Symplectic Topology of Weinstein Manifolds and Related Topics

温斯坦流形的辛拓扑及相关主题

• 批准号: 1807270
• 负责人: Yakov Eliashberg
• 金额: $ 40.73万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-07-01 至 2022-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1807270&HistoricalAwards=false
• 关键词: Symplectic; Topology; Weinstein; Manifolds; Related

Symplectic and contact topology emerged in attempt to answer qualitative problems of Classical Mechanics and Optics. Since its inception in 1980s there were discovered many new connections of symplectic topology with other areas of Mathematics and Physics. The primary techniques used in the subject go back to Gromov's theory of holomorphic curves in symplectic manifolds and its many ramifications, such as Floer homology, Fukaya categories and Symplectic Field Theory. However, there remains a large class of open problems where holomorphic curve techniques seems not to be sufficient for proving expected results. The current project attempts to develop alternative techniques, or in case they do not exist to develop new methods of construction which could show that everything which is not prohibited by holomorphic curve techniques can indeed happen. In particular, it provides a reformulation of symplectic topology of affine symplectic manifolds as differential topology of singular spaces with a certain well defined list of singularities. This approach can bring new tools from differential to symplectic topology.Weinstein symplectic manifolds recently moved to the forefront of the development of symplectic and contact topology. Building on recent developments, both on the rigid and flexible side of symplectic topology of Weinstein manifolds, the project is designed to advance several central problems of the theory, such as symplectic topology of Lagrangian submanifolds. Among the main objectives of the project are: exploration of techniques for simplification of singularities of Lagrangian skeleta of Weinstein manifolds, and in particular for proving exploration of methods for attacking the regularity conjecture for exact Lagrangian submanifolds of Weinstein manifolds; exploration of a new approach to singularity theory: h-principle without pre-conditions; further exploration of flexibility phenomena for Weinstein manifolds and their Lagrangian submanifolds; and further development of Symplectic Field Theory. The project is designed to bridge the gap between the negative results establishing limits for possible symplectic constructions, and positive results involving the advanced symplectic constructions on the frontier of possibilities. The new methods developed for the arborealization project may find applications elsewhere in singularity theory. The work on the project will involve several graduate students and postdocs and there will be written a graduate student level book devoted to new advances in symplectic flexibility. The obtained results and developed methods may find applications in other areas of mathematics and theoretical physics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

出现符号和接触拓扑，以试图回答经典力学和光学的定性问题。自1980年代成立以来，发现了许多与其他数学和物理学领域的符号拓扑之间的新联系。该主题中使用的主要技术可以追溯到格罗莫夫（Gromov）在符号歧管及其许多后果中的全体形态曲线理论，例如浮动同源性，福卡亚类别和符号田地理论。但是，仍然存在大量的开放式问题，其中塑形曲线技术似乎不足以证明预期的结果。当前的项目试图开发替代技术，或者如果它们不存在开发新的构造方法，这可能表明，所有霍明态曲线技术所不禁止的一切确实会发生。特别是，它提供了仿射象征性歧管的象征性拓扑的重新构造，因为它们是奇异空间的差异拓扑，并具有一定定义明确的奇异性列表。这种方法可以带来从差分到符号拓扑的新工具。WEINSTEINSIMBLECTIC歧管最近移至Symblectic和接触拓扑的发展的最前沿。在最近的发展的基础上，无论是在温斯坦歧管的象征性拓扑的僵化而灵活的一面，该项目旨在推进该理论的几个核心问题，例如拉格朗日亚曼菲尔德的象征性拓扑。 该项目的主要目标之一是：探索Weinstein歧管拉格朗日骨骼的奇异性技术的探索，尤其是为了证明探索攻击Weinstein歧管的精确拉格朗日submanifolds的方法的方法；探索一种新的奇异理论方法：没有预先条件的H原理；进一步探索温斯坦歧管及其拉格朗日亚曼叶的灵活性现象；并进一步发展符号领域理论。该项目旨在弥合负面结果之间的差距，以建立可能的符号结构，以及涉及可能性前沿的先进符号结构的积极结果。为树木化项目开发的新方法可能会在奇异理论中其他地方找到应用。该项目的工作将涉及几个研究生和博士后，并将写一本研究生级书籍，旨在致力于符合性灵活性的新进步。 获得的结果和开发的方法可能会在数学和理论物理学的其他领域中找到应用。该奖项反映了NSF的法定任务，并且使用基金会的知识分子优点和更广泛的影响审查标准，认为值得通过评估来获得支持。

项目成果

The simplification of singularities of Lagrangian and Legendrian fronts

拉格朗日和勒让德锋面奇点的简化

• DOI: 10.1007/s00222-018-0811-3
• 发表时间: 2018
• 期刊: Inventiones mathematicae
• 影响因子: 3.1
• 作者: Álvarez-Gavela, Daniel

On linking of Lagrangian tori in $\mathbb{R}^4$

关于 $mathbb{R}^4$ 中拉格朗日环面的链接

• DOI: 10.4310/jsg.2020.v18.n2.a3
• 发表时间: 2020
• 期刊: Journal of Symplectic Geometry
• 影响因子: 0.7
• 作者: Côté, Laurent

Stabilized convex symplectic manifolds are Weinstein

稳定凸辛流形是 Weinstein

• 发表时间: 2022
• 作者: 小川竜

Contact structures and cones of structure currents

接触结构和结构电流锥

• DOI: 10.4310/jsg.2018.v16.n4.a5
• 发表时间: 2018
• 期刊: Journal of Symplectic Geometry
• 影响因子: 0.7
• 作者: Bertelson, Mélanie;De Groote, Cédric
• 通讯作者: De Groote, Cédric

New Applications of Symplectic Topology in Several Complex Variables

辛拓扑在多复变量中的新应用

• DOI: 10.1007/s12220-020-00395-1
• 发表时间: 2021
• 期刊: The Journal of Geometric Analysis
• 作者: Cieliebak, Kai;Eliashberg, Yakov
• 通讯作者: Eliashberg, Yakov