Surgery in Contact Geometry

接触几何外科手术

• 批准号: 2203312
• 负责人: John Etnyre
• 金额: $ 63.55万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-09-01 至 2025-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2203312&HistoricalAwards=false
• 关键词: Surgery; Contact; Geometry

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.

该项目将集中于低维拓扑的几个造型。除了这些结构的特性如何与更熟悉的几何形状相关联，他们的某些子集也将对研究生和博士后研究员进行教育，并将其交往。 “代数和几何拓扑结构”的编辑，并开始了一个书籍项目，旨在为联系几何的某些关键技术提供汇编资源。与Riemonian指标的联系。 R将在所有SIFERTS纤维空间属性上启动一个项目，以便在ttontact属性上进行。 ）反映在Riemannian指标中。