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.

该项目将重点关注低维触点拓扑中的几个中心问题。接触拓扑研究奇数尺寸歧管上的几何结构，称为接触结构，在过去的二十到三十年中，已证明与三个和四个维空间具有深厚的连接。首席调查员将继续他的工作，以第三维的联系结构分类，其中一些特殊的子集被称为legendrian结。此外，他将研究这些结构的特性如何相互作用，并与称为Riemannian几何形状的几何类型相关。 PI还将继续致力于对本科生和研究生以及博士后研究员的教育。他将组织会议和讲习班，并成为“代数和几何拓扑”的执行编辑，并开始一个书籍项目，以为接触几何的某些关键技术提供全面的资源。在维度中，三个理解触点歧管中的传奇和横向结和横向结已经与我们对接触结构的理解及其与拓扑的微妙联系的进步汇友。首席调查员将继续在三个流形中对此类结的投资，重点关注它们的定性特征以及分类导致新的情况。他将研究接触结构的各种特性，例如Giroux Torsion，填充性和虚拟范围，在手术中表现出色。首席研究者还将开始一个项目，以对所有小型Seifert纤维空间（和一些大型空间）上的接触结构进行分类，并研究其接触几何特性。长期以来，人们已经众所周知，里曼尼亚的指标与流形的平滑拓扑具有很深的联系，最近已经表明，接触结构也可以。主要研究者将继续探索这两个几何结构之间的关系，目的是看到触点结构的关键特性（例如紧密度）反映在适应它们的Riemannian指标中。希望这将使人们对3个流形的接触结构的一般情况有更全面的了解，并创建新的工具来研究更高的维度接触歧管。该奖项反映了NSF的法定任务，并通过基金会的智力优点和更广泛的影响通过评估来诚实地支持支持。