Gauge Theory and Low Dimensional Topology

本项目的主要内容是利用Heegaard Floer同调、规范理论和辛几何来研究纽结、三维流形和光滑四维流形。潜在的应用包括构造纽结和链环的新的不变量，更好地理解那些允许透镜空间手术的纽结，以及在4-流形上构造新的奇异光滑结构。PI还建议研究Floerhomology不变量的结构及其与光滑4-流形不变量的关系。该方案研究了Seiberg-Witten同调与HeegaardFloer同调之间的关系及其应用，HeegaardFloer同调理论使用的是拓扑工具和辛几何，而Seiberg-Witten同调理论使用的是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应导致更好地理解低维拓扑和规范理论之间的连接。

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.