Surgery in Contact Geometry

本计画将著重于低维接触拓扑学中的几个核心问题。接触拓扑学研究奇维流形上的几何结构，称为接触结构，在过去的20到30年中，它已经被证明与三维和四维空间有着深刻的联系。首席研究员将继续他的工作分类接触结构的三维和一些特殊的子集，他们称为勒让德结。此外，他将研究这些结构的属性如何相互作用，并涉及到一个更熟悉的类型的几何称为黎曼几何。PI还将继续致力于本科生和研究生以及博士后研究员的教育。他将组织会议和研讨会，并担任《代数学和几何拓扑学》的执行编辑，以及开始一个图书项目，为接触几何中的某些关键技术提供全面的资源。首席研究员将通过各种技术研究接触和辛结构，但重点是手术技术和与黎曼度量的联系。在三维理解勒让德和横向结在接触流形已经携手前进，我们的理解接触结构和它们的微妙联系与拓扑。首席研究员将继续在三个流形上研究这些结，重点是它们的定性特征以及新情况下的分类结果。他将研究如何接触结构的各种属性，如吉鲁扭转，可填充性和虚拟过扭曲，手术下的行为。首席研究员还将启动一个项目，对所有小塞弗特纤维空间（和一些大的）上的接触结构进行分类，并研究它们的接触几何特性。黎曼度量长期以来被认为与流形的光滑拓扑有很深的联系，最近又被证明接触结构也有很深的联系。主要研究者将继续探索这两种几何结构之间的关系，目标是看到适应它们的黎曼度量中反映的接触结构的关键属性（如紧密度）。这将有望导致一个更完整的了解的一般图片上的3流形接触结构，并创建新的工具，研究高维接触流形。这个奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。

This project will focus on several central questions in low dimensional contact topology. Contact topology studies a geometric structure on odd dimensional manifolds, called a contact structure, that over the last twenty to thirty years has been shown to have deep connections with three and four dimensional spaces. The Principal Investigator will continue his work classifying contact structures in dimension three and some special subsets of them called Legendrian knots. In addition he will investigate how properties of these structures interact and relate to a more familiar type of geometry called Riemannian geometry. The PI will also continue his commitment to the education of undergraduate and graduate students and postdoctoral fellows. He will organize conferences and workshops, and be a managing editor for “Algebraic and Geometric Topology”, as well as begin a book project to provide a comprehensive resource for certain key techniques in contact geometry.The Principal Investigator will investigate contact and symplectic structures through a variety of techniques, but focusing on surgery techniques and connections to Riemannian metrics. In dimension three understanding Legendrian and transverse knots in a contact manifold has gone hand in hand with advances in our understanding of contact structures and their subtle links with topology. The Principal Investigator will continue his investigations of such knots in three manifolds, focusing on qualitative features of them as well as classification results in novel situations. He will study how various properties of a contact structure, such as Giroux torsion, fillability, and virtual overtwistedness, behave under surgery. The Principal Investigator will also start a project to classify contact structures on all small Seifert fibered spaces (and some large ones) and study their contact geometric properties. Riemannian metrics have long been known to have deep connections with the smooth topology of manifolds and more recently it has been shown that contact structures do as well. The Principal Investigator will continue to explore relations between these two geometric structures with the goal of seeing key properties of a contact structure (such as tightness) reflected in Riemannian metrics that are adapted to them. This will hopefully lead to a more complete understanding of the general picture of contact structures on 3 manifolds and create new tools for studying higher dimensional contact manifolds.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.