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.

辛拓扑和接触拓扑出现在试图回答经典力学和光学的定性问题。自20世纪80年代辛拓扑学诞生以来，人们发现了辛拓扑学与数学和物理学其他领域的许多新联系。在这个主题中使用的主要技术可以追溯到Gromov的辛流形中的全纯曲线理论及其许多分支，如Floer同调，福谷范畴和辛场论。然而，仍然有一大类开放的问题，全纯曲线技术似乎不足以证明预期的结果。目前的项目试图开发替代技术，或者在它们不存在的情况下开发新的构造方法，这些方法可以表明全纯曲线技术不禁止的一切都确实可以发生。特别地，它提供了仿射辛流形的辛拓扑的重新表述为： 奇异空间的微分拓扑，有一个明确定义的奇异点列表。这种方法可以带来新的工具，从微分辛拓扑。Weinstein辛流形最近移动到辛和接触拓扑的发展的最前沿。基于最近的发展，在刚性和柔性方面的辛拓扑的温斯坦流形，该项目旨在推进几个中心问题的理论，如辛拓扑的拉格朗日子流形。 该项目的主要目标包括：探索用于简化Weinstein流形的拉格朗日子流形的奇点的技术，特别是用于证明探索攻击Weinstein流形的精确拉格朗日子流形的正则性猜想的方法;探索奇点理论的新方法：无先决条件的h-原理;进一步探索Weinstein流形及其拉格朗日子流形的柔性现象，进一步发展辛场论。该项目旨在弥合负面结果之间的差距差距建立可能的辛结构的限制，积极的结果涉及先进的辛结构的前沿的可能性。为树实现项目开发的新方法可能在奇点理论的其他地方找到应用。该项目的工作将涉及几个研究生和博士后，并将写一本研究生水平的书，致力于辛灵活性的新进展。 获得的结果和开发的方法可能会在数学和理论物理的其他领域找到应用。该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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