Conformal Symplectic Structures, Contact Structures, Foliations, and Their Interactions

这个项目探索了最近出现的数学各个领域之间的新联系，例如辛拓扑和接触拓扑学以及叶层理论。它建立在先前研究的最新进展和结果的基础上，试图推进这些领域和相关领域的一些长期存在的问题。一个相关的目标是在目前使用的技术之外开发新的和替代的工具，虽然这些技术被证明对一些应用程序非常有效，但对于该学科中的一大类未决问题却失败了。这项工作将涉及几名研究生和博士后研究人员。将组织一次研究生讲习班，致力于传播新的想法、方法和结果。首席研究员还将撰写一本研究生水平的书，致力于辛灵活性方面的新进展，包括拟议研究的主要发现。自20世纪80年代这些学科诞生以来，辛拓扑与接触拓扑学之间的丰富联系已为人们所知。十年后，人们发现了接触拓扑学与叶层理论的联系。近年来，人们认识到，在上述混合中加入共形辛结构理论是很重要的。该项目以先前研究的最新进展和结果为基础，试图探索其中每一个领域的发展，以推进其他领域中一些长期存在的问题。一个相关的目标是在当前使用的技术之外发展新的工具，例如Gromov的全纯曲线理论及其分支，例如Floer同调、Fukaya范畴和辛场理论。虽然这些技术被证明对一些应用程序非常有效，但对于该学科中的一大类开放问题来说，它们却失败了。这个项目旨在开发替代工具，或者在它们不存在的情况下证明h-原理类型的结果，断言任何不被全纯曲线方法禁止的东西实际上是可能的。该项目的主要目标是：1)完成Weinstein流形的拉格朗日骨架奇点化简的树形实现程序，特别是建立树形同伦的组合概念，即找到连接同伦Weinstein结构的两个树形骨架的Reidemister型运动的最小集；2)找到共形辛结构的适当的超扭性概念，并证明相应的参数h-原理；3)找到适用于一般极化情形以外的一般Weinstein流形的树形奇点的概念；4)在余维1的叶式上找到允许叶状共形辛结构的条件；探索将叶状结构变形为接触结构这一问题的条件；以及5)为开放接触流形开发有效的不变量，并证明其计算的外科类型公式。该奖项反映了NSF的法定使命，并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，被认为值得支持。

This project explores the new recently emerged links between various areas of mathematics, such as symplectic and contact topology and the theory of foliations. It builds on recent progress and results of prior research in an attempt to advance some of the long-standing problems in these and related areas. A related goal is the development of new and alternative tools beyond the currently used techniques that, while they proved to be quite effective for some applications, fail for a large class of open problems in the subject. The work will involve several graduate students and postdoctoral researchers. A graduate student workshop devoted to dissemination of new ideas, methods and results will be organized. The principal investigator will also write a graduate student level book devoted to new advances in symplectic flexibility, including main findings of the proposed research. Rich links between symplectic and contact topology were known since the inception of these subjects in 1980s. A decade later there were discovered connections of contact topology with the theory of foliations. In recent years it was understood that it is important to add to the above mix the theory of conformal symplectic structures. This project builds on recent progress and results of prior research in an attempt to explore developments in each of these areas for advancing some of the long-standing problems in the others. A related goal is the development of new tools beyond the currently used techniques, such as Gromov's theory of holomorphic curves and its ramifications, e.g., Floer homology, Fukaya categories and Symplectic Field Theory. While these techniques proved to be very effective for some applications, they fail for a large class of open problems in the subject. This project aims to develop alternative tools, or in case they do not exist to prove h-principle type results asserting that whatever is not prohibited by holomorphic curve method is, in fact, possible. The main objectives of the project are: 1) Completion of the arborealization program of simplification of singularities of Lagrangian skeleta of Weinstein manifolds, and, in particular, establishing the combinatorial notion of arboreal homotopy, i.e., finding the minimal set of Reidemeister type moves necessary and sufficient to connect two arboreal skeleta of homotopic Weinstein structures; 2) Finding an appropriate notion of overtwistedness for conformal symplectic structures and proving the corresponding parametric h-principle; 3) Finding a generalization of the notion of arboreal singularities applicable to general Weinstein manifolds beyond the polarized case; 4) Finding conditions on a codimension 1 foliation to admit a leafwise conformal symplectic structure; explorations of this condition for the problem of deformation of foliations into contact structures; and 5) Developing effective invariants for open contact manifolds and proving surgery type formulas for their computations.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.