Lagrangean Isotopy Problem and Special Lagrangian Submanifolds in Symplectic Geometry

M. Gromov关于伪全纯曲线的研究开创了辛几何的新纪元。在那里发展的理论，以及唐纳森和塞贝尔-威滕理论，可能是现代微分几何以下原则的最壮观的表现：研究与问题相关的某些非线性椭圆偏微分方程的整体解的性质，使人们深入了解基本流形的几何。该研究项目的目的是将这一原理应用于辛流形中的拉格朗日子流形理论，其中相应的PDE由特殊的拉格朗日条件施加，然后将所开发的技术用于拉格朗日合痕问题的研究。

The seminar paper of M. Gromov on pseudoholomorphic curves had started a new era in the symplectic geometry. The theory developed there, as also the Donaldson and Seiber-Witten theories, are probably the most spectacular manifestation of the following principle of modern differential geometry: The study of properties of global solutions of certain non-linear elliptic PDEs related to the problem gives deep insight into the geometry of the underlying manifold. The aim of the proposed research project is to apply this principle to the theory of Lagrangean submanifolds in symplectic manifolds, in which the corresponding PDE is imposed by the special Lagrangean condition, and then use the developed techniques in the study of the Lagrangean isotopy problem.