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1. Lagrangian Floer theory of symmetric products and Hilbert schemes

This section includes problems inspired by Heegaard Floer theory, symplectic Khonvanov homology, and Aganagic’s recent proposal concerning geometric models for knot homology theories.


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      Problem 1.05.

      [Peng Zhou] For a symplectic manifold $(X, \omega)$, what is the relationship between $\mathcal{F}^{S_n}(X^n)$ and $\mathcal{F}(\text{Sym}^n(X))$? Here $\mathcal{F}^{S_n}(X^n)$ is (some sort of) $S_n$-equivariant Fukaya category, while $\mathcal{F}(\text{Sym}^n(X))$ is (some kind of) orbifold Fukaya category.


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          Problem 1.1.

          [Peng Zhou] Continuing the previous problem, if $X$ is a Weinstein manifold of dimension at least $4$, are there (canonical) resolutions of $\text{Sym}^n(X)$ that have well-behaved Floer theory?
            1. Remark. As an explicit instance, one could ask if it is possible to generalize [MR4211943] concerning symmetric powers of discs with stops to dimension $4$.


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                  Problem 1.15.

                  [Mina Aganagic] Study resolution of the diagonal on symmetric products (equipped with potentials) and applications to fixed point Floer cohomology models for link invariants.


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                      Problem 1.2.

                      Continuing the previous problem, understand explicitly the resolution of diagonals in Landau–Ginzburg models.
                        1. Remark. For the above two problems, the diagonal stands for the diagonal bimodule of the Fukaya category. It can be realized geometrically as (a perturbation of) the diagonal Lagrangian in the product symplectic manifold.


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                              Problem 1.25.

                              [Mohammed Abouzaid] Give a mathematical construction of a link invariant using the Fukaya category associated to potentials on $\text{Hilb}^n(\mathbb{C}^* \times \mathbb{C}^*)_{hor}$ and Mak–Smith’s cylindrical model [MR4422212].


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                                  Problem 1.3.

                                  [Mohammed Abouzaid] Barmeier–Wang [arXiv:2211.03354] recently exhibited a nontrivial deformation of the extended arc algebra via Hochschild cohomology computations. What is the symplectic meaning of this?
                                    1. Remark. [Ivan Smith] This deformation may correspond to adding back the horizontal part of the Hilbert schemes that define (annular) symplectic Khonanov homology. Because this class lies in $HH^2$, the grading structure on the Fukaya category is preserved under the induced deformation.


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                                          Problem 1.35.

                                          [Mohammed Abouzaid] What is the B-model interpretation of the deformation introduced in the previous problem?


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                                              Problem 1.4.

                                              [Ben Webster] How can we explicitly understand the mirror symmetry equivalence between the A model and B model knot homology constructions in a way which allows one to compare (e.g., functorial) structures?


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                                                  Problem 1.45.

                                                  [Robert Lipshitz] Can one adapt any of the symplectic Khovanov homology constructions to the case of knots in lens spaces? Specifically, what about Aganagic’s construction of knot Floer homology?


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                                                      Problem 1.5.

                                                      [Alexei Oblomkov] Can one establish a surgery formula (or exact triangle) in Aganagic’s construction?
                                                        1. Remark. This would require having definitions in place for links in other $3$-manifolds.


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                                                              Problem 1.55.

                                                              [Robert Lipshitz] Can one introduce and then apply the computational tool of “nice diagrams" of [MR2630063] to Aganagic’s construction and use it to simplify computations?

                                                                  Cite this as: AimPL: Floer theory of symmetric products and Hilbert schemes , available at http://aimpl.org/floerhilbert.