Some Questions in Low-Dimensional and Contact Topology

低维接触拓扑的一些问题

• 批准号: 1510091
• 负责人: Olga Plamenevskaya
• 金额: $ 16.99万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-09-01 至 2019-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1510091&HistoricalAwards=false
• 关键词: Some; Questions; Low; Dimensional; Contact

This project aims to carry out research on several interrelated topics in contact and low-dimensional topology. A "contact structure'' is an extra geometric structure on a manifold (these structures originate from physics); the study of contact structures and knots in contact manifolds draws on ideas from different fields (geometry, algebra, combinatorics). This project will contribute to the development of these areas, especially to geometry of manifolds. The main part of the project concerns the investigation of "right-handed'' and "left-handed'' properties of certain knots. Related ideas often appear in the broader context of geometry, algebra, and physics.This project will focus on two different topics. The first topic concerns knot theory and contact topology in dimension 3, and is related to knot invariants of gauge-theoretic and combinatorial nature. The project's goal is to develop properties and applications for a special class of transverse knots (those with certain "positivity" properties); the PI expects to find relations between algebraic properties of corresponding braids, behavior of invariants from knot Floer and Khovanov homology (including an invariant introduced by the PI in 2006), properties of symplectic surfaces (cobordisms) connecting the knots, and rigidity of their branched double covers. This investigation develops a novel approach (with a focus on braid group orderings and the braid monodromy) but also continues the PI's past research. The PI also plans to continue studying flexible phenomena in higher-dimensional contact topology and expects to contribute to this rapidly developing area. The goal is to better understand properties of rigid/flexible contact manifolds, developing analogs of existing results in dimension 3.

该项目旨在对接触和低维拓扑中的几个相关主题进行研究。 “接触结构”是流形上的额外几何结构（这些结构源自物理学）；接触流形中的接触结构和结的研究借鉴了不同领域（几何、代数、组合学）的思想。该项目将贡献该项目的主要部分涉及某些纽结的“右手”和“左手”性质的研究，相关想法经常出现在更广泛的背景下。几何学，该项目将重点关注两个不同的主题。第一个主题涉及维度 3 中的结理论和接触拓扑，并且与规范理论和组合性质的结不变量相关。该项目的目标是开发属性和应用。对于一类特殊的横向结（具有某些“正性”属性的横向结），PI 期望找到相应辫子的代数属性、结 Floer 和 Khovanov 同源性的不变量行为之间的关系； PI 在 2006 年引入的不变量）、连接结的辛曲面（配边）的属性以及分支双覆盖层的刚性。这项研究开发了一种新颖的方法（重点关注辫子组排序和辫子单一性），但也延续了 PI 过去的研究。 PI 还计划继续研究高维接触拓扑中的柔性现象，并期望为这个快速发展的领域做出贡献。目标是更好地理解刚性/柔性接触流形的特性，开发 3 维现有结果的类似物。