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

该项目探讨了最近出现的数学各个领域之间的联系，如辛和接触拓扑以及叶理理论。它建立在最近的进展和先前的研究成果，试图推进一些长期存在的问题，在这些和相关领域。一个相关的目标是在目前使用的技术之外开发新的和可替代的工具，尽管它们被证明对某些应用程序非常有效，但对于该主题中的一大类开放问题却失败了。这项工作将涉及几名研究生和博士后研究人员。将组织一个研究生讲习班，专门传播新思想、新方法和新结果。首席研究员还将撰写一本研究生水平的书，专门介绍辛柔韧性的新进展，包括拟议研究的主要发现。自20世纪80年代这些学科开始以来，辛拓扑和接触拓扑之间存在丰富的联系。十年后，人们发现了接触拓扑学与叶理理论之间的联系。近年来，人们认识到在上述混合理论中加入共形辛结构理论是很重要的。这个项目建立在最近的进展和先前的研究结果的基础上，试图探索这些领域的发展，以推进其他领域的一些长期问题。一个相关的目标是开发超越当前使用的技术的新工具，例如Gromov的全纯曲线理论及其分支，例如Floer同调，Fukaya范畴和辛场论。虽然这些技术在某些应用中被证明是非常有效的，但对于该学科中大量的开放问题来说，它们是失败的。该项目旨在开发替代工具，或者在不存在的情况下证明h原理型结果，断言任何不被全纯曲线方法禁止的结果实际上都是可能的。本课题的主要目标是：1)完成了简化Weinstein流形lagrange骨架奇异点的树形实现方案，特别是建立了树形同伦的组合概念，即找到了连接两个同伦Weinstein结构的树形骨架所必需且充分的Reidemeister型移动的最小集；2)找到了适形辛结构的适当的过扭度概念，并证明了相应的参数h原理；3)找到了广义Weinstein流形在极化情况之外的树状奇异性概念的推广；4)寻找余维1叶理上存在叶形共形辛结构的条件；对叶理变形为接触结构问题的这一条件的探讨；5)建立开放接触流形的有效不变量，并证明其计算的手术式公式。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识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.