I know vaguely from a few friends that there is some cross pollination between algebraic geometry and PDE's but both fields are so huge that this doesn't characterize either field. I think the gist of it is that the moduli space of PDE solutions are frequently interesting as varieties.

There's this rare unclosed survey-like mathoverflow question that highlights some of these points: https://mathoverflow.net/questions/66046/which-nonlinear-pde...