Gauge Theory and Low Dimensional Topology

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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.