Metrics and intersections in symplectic and contact topology

辛拓扑和接触拓扑是现代数学中一个快速发展的领域，它起源于经典物理学--经典力学和光学--但已经成为一个与许多其他学科--代数几何、微分几何、奇点理论、代数拓扑、动力系统等--有联系的广泛领域。它主要基于测量偶数维流形中的二维面积，而不是黎曼几何中测量的长度。所有的辛流形局部看起来都是一样的--就像力学系统的经典相空间中一点的邻域。 因此，它是一个全局的拓扑理论。在这个理论中的自然对称性，即所谓的哈密尔顿同构，直接概括了力学系统相空间中的时间演化。接触拓扑是辛拓扑的奇维类似物-在扩展相空间上局部建模-与描述光传播的黎曼几何部分密切相关。

Symplectic and contact topology is a rapidly developing area of modern mathematics that has its roots in classical physics - classical mechanics and optics - but has already become an established broad field with ties to many other disciplines - algebraic geometry, differential geometry, singularity theory, algebraic topology, dynamical systems, and others. It is primarily based on measuring two-dimensional areas in even-dimensional manifolds, instead of the lengths measured in Riemannian geometry. All symplectic manifolds locally look the same - like a neighborhood of a point in the classical phase-space of a mechanical system. It is therefore a global, topological theory. The natural symmetries in this theory, the so-called Hamiltonian diffeomorphisms, directly generalize the time-evolution in phase-space of a mechanical system. Contact topology is the odd-dimensional analogue of symplectic topology - locally modelled on the extended phase-space - that is closely related to the part of Riemannian geometry that describes the propagation of light.