Metrics and intersections in symplectic and contact topology

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。******我的研究主要集中在度量和拓扑研究的无限维群和空间，自然出现在辛和接触拓扑。在这项研究中，我使用了各种工具：主要过滤了Floer理论（基于非线性Cauchy-Riemann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），这是一种从辛拓扑中的几何相交模式中获得定量不变信息的方法，以及几何量化（基于Spin^c-Dirac算子）和几何群论的概念，从远处研究群的几何（特别是准态射和准等距嵌入的概念）。******本提案主要包括上述统一研究计划下的三个研究方向，并分享了过滤花理论和持久性的方法，每个方向都包含了一些短期和长期的方面。它们是：1；关于hamilton组的度量、持久性模块和相关主题，2。拉格朗日子流形空间上的度量，协同范畴，和Fukaya范畴的版本，3。接触形态群及相关主体的度量。此外，我还预览了一些项目，一些是长期的，一些是短期的，这些项目都与我喜欢使用的其他工具有关。******作为我项目的一部分，在未来5年，我计划指导5名本科生，约2名硕士，2名博士，其中1名与M大学的同事共同指导，1名博士后，与M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，方法和研究方向，并解决该领域的开放性问题和猜想。它将在国际上可见，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.