In type A, affine Springer fibers in the affine Grassmannian are moduli spaces of torsion-free sheaves on singular plane curves, which are closely related to the quantum topology of the corresponding singularities. On the other hand, conjectures of Gorsky-Negut-Rasmussen and Oblomkov-Rozansky (stemming from three-dimensional supersymmetric field theories) relate quantum topology of links to the Hilbert schemes of points on the plane and its generalizations. I will explain some indirect connections between the two setups, using commutative algebra and representation theory. The talk is partially based on joint work in progress with Gorsky and Oblomkov.