇-安培方程的整体结构和奇异性。我们还得到了高斯映射和对偶变异体的奇异性的结果，这些奇异性出现在普通形式几何和格拉斯曼几何中，并应用于微分方程。利用最小子流形、调和映射和定标构造了可展子流形。

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.