Problems in Floer homology

Let $Y$ be a closed 3-manifold. Let $S_Y$ be the set of compact, oriented, connected surfaces with boundary in $Y$. Let $\sim$ denote the equivalence relation generated by isotopy and positive Hopf plumbing, and consider the set $$\Sigma_Y= S_Y/\sim$$ of equivalence classes under this relation. Problem: is there a natural geometric description of the set $\Sigma_Y$? Let $S_Y^f\subset S_Y$ be the subset whose elements are fiber surfaces for fibered links in $Y$, and let $\Sigma_Y^f$ be the corresponding set of equivalence classes. Then the elements of $\Sigma_Y^f$ are in bijection with isotopy classes of contact structures on $Y$. The problem is asking about the geometric content of equivalence classes of non-fiber surfaces.  Problem: define a Floer-theoretic invariant of elements in $\Sigma_Y$. The Ozsvath-Szabo contact invariant assigns to each $\xi \in \Sigma_Y^f$ (aka each isotopy class of contact structures on $Y$) an element  $$c(\xi)\in \widehat{\mathit{HF}}(-Y).$$ Perh...

Problem:  How is the knot Floer homology of a hyperbolic knot in its next-to-top Alexander grading related to the orbits of pseudo-Anosov flows on its complement which generate the first homology of the complement? If $K\subset Y$ is a hyperbolic knot then its complement admits a pseudo-Anosov flow. This follows from an unpublished result of Gabai, a proof of which was outlined by Mosher; see this paper of Landry--Tsang for some of the history. If $K$ is fibered, then the suspension flow of its monodromy is pseudo-Anosov, and in this case, the problem has been answered . Indeed, work of Ni and independently Ghiggini--Spano , which relies on (1) a relationship between periodic Floer homology and monopole Floer homology due to Lee--Taubes , and (2) an isomorphism between monopole and Heegaard Floer homology due to Kutluhan--Lee--Taubes , shows that if $K$ is fibered then $$\widehat{\mathit{HFK}}(Y,K,g-1)-1$$ is an upper bound for the number of fixed points of the pseudo-Anosov repr...