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。我的研究重点是对自然出现在辛和接触拓扑中的无限维群和空间的度量和拓扑研究。在这项研究中，我使用了各种工具：初步过滤弗洛尔理论（基于对非线性Cauchy-Riemann算子的分析）及其新引入的与持久模的关系（起源于数据科学），这是一种从辛拓扑中的几何相交模式获得定量不变信息的方法，也是几何量化的概念（基于自旋^c-Dirac算子）和几何群论，研究从远处观察的群的几何（特别是拟态射和拟等距嵌入的概念）。该建议主要包括三个研究方向，这些研究方向属于上述统一的研究计划，并分享过滤弗洛尔理论和持久性的方法，每个方法都包含一些短期和长期方面。这些是：1.关于Hamilton群、持久性模及相关主题的讨论，2。3.在拉格朗日子流形的空间上，协边范畴，和福谷范畴的版本。接触同构群及相关学科。此外，我还预览了一些项目，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，我计划指导5名本科生，大约2名硕士生，2名博士生。学生，其中一人与密歇根大学的一位同事共同监督，一名博士后研究员与我在密歇根大学的两位同事共同监督。我希望我的计划产生新的结果，方法和研究方向，并解决该领域的开放性问题和挑战。它将在国际上可见，并有助于加拿大的数学。