Gauge theory, symplectic geometry and fibered three-manifolds

规范理论、辛几何和纤维三流形

• 批准号: 1021956
• 负责人: Yi Ni
• 金额: $ 5.74万
• 依托单位: California Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-01 至 2012-02-29
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1021956&HistoricalAwards=false
• 关键词: Gauge; theory; symplectic; geometry; fibered

This project deals with invariants in low-dimensional topology which come from gauge theory and symplectic geometry. The focus will be the relation between these invariants and fibered manifolds, in particular, whether these invariants detect fibered 3-manifolds. It is hoped to prove that Monopole Floer homology detects fibered 3-manifolds, which is the analogue of a known result in Heegaard Floer homology. Another goal of this project is to study a problem of Boileau concerning surgeries on null-homotopic knots.The shape of the universe is modelled on three and four dimensional manifolds, which are the objects studied in this project. Understanding such shapes is an important step towards the understanding of the universe we live in. In the microscopic world, macromolecules are often visualized as knots and links in the three space. The knot invariants studied in this project thus provide important tools in analyzing the structures of macromolecules, which turn out to be extremely significant in pharmacology.

这个项目研究低维拓扑中的不变量，这些不变量来自规范理论和辛几何。重点将是这些不变量和纤维流形之间的关系，特别是这些不变量是否检测纤维3-流形。我们希望证明单极Floer同调检测到纤维的3-流形，这类似于Heegaard Floer同调中的一个已知结果。这个项目的另一个目标是研究Boileau关于零同伦纽结上的手术的问题。宇宙的形状是在三维和四维流形上模拟的，这也是本项目研究的对象。了解这些形状是理解我们生活的宇宙的重要一步。在微观世界中，大分子常常被想象成三个空间中的结和环。因此，本项目中研究的纽结不变量为分析大分子的结构提供了重要的工具，这在药理学中具有极其重要的意义。