Heegaard Floer homology $\widehat{HF}(Y)$, defined by Ozsváth-Szabó, is a 3-manifold invariant constructed by counting pseudo-holomorphic discs in the symmetric product $\mathrm{Sym}^g(\Sigma)$ associated to a Heegaard decomposition of $Y$. It categorifies the Alexander polynomial and detects the Thurston norm, genus of knots, and fiberedness.

Floer's original construction (1988): for a homology 3-sphere $Y$, instanton Floer homology $I_*(Y)$ is the Morse homology of the Chern-Simons functional on the space of $SU(2)$-connections on $Y$. Critical points are flat connections, and gradient flow lines are anti-self-dual instantons on $Y \times \mathbb{R}$.

Filtered Floer homology $HF_*^{<a}(H)$ uses the action filtration to create a persistence module. The barcode of this persistence module is a symplectic invariant that captures information beyond the unfiltered Floer homology. Usher-Zhang and Polterovich-Shelukhin developed this into the theory of symplectic barcodes, connecting symplectic topology to topological data analysis.