Great question! 
[Gromov's proof of the nonexistence of compact exact Lagrangian submanifolds 
$L \subset \mathbb{R}^{2n}$](http://math.mit.edu/conferences/geometryworkshop/references/gromov-pseudoholomorphic-curves-in-symplectic-manifolds.pdf) (as well as few other non-existence results proved there). Gromov'es work in general could be mentioned as the starting of modern symplectic geometry - but the result itself showed that Lagrangian submanifolds exhibit special intersection \ non-existence properties (toghether with the Lagrangian Arnold conjectures, and other results from that time) paving the way to ideas such as Lagrangian Floer homology, Fukaya categories, etc...