Symplectic Topology of Weinstein Manifolds and Related Topics

辛拓扑和接触拓扑学的出现是为了回答经典力学和光学的定性问题。自20世纪80年代建立以来，辛拓扑与数学和物理的其他领域有了许多新的联系。本课题所用的主要技巧可以追溯到Gromov的辛流形上的全纯曲线理论及其许多分支，如Floer同调、Fukaya范畴和辛场论。然而，仍然有一大类开放的问题，其中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建方法，这种方法可以表明，全纯曲线技术不禁止的一切确实都可以发生。特别地，它将仿射辛流形的辛拓扑重新表示为具有特定定义的奇点列表的奇异空间的微分拓扑。这种方法可以带来从微分拓扑学到辛拓扑学的新工具。Weinstein辛流形最近发展到辛拓扑和接触拓扑学的前沿。基于Weinstein流形的辛拓扑的刚性和柔性方面的最新发展，该项目旨在提出该理论的几个核心问题，例如拉格朗日子流形的辛拓扑。该项目的主要目标包括：探索Weinstein流形的拉格朗日骨架奇点的简化技术，特别是用于证明Weinstein流形的精确Lagrangian子流形的正则性猜想的方法的探索；探索奇点理论的新途径：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度现象；以及进一步发展辛场理论。该项目旨在弥合为可能的辛构造建立极限的负面结果和涉及可能性前沿的高级辛构造的积极结果之间的差距。为树木实现项目开发的新方法可能会在奇点理论的其他地方找到应用。该项目的工作将涉及几名研究生和博士后，并将撰写一本研究生水平的书籍，致力于辛弹性方面的新进展。所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得到应用。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.