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: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=711737
• 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. My research focuses on the metric and topological study of the infinite-dimensional groups and spaces that appear naturally in symplectic and contact topology. In this study I employ various tools: primarily filtered Floer theory (based on the analysis of nonlinear Cauchy-Riemann operators) and its newly introduced relation to persistence modules (originating in data sciences), which is a way of obtaining quantitative, invariant information from geometric intersection patterns in symplectic topology, and also notions of geometric quantization (based on Spin^c-Dirac operators), and geometric group theory, studying the geometry of groups viewed from afar (the notion of quasi-morphisms and quasi-isometric embeddings, in particular). This proposal consists primarily of three directions of research which fall under the above unified research program, and share methods of filtered Floer theory and persistence, containing each a number of shorter term and longer term aspects. These are: 1. Metrics on the Hamiltonian group, persistence modules, and related topics, 2. Metrics on the space of Lagrangian submanifolds, the cobordism category, and versions of the Fukaya category, 3. Metrics on groups of contactomorphisms and related subjects. In addition, I preview a few projects, some long term and some short term, having to do with the other tools that I like to use. As part of my program, in the next 5 years, I plan to supervise 5 undergraduate students, about 2 Masters students, 2 Ph.D. students, one of which co-supervised with a colleague in U de M, and one post-doctoral fellow, co-supervised with two of my colleagues in U de M. I expect my program to yield new results, methods, and directions of research, and to resolve open questions and conjectures in the field. It would be visible internationally and contribute to mathematics in Canada.

辛和接触拓扑学是现代数学中一个迅速发展的领域，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、代数拓扑学、动力系统等——联系在一起的广泛领域。它主要基于测量偶数维流形中的二维区域，而不是黎曼几何中测量的长度。所有辛流形局部看起来都是一样的——就像机械系统经典相空间中一个点的邻域。因此，它是一个全局的拓扑理论。这个理论中的自然对称性，即所谓的哈密顿微分同态，直接概括了机械系统在相空间中的时间演化。接触拓扑是辛拓扑的奇维模拟——局部模拟扩展相空间——它与描述光传播的黎曼几何部分密切相关。