Surface operators are two-dimensional defects in a gauge theory. Certain (half-BPS) surface operators in a four-dimensional supersymmetric $N=2$ gauge theory admit an alternative description as a two-dimensional supersymmetric $N=(2,2)$ gauge theory that is coupled to the four-dimensional gauge theory. The two-dimensional gauge theory itself localizes onto the moduli space of vortex equations. In this seminar we study a duality between vortex counting in the two-dimensional gauge theory and open BPS invariants associated to Lagrangian $3$-manifolds. We employ this duality to compute equivariant vortex partition functions and to learn more about surface operators. For example, we find that the geometric transition yields an alternative description of these surface operators as degenerate insertions in a certain two-dimensional CFT.