Abstract: One can often solve counting problems depending on a discrete parameter by assembling the different counts as coefficients of a generating function, deriving a differential or algebraic equation for this generating function, and finally solving that equation. I will explain how this method can be applied to a problem of counting holomorphic maps from Riemann surfaces with boundary into $\mathbb{C}^3$, such that the boundary lands on a fixed real (Lagrangian) 3-manifold $L$. It turns out that the equations satisfied by the corresponding generating functions associated to different choices of boundary conditions $L$ can be encoded via a single algebro-geometric object known as a cluster Poisson variety. The rigid combinatorial structure of this auxiliary geometry can then be used to find an explicit formula for the solution of these equations, making manifest nontrivial properties of the original curve counts.
Based on joint works with Mingyuan Hu, Linhui Shen, and Eric Zaslow.