Metrics and intersections in symplectic and contact topology

辛和接触拓扑中的度量和交集

• 批准号: RGPIN-2017-05596
• 负责人: Shelukhin, Egor
• 金额: $ 1.53万
• 依托单位: Université de Montréal
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635315
• 关键词: Metrics; intersections; symplectic; contact; topology

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.

辛和接触拓扑是现代数学的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联系在一起的广泛领域——代数几何、微分几何、奇点理论、代数数学拓扑、动力系统等。它主要基于测量偶维流形中的二维面积，而不是黎曼几何中测量的长度。所有辛流形在局部看起来都是相同的 - 就像机械系统的经典相空间中的点的邻域一样。 因此，它是一个全局的拓扑理论。该理论中的自然对称性，即所谓的哈密顿微分同胚，直接概括了机械系统相空间中的时间演化。接触拓扑是辛拓扑的奇维模拟 - 在扩展相空间上进行局部建模 - 与黎曼几何中描述光传播的部分密切相关。