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: $ 3.06万
• 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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759382
• 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 - 在扩展相空间上局部建模 - 与描述光传播的黎曼几何部分密切相关。我的研究重点是无限的度量和拓扑研究- 在辛和接触拓扑中自然出现的维群和空间。在这项研究中，我使用了各种工具：主要是过滤弗洛尔理论（基于非线性柯西-黎曼算子的分析）及其新引入的与持久性模块的关系（起源于数据科学），这是一种从数据中获取定量、不变信息的方法。辛拓扑中的几何交叉模式，以及几何量化的概念（基于 Spin^c-Dirac 算子）和几何群论，研究从远处观察的群的几何形状（特别是拟态射和拟等距嵌入）。该提案主要包括属于上述统一研究计划的三个研究方向，并共享过滤弗洛尔理论和持久性的方法，每个方向都包含多个短期和长期术语方面。这些是： 1. 哈密顿群、持久性模块和相关主题的度量， 2. 拉格朗日子流形空间、共边范畴和 Fukaya 范畴的版本的度量， 3. 接触同态群和相关主题的度量。此外，我预览了一些项目，一些是长期的，一些是短期的，与我喜欢使用的其他工具有关。作为我计划的一部分，在未来 5 年中，我计划监督 5 名本科生，硕士生2人左右，博士生2人左右学生，其中一名与密歇根大学的一位同事共同指导，一名博士后研究员与我在密歇根大学的两名同事共同指导。我希望我的项目能够产生新的结果、方法和方向研究，并解决该领域的悬而未决的问题和猜想。它将在国际上引起关注，并为加拿大的数学做出贡献。