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

这个项目探讨了最近出现的新的数学各个领域之间的联系，如辛和接触拓扑和理论的叶理。它建立在最近的进展和先前的研究成果，试图推进在这些和相关领域的一些长期存在的问题。一个相关的目标是开发新的和替代的工具，超越目前使用的技术，虽然他们被证明是相当有效的一些应用程序，失败的一大类开放的问题的主题。 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，专门传播新的思想、方法和成果。首席研究员还将写一本研究生水平的书，致力于辛弹性的新进展，包括 拟议研究的主要结果。自20世纪80年代辛拓扑和接触拓扑开始以来，它们之间就有着丰富的联系。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，重要的是要添加到上述混合共形辛结构的理论。该项目建立在最近的进展和先前研究的结果，试图探索这些领域的发展，以推进其他领域的一些长期存在的问题。 一个相关的目标是开发新的工具，超越目前使用的技术，如格罗莫夫的全纯曲线理论及其分支，例如，弗洛尔同调、福谷范畴与辛场论。虽然这些技术被证明是非常有效的一些应用程序，他们失败的一大类开放的问题的主题。这个项目的目的是开发替代工具，或者在它们不存在的情况下证明h-原理类型的结果，断言全纯曲线方法不禁止的任何东西实际上都是可能的。该项目的主要目标是：1）完成简化Weinstein流形的Lagrange流形奇点的树实现程序，特别是建立树同伦的组合概念，即，本文的主要工作是：（1）找到连通同伦Weinstein结构的两个树体流形的Reidemeister型移动的最小集合;（2）找到共形辛结构的过扭性的概念，并证明了相应的参数h-原理;（3）找到树体奇点概念的推广，它适用于极化情形以外的一般Weinstein流形; 4）寻找余维1叶理上允许叶状共形辛结构的条件;探索叶理变形为接触结构问题的该条件;和5）开发有效的开放接触流形的不变量，并证明其计算的外科手术式公式。该奖项反映了NSF的法定使命，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，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.