Gauge theory, symplectic geometry and fibered three-manifolds

本项目主要研究来自规范理论和辛几何的低维拓扑中的不变量。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。希望证明单极Floer同调检测纤维3-流形，这是Heegaard Floer同调的一个已知结果的类似。本项目的另一个目标是研究Boileau关于零同伦结手术的问题。宇宙的形状是在三维和四维流形上建模的，这是本项目研究的对象。理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，大分子通常被视为三个空间中的节点和链接。因此，本项目研究的结不变量为分析大分子结构提供了重要工具，这在药理学中具有极其重要的意义。

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.