GRK 1269: Global Structures in Geometry and Analysis

流形是由非线性方程组描述的几何空间。它们几乎出现在数学及其应用的所有领域，如物理学和应用数学中的相空间或模型空间，微分几何中的极小曲面和具有某些曲率性质的空间，或数论中的模变异体。在实践中遇到的流形将始终携带一个附加的全局结构，该结构编码关于它们被定义的上下文的信息。例如，力学或光学的相空间具有辛结构或接触结构，某些黎曼流形上的扭转空间是复杂的和不规则的。

Manifolds are geometric spaces, which are described by systems of non-linear equations. They appear in almost all fields of mathematics and its applications, e.g. as phase or model spaces in physics and applied mathematics, as minimal surfaces and spaces with certain curvature properties in differential geometry, or as modular varieties in number theory. The manifolds encountered in practice will always carry an additional global structure that encodes information about the context in which they were defined. For instance, the phase spaces of mechanics or optics carry a symplectic or a contact structure, twistor spaces over certain Riemannian manifolds are complex and uniruled.