Surgery in Contact Geometry

本项目将重点研究低维接触拓扑学中的几个核心问题。接触拓扑学研究奇数维流形上的一种几何结构，称为接触结构，在过去的二三十年中，这种结构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，分类三维接触结构和一些特殊的子集，称为Legendrian结。此外，他将研究这些结构的性质如何相互作用，并与一种更熟悉的几何类型——黎曼几何——相关联。PI还将继续致力于本科生、研究生和博士后的教育。他将组织会议和研讨会，并担任“代数和几何拓扑”的执行编辑，同时开始一个图书项目，为接触几何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。在三维空间中，对接触流形中的勒让德结和横结的理解与我们对接触结构及其与拓扑学的微妙联系的理解携手并进。首席研究员将继续在三个流形中研究这些结，重点关注它们的定性特征以及在新情况下的分类结果。他将研究接触结构的各种特性，如吉鲁扭转、填充性和虚拟过扭度，在手术下的表现。首席研究员还将启动一个项目，对所有小塞弗特纤维空间（以及一些大空间）的接触结构进行分类，并研究它们的接触几何特性。黎曼度量长期以来一直被认为与流形的光滑拓扑有很深的联系，最近也证明了接触结构也是如此。首席研究员将继续探索这两种几何结构之间的关系，目的是看到接触结构的关键属性（如紧密性）反映在适应它们的黎曼度量中。这将有希望使我们对三维流形上的接触结构有一个更全面的了解，并为研究高维接触流形创造新的工具。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 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.