Gauge Theory and Low Dimensional Topology

这个项目的主要主题是利用Heegaard flower同调、规范理论和辛几何来研究结、三维流形和光滑4流形。潜在的应用将包括为结和链接构造新的不变量，更好地理解那些允许透镜空间手术的结，以及在4流形上构造新的奇异光滑结构。PI还提出研究花同调不变量的结构及其与光滑4流形不变量的关系。本文研究了Seiberg-Witten和heegaardflower同源关系及其应用。后一种理论使用拓扑工具和辛几何，而前者使用与规范理论和数学物理相关的Seiberg-Witten方程。在这个方向上的进展将导致对低维拓扑和规范理论之间联系的更好理解。

The main theme of this project is the study of knots, three dimensionalmanifolds and smooth 4-manifolds by using Heegaard Floer homology, gauge theory and symplectic geometry. Potential applications wouldinclude the construction of new invariants for knots and links, better understanding of those knots that admit lens space surgeries andand the construction of new exotic smooth structures on 4-manifolds. The PI also proposes to study the structure of the Floerhomology invariants and their relationship with smooth 4-manifoldinvariants. The proposal studies the relationship between Seiberg-Witten and HeegaardFloer homologies and their applications.While the latter theory uses topological tools and symplectic geometry, the former uses the Seiberg-Witten equations thatare related to gauge theory and mathematical physics. Progress in this direction should lead to a better understanding of connections betweenlow dimensional topology and gauge theory.