Two results in Heegaard Floer homology inspired by gauge theory

Abstract: Heegaard Floer homology and instanton Floer homology are packages of invariants in low dimensional topology constructed via symplectic topology and gauge theory respectively. Kronheimer and Mrowka conjecture that appropriate versions of the two invariants are equivalent. I will discuss two results motivated by this conjecture. The first is a classification result for Heegaard Floer simple knots in sutured manifolds, analogous to a result of Li-Xie-Zhang in the instanton setting. The second is a rank inequality for Heegaard Floer homology under a pinching operation on certain rational homology spheres, extending work of Hanselman-Rasmussen-Watson. Both results are work in progress, and the second is joint work with Sudipta Ghosh.