Metrics and intersections in symplectic and contact topology

辛拓扑和接触拓扑学是现代数学中一个发展迅速的领域，它起源于经典物理-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、代数拓扑学、动力系统等。它主要基于测量偶数维流形中的二维面积，而不是以黎曼几何测量的长度。所有的辛流形在局部上看起来都是一样的--就像一个力学系统的经典相空间中的一个点的邻域。因此，这是一种全球性的拓扑学理论。该理论中的自然对称性，即所谓的哈密顿微分同胚，直接推广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模拟--局部模拟在扩展的相空间上--与描述光传播的黎曼几何部分密切相关。我的研究集中在自然出现在辛拓扑和接触拓扑学中的无限维群和空间的度规和拓扑研究。在这项研究中，我使用了各种工具：主要的滤子Floer理论(基于对非线性Cauchy-Riemann算子的分析)及其与持久模的新引入的关系(源于数据科学)，以及几何量子化的概念(基于自旋、c-Dirac算子)和几何群论，从远处研究群的几何(特别是拟态射和准等距嵌入的概念)。本建议主要包括三个研究方向，属于上述统一研究方案，并分享了滤子Floer理论和持久、持久的方法每个都包含一些较短期和较长期的方面。它们是：1.关于哈密顿群、持久模和相关主题的度量；2.关于拉格朗日子流形空间、余边范畴和Fukaya范畴的度量；3.关于联络同构群和相关主题的度量。此外，我还预览了几个项目，有些是长期的，有些是短期的，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名硕士学生，2名博士生，其中一人与密歇根大学的一名同事共同指导，以及一名博士后研究员，与我在密歇根大学的两名同事共同指导。我希望我的计划能产生新的结果、方法和研究方向，并解决该领域的公开问题和猜想。它将在国际上可见一斑，并为加拿大的数学做出贡献。

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.