Surgery in Contact Geometry

这个项目将集中在低维接触拓扑中的几个核心问题上。接触拓扑学研究奇维流形上的一种几何结构，称为接触结构，在过去的二三十年里，它被证明与三维和四维空间有着深刻的联系。首席调查员将继续他的工作，将接触结构分类为第三维，以及它们的一些特殊子集，称为传奇结。此外，他还将研究这些结构的性质如何与一种更熟悉的几何类型相互作用和联系，这种类型的几何称为黎曼几何。国际和平协会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《代数和几何拓扑学》的执行主编，并开始一个图书项目，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，但重点是外科技术和与黎曼度量的联系。在三维理解中，接触流形中的勒让德结和横向结与我们对接触结构及其与拓扑的微妙联系的理解是齐头并进的。首席调查员将继续对三个流形中的这种结进行调查，重点是它们的质量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属性，如吉鲁克斯扭转、可填充性和虚拟过载，在手术中如何表现。首席调查员还将启动一个项目，对所有小的Seifert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性质。黎曼度量与流形的光滑拓扑有着很深的联系，最近的研究表明，接触结构也是如此。首席调查员将继续探索这两个几何结构之间的关系，目标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。这将有望导致对3个流形上的接触结构的总体情况的更全面的了解，并为研究更高维度的接触流形创造新的工具。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.