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。它们出现在几乎所有的数学领域及其应用中，例如物理学和应用数学中的相空间或模型空间，微分几何中具有一定曲率性质的极小曲面和空间，或者数论中的模变体。在实践中遇到的流形总是带有一个额外的全局结构，它编码了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或接触结构，某些黎曼流形上的扭量空间是复的和单圆的。

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.